A couple months ago, at Indiana University, David Fisher, Nets Katz, and I  organized a summer school on Analysis and geometry in the theory of computation.  This school is one in a series organized by David and funded by NSF grant DMS-0643546 (see, e.g. last year’s school). What follows is a brief synopsis of what the school covered.  All the lectures were given by the participants, and there are links to their lecture notes below.  This is essentially an extended version of an introductory document I wrote for the participants, who were a mix of mathematicians and theoretical computer scientists.

Approximation Algorithms

In the following discussion, we will use the word efficient to describe an algorithm that runs in time polynomial in the size of its input. For a graph , we use to denote the “MAX-CUT value,” i.e. the quantity

where denotes the set of edges between and its complement. It is well-known that computing is -complete, and thus assuming , there is no efficient algorithm that, given , outputs .

Given this state of affairs, it is natural to ask how well we can approximate the value with an efficient algorithm. For an algorithm , we use to denote its output when run on the graph . If satisfies for all , we define its approximation ratio as

Clearly . Now we are interested in the best approximation ratio achievable by an efficient algorithm , i.e. the quantity

It should be clear that similar questions arise for all sorts of other values which are NP-hard to compute (e.g. the chromatic number of a graph, or the length of its shortest tour, or the length of the longest simple path, etc.) An algorithm of Goemans and Williamson (based on a form of convex optimization known as semi-definite programming) shows that

On the other hand, Håstad proved that, as a consequence of the PCP Theorem, it is NP-complete to obtain an approximation ratio better than , i.e. if , then

How does one prove such a theorem? Well, the -hardness of MAX-CUT is based on constructing graphs where every optimal solution has a particular structure (which eventually encodes the solution to another NP-hard problem like SATISFIABILITY). Similarly, the NP-hardness of of obtaining even “near-optimal” solutions is proved, in part, by constructing graphs where every solution whose value is close to optimal has some very specific structure (e.g. is close—in some stronger sense—to an optimal solution).

In this way, one of the main steps in proving the inapproximability of -hard problems involves constructing objects which have such a “rigidity” property. This summer school is about how one can use the rigidity of analytic and geometric objects to obtain combinatorial objects with the same property. In fact, assuming something called the “Unique Games Conjecture” (which we will see later), the approximability of many constraint satisfaction problems can be tied directly to the existence of certain geometric configurations.

The Lectures

The first series of lectures will concern the Sparsest Cut problem in graphs and its relationship to bi-lipschitz embeddings of finite metric spaces. In particular, we will look at rigidity properties of  “nice” subsets of the Heisenberg group, and how these can be used to prove limitations on a semi-definite programming approach to Sparsest Cut. In the second series, we will see how—assuming the Unique Games Conjecture (UGC)—proving lower bounds on certain simple semi-definite programs actually proves lower bounds against all efficient algorithms. This will entail, among other things, an analytic view of -valued functions, primarily through harmonic analysis.

Sparsest Cut and embeddings

The Sparsest Cut problem is classically described as follows. We have a graph and two functions and , with . The goal is to compute

where we use and . The problem has a number of important applications in computer science.

Computing is NP-hard, but again we can ask for approximation algorithms. The best-known approach is based on computing the value of the Goemans-Linial semi-definite program, , which is

This value can be computed by a semi-definite program (SDP), as we will see. It is an easy exercise to check that , and we can ask for the smallest such that for all -node graphs and all functions , we have

(E.g. it is now known that , with the upper bound proved here, and the lower bound proved here.)

By some duality arguments, one can characterize in a different way. For a metric space , write for the infimal constant such that there exists a mapping satisfying, for all ,

It turns out that

This shows that determining the power of the preceding SDP is intimately connected to understanding bi-lipschitz embeddings into . This is what we will study in the first 6 lectures.

1. (Arnaud de Mesmay) In the first lecture, we will be introduced to the basic geometry of the 3-dimensional Heisenberg group , and how differentiation plays a roll in proving lower bounds on bi-lipschitz distortion. In particular, we will see Pansu’s approach for finite-dimensional targets and a generalization to spaces with the RNP, and also why a straightforward generalization would fail for .
2. (Mohammad Moharrami) Next, we will see how a differentiation approach to embeddings might work in a toy setting that uses only finite graphs. The study of “monotone subsets” (which is elementary here) also arises in the work of Cheeger and Kleiner in lectures 4 and 5.  (See also this post.)
3. (Sean Li) Here, we will see that there is an equivalent metric on the Heisenberg group for which embeds isometrically into . This is one half of proving lower bounds on using (1).
4. (Jeehyeon Seo and John Mackay) In Lectures 4-5, we’ll look at the approach of Cheeger and Kleiner for proving that does not bi-lipschitz embed into .  (Note that these authors previously offered a different approach to non-embeddability, though the one presented in these lectures is somewhat simpler.)
5. (Florent Baudier) Finally, in Lecture 6, we see some embedding theorems for finite metric spaces that allow us to prove upper bounds on .

The UGC, semi-definite programs, and constraint satisfaction

In the second series of lectures, we’ll see how rigidity of geometric objects can possibly say something, not just about a single algorithm (like a semi-definite program), but about all efficient algorithms for solving a particular problem.

1. (An-Sheng Jhang) First, we’ll review basic Fourier analysis on the discrete cube, and how this leads to some global rigidity theorems for cuts. These tools will be essential later.  (See also these lecture notes from Ryan O’Donnell.)
2. (Igor Gorodezky) Next, we’ll see a semi-definite program (SDP) for the MAX-CUT problem, and a tight analysis of its approximation ratio (which turns out to be the value we saw earlier).
3. (Sam Daitch) In the third lecture, we’ll see the definition of the Unique Games Conjecture, and how it can be used (in an ad-hoc manner, for now) to transform our SDP analysis into a proof that the SDP-based algorithm is optimal (among all efficient algorithms) under some complexity-theoretic assumptions.
4. (Deanna Needell) A key technical component of the preceding lecture is something called the Majority is Stablest Theorem that relates sufficiently nice functions on the discrete cube to functions on Gaussian space.
5. (Sushant Sachdeva) In the final lecture, we’ll see Raghavendra’s work which shows that, for a certain broad class of NP-hard constraint satisfaction problems, assuming the UGC, the best-possible algorithm is the “canonical” semi-definite program. In other words, the approximation ratio for these problems is completely determined by the existence (or lack thereof) of certain vector configurations in .  (See also this post.)

Here are Prasad Raghavendra’s notes on one of two guest lectures he gave for CSE 599S. Prasad will be a faculty member at Georgia Tech after he finishes his postdoc at MSR New England.

In yet another excursion in to applications of discrete harmonic analysis, we will now see an application of the invariance principle, to hardness of approximation. We will complement this discussion with the proof of the invariance principle in the next lecture.

1. Invariance Principle

In its simplest form, the central limit theorem asserts the following: “As increases, the sum of independent bernoulli random variables ( random variables) has approximately the same distribution as the sum of independent normal (Gaussian with mean and variance ) random variables”

Alternatively, as increases, the value of the polynomial has approximately the same distribution whether the random variables are i.i.d Bernoulli random variables or i.i.d normal random variables. More generally, the distribution of is the approximately the same as long as the random variables are independent with mean , variance and satisfy certain mild regularity assumptions.

A phenomenon of this nature where the distribution of a function of random variables, depends solely on a small number of their moments is referred to as invariance.

A natural approach to generalize of the above stated central limit theorem, is to replace the “sum” () by other multivariate polynomials.   As we will see in the next lecture, the invariance principle indeed holds for low degree multilinear polynomials that are not influenced heavily by any single coordinate. Formally, define the influence of a coordinate on a polynomial as follows:

Definition 1 For a multilinear polynomial define the influence of the coordinate as follows:

Here denotes the variance of over random choice of .

The invariance principle for low degree polynomials was first shown by Rotar in 1979. More recently, invariance principles for low degree polynomials were shown in different settings in the work of Mossel-O’Donnell-Olekszciewicz and Chatterjee. The former of the two works also showed the Majority is Stablest conjecture, and has been influential in introducing the powerful tool of invariance to hardness of approximation.

Here we state a special case of the invariance principle tailored to the application at hand.  To this end, let us first define a rounding function as follows:

Theorem 2 (Invariance Principle [Mossel-ODonnell-Oleszkiewicz]) Let and be sets of Bernoulli and Gaussian random variables respectively. Furthermore, let

and . Let denote independent copies of the random variables and .

Let be a multilinear polynomial given by , and let . If for all , then the following statements hold:

1. For every function which is thrice differentiable with all its partial derivatives up to order bounded uniformly by ,
   

   

2. Define the function as Then, we have

2. Dictatorship Testing

A boolean function on bits is a dictator or a long code if for some . Given the truth table of a function , a dictatorship test is a randomized procedure that queries a few locations (say ) in the truth table of , and tests a predicate on the values it queried. If the queried values satisfy the predicate , the test outputs ACCEPT else it outputs REJECT. The goal of the test is to determine if is a dictator or is far from being a dictator. The main parameters of interest in a dictatorship test are :

• Completeness Every dictator function is accepted with probability at least .
• Soundness Any function which is far from every dictator is accepted with probability at most .
• Number of queries made, and the predicate used by the test.

2.1. Motivation

The motivation for the problem of dictatorship testing arises from hardness of approximation and PCP constructions. To show that an optimization problem is -hard to approximate, one constructs a reduction from a well-known -hard problem such as Label Cover to . Given an instance of the Label Cover problem, a hardness reduction produces an instance of the problem . The instance has a large optimum value if and only if has a high optimum. Dictatorship tests serve as “gadgets” that encode solutions to the Label Cover, as solutions to the problem . In fact, constructing an appropriate dictatorship test almost always translates in to a corresponding hardness result based on the Unique Games Conjecture.

Dictatorship tests or long code tests as they are also referred to, were originally conceived purely from the insight of error correcting codes. Let us suppose we are to encode a message that could take one of different values . The long code encoding of the message is a bit string of length , consisting of the truth table of the function . This encoding is maximally redundant in that any binary encoding with more than bits would contain bits that are identical for all messages. Intuitively, greater the redundancy in the encoding, the easier it is to perform the reduction.

While long code tests/dictatorship tests were originally conceived from a coding theoretic perspective, somewhat surprisingly these objects are intimately connected to semidefinite programs. This connection between semidefinite programs and dictatorship tests is the subject of today’s lecture. In particular, we will see a black-box reduction from SDP integrality gaps for Max Cut to dictatorship tests that can be used to show hardness of approximating Max Cut.

2.2. The case of Max Cut

The nature of dictatorship test needed for a hardness reduction varies with the specific problem one is trying to show is hard. To keep things concrete and simple, we will restrict our attention to the Max Cut problem.

A dictatorship test for the Max Cut problem consists of a graph on the set of vertices . By convention, the graph is a weighted graph where the edge weights form a probability distribution (sum up to ). We will write to denote an edge sampled from the graph (here ).

A cut of the graph can be thought of as a boolean function . For a boolean function , let denote the value of the cut. The value of a cut is given by

It is useful to define , for non-boolean functions that take values in the interval . To this end, we will interpret a value as a random variable that takes values. Specifically, we think of a number as the following random variable

$ latex a = -1 $ with probability and with probability  $\frac{1+a}{2}$.

With this interpretation, the natural definition of is as follows:

Indeed, the above expression is equal to the expected value of the cut obtained by randomly rounding the values of the function to as described in Equation 2.

A Dictator Cut

The dictator cuts are given by the functions for some . The of the test is the minimum value of a dictator cut, i.e.,

The soundness of the dictatorship test is the value of cuts that are far from every dictator. We will formalize the notion of being far from every dictator is formalized using influences as follows:

Cut Far From Every Dictator

Definition 3 (-quasirandom) A function is said to be -quasirandom if for all , .

Definition 4 () For a dictatorship test over and , define the soundness of as

2.3. SDP Relaxation

Let be an an instance of the Max Cut problem. Specifically, is a weighted graph over a set of vertices , whose edge weights sum up to (by convention). Thus, the set of edges will also be thought of as a probability distribution over edges. Let us the Goemans-Williamson semidefinite programming relaxation for Max Cut. The variables of the GW SDP consist of a set of vectors , one vector for each vertex in the graph .

GW SDP Relaxation
Maximize (Average Squared Length of Edges)

Subject to  (all vectors  $\latex \mathbf v_i\ $ are unit vectors)

3. Intuition

Let be an an arbitrary instance of the Max Cut problem and let denote a feasible solution to the GW SDP relaxation. We begin by presenting the intuition behind the black box reduction.

3.1. Dimension Reduction

Without loss of generality, the SDP solution can be assumed to lie in a large constant dimensional space. Specifically, given an arbitrary SDP solution in -dimensional space, project it in to a random subspace of dimension – a large constant. Random projections approximately preserve the lengths of vectors and distances between them. Hence, roughly speaking, the vectors produced after random projection yield a low-dimensional SDP solution to the same graph .

Formally, sample random Gaussian vectors of the same dimension as the SDP vectors . Here is to be thought of as a large constant independent of the size of the graph . Define a solution to the GW SDP relaxation as follows:

      for all vertices     in graph $latex  G$

The vector is just the projection of the vector along directions , normalized to unit length. Clearly, the vectors form a feasible SDP solution to GW SDP.

For every , by choosing to be a sufficiently large constant, the following can be ensured: the distance between any two vectors and is preserved up to -multiplicative factor with probability at least . Therefore, there exists some choice of such that the vectors form a low dimensional SDP solution with roughly the same value as , i.e., . Henceforth, without loss of generality, let us assume that the SDP solution consist of -dimensional vectors for a large enough constant .

3.2. Sphere Graph

A graph on the unit sphere, will consist of a set of unit vectors, and weigthed edges between them. As usual, the weights of the graph form a probability distribution, in that they sum up to .

The SDP solution for a graph , yields a graph on the -dimensional unit sphere, which is isomorphic to . Recall that the objective value of the GW SDP is the average squared length of the edges. Hence, the SDP value remains unchanged under the following transformations:

• Rotation Any rotation of the SDP vectors about the origin, preserves the lengths of edges, and the distances between them. Thus, rotating the SDP solution yields another feasible solution with the same objective value.
• Union of Rotations Let and be two solutions obtained by applying rotations to the SDP vectors . Let be the associated graphs on the unit sphere. Let denote the union of the two graphs, i.e., . The set of all distinct vectors in are the vertices of . The edge distribution of is the average of the edge distributions of and .The average squared lengths of edges in both and are equal to . Hence, the average squared edge length in is also equal to . Thus, taking the union of two rotations of a graph preserves the SDP value.

Define the sphere graph as follows:

Constructing the Sphere Graph

Sphere Graph : Union of all possible rotations of the graph (on the set of vectors ) on the -dimensional unit sphere.

Clearly the sphere graph is an infinite graph. The sphere graph is solely a conceptual tool, and an explicit representation is never needed in the reduction. Nevertheless, due to its symmetry, indeed the sphere graph can be represented succinctly. By construction, the SDP value of the sphere graph is the same as that of the original graph . However, we will argue that is a harder instance of Max Cut than the original graph . In fact, given a cut for the sphere graph , it is possible to retrieve a cut for the original graph with the same objective value. Let us suppose is a cut of the sphere graph that cuts a -fraction of the edges. Notice that consists of a union of infinitely many copies (or rotations) of the graph . Therefore, on at least one of the copies of , the cut must cut a -fraction of the edges. Indeed, if we have oracle access to the cut function , we can efficiently construct a cut of the graph with the same value as using the following rounding procedure:

• Sample a rotation of the unit sphere, uniformly at random.
• Output the cut induced by on the copy of the graph .

The expected value of the cut output by the procedure is equal to the value of the cut on the sphere graph . An immediate consequence is that,

The sphere graph inherits the same SDP value as , while the optimum value is at most that of the graph . In this light, the sphere graph is a harder instance of Max Cut than the original graph .

It is easy to see that the following is an equivalent definition for the sphere graph .

Definition 5 (Sphere Graph ) The set of vertices of is the set of all points on the -dimensional unit sphere. To sample an edge of use the following procedure,

• Sample an edge in the graph ,
• Sample two points on the sphere at a squared distance uniformly at random.
• Output the edge between .

3.3. Hypercube Graph

Finally, we are ready to describe the construction of the graph on the -dimensional hypercube. Here we refer to the hypercube suitably normalized to make all its points lie on the unit sphere.

The set of vertices of are points in . An edge of can be sampled as follows:

• Sample an edge in the graph .
• Sample two points in , at squared distance uniformly at random.
• Output the edge between .

It is likely that there are no pair of points on the hypercube at a distance exactly equal to . For the sake of exposition, let us ignore this issue for now. To remedy this issue, in the final construction, for each edge just introduce a probability distribution over edges such that the expected length of an edge is indeed .

Completeness:

Consider the th dictator cut given . This corresponds to the axis-parallel cut of the graph along the th axis of the hypercube. Let us estimate the value of the cut . An edge is cut by the th dictator cut if and only if . Therefore, the value of the th dictator cut is given by:

Notice that two points at a squared distance differ on exactly coordinates. Hence, two random points at distance at a squared distance differ on a given coordinate with probability . Therefore, let us rewrite the expression for as follows:

Hence,   is at least .

Soundness:

Consider a cut that is far from every dictator. Intuitively, the cut is not parallel to any of the axis of the hypercube. Note the strong similarity in the construction of the sphere graph and the hypercube graph . In both cases, we sampled two random points at a distance equal to the edge length. In fact, the hypercube graph is a subgraph of the sphere graph . The existence of special directions (the axis of the hypercube) is what distinguishes the hypercube graph from the sphere graph . Thus, roughly speaking, a cut which is not paralel to any axis must be unable to distinguish between the sphere graph and the hypercube graph . If we visualize the cut of as a geometric surface not parallel to any axis, then the same geometric surface viewed as a cut

Extending the cut from Hypercube to Sphere graph

of the sphere graph must separate roughly the same fraction of edges.

Indeed, the above intuition can be made precise if the cut is sufficiently smooth (low degree). The cut can be expressed as a multilinear polynomial (by Fourier expansion), thus extending the cut function from to . The function is smooth if the corresponding polynomial polynomial is low degree. If is smooth and far from every dictator, then one can show that,

By Equation 1, the maximum value of a cut of the sphere graph is at most . Therefore, for any cut that is smooth and far from every dictator, we get .

Ignoring the smoothness condition for now, the above argument shows that the soundness of the dictatorship test is at most . Summarizing the above discussion, starting from a SDP solution for a graph , we constructed hypercube graph (dictatorship test) such that , and .

By suitably modifying the construction of , the smoothness requirement for the cut can be dropped. The basic idea is fairly simple yet powerful. In the definition of , while introducing an edge between , perturb each coordinate of and with a tiny probability to obtain and respectively, then introduce the edge instead of . The introduction of noise to the vertices and has an averaging effect on the cut function, thus making it smooth.

4. Formal Proof

Let be an arbitrary instance of Max Cut. Let be a feasible solution to the SDP relaxation.

Locally, for every edge in , there exists a distribution over assignments that match the SDP inner products. In other words, there exists valued random variables such that

For each edge , let denote the local integral distribution over assignments.

The details of the construction of dictatorship test are as follows:

The set of vertices of consist of the -dimensional hypercube . The distribution of edges in is the one induced by the following sampling procedure:

• Sample an edge in the graph .
• Sample times independently from the distribution to obtain and , both in .
• Perturb each coordinate of and independently with probability to obtain respectively. Formally, for each ,

  

• Output the edge .

Theorem 6 There exists absolute constants such that for all , for any graph and an SDP solution for the GW-SDP relaxation of ,

• 
• 

Let be a cut of the graph. The fraction of edges cut by is given by

4.1. Completeness

Consider the th dictator cut given by . With probability , the perturbation does not affect the th coordinate of and . In other words, with probability , the following hold: and . Hence,

Observe that if the edge in is sampled, then the distribution is used to generate each coordinates of and . Specifically, this means that the coordinates and satisfy,

Therefore, .

4.2. Soundness

For the sake of analysis, let us construct a graph , roughly similar to the sphere graph described earlier. Gaussian Graph

The vertices of are points in . The graph is the union of all random projections of the SDP solution in to dimensions. Formally, an edge of can be sampled as follows:

• Sample random Gaussian vectors of the same dimension as the SDP solution .
• Project the SDP vectors along directions to obtain a copy of in a . Formally define,

  

• Sample an edge in , and output the corresponding edge in

As lengths of vectors are approximately preserved under random projections, most of the vectors are are roughly unit vectors. Hence, the Gaussian graph is a slightly fudged version of the sphere graph described earlier.

As the graph consists of a union of several isomorphic copies of , the following claim is an immediate consequence.

Claim 1

Let us suppose be a -quasirandom function. For the sake of succinctness, let us denote . Essentially, is the expected value returned by on querying a perturbation of the input . Thus the function is a smooth version of , obtained by averaging the values of .

Now we will extend the cut from the hypercube graph to the Gaussian graph . To this end, write the functions as written as a multilinear polynomials in the coordinates of . In particular, the Fourier expansion of and yields the intended multilinear polynomials.

The polynomials and yield natural extension of the cut functions and from to . However, unlike the original cut function , the range of its extension need not be restricted to . To ensure that the extension defines a cut of the graph , let us round the extension in the most natural fashion using .   Specifically, the extension of the cut to is given by

  where 

Let denote the value of the cut of the graph . Now we can show the following claim.

Claim 2 For a -quasirandom function ,

By definition of , we have . Along with Claim 1, this implies that , completing the proof of Theorem 6.

Proof: [Proof of Claim 2] Returning to the definition of , notice that the random variable depends only on . Thus, the value of a cut can be rewritten as,

By the definition of the noise operator ,. Hence can be rewritten as

By definition of the Gaussian graph , we have

Firstly, let us denote by the function given by . Let us restrict our attention to a particular edge . For this edge, we will show that

By averaging the above equality over all edges in the graph , the claim follows. We will use the invariance principle to show the above claim.

By design, for each edge the pairs of random variables and satisfy,

The predicate/payoff is currently defined as in the domain . Extend the payoff to a smooth function over the entire space , with all its partial derivatives up to order bounded uniformly throughout . Notice that the function by itself does not have uniformly bounded derivatives in . Further, it is easy to ensure that the extension satisfies the following Lipschitz condition for some large enough constant ,

for all

We will prove Equation 4 in two steps.

Step I : Apply the invariance principle with the ensembles and , for the vector of multilinear polynomials and the smooth function . This yields,

Step II : In this step, let us bound the effect of the rounding operation used in extending the cut from to .

As is a cut of , its range is . Hence, the corresponding polynomial takes values on inputs from . As is an average of the values of , the values and are always in the range .

By the invariance principle, the random variable has approximately the same behaviour as . Roughly speaking, this implies that the values are also nearly always in the range . Hence, intuitively the rounding operation must have little effect on the value of the cut.

This intuition is formalized by the second claim in the invariance principle. The function measures the squared deviation from the range . For random variables , clearly we have . By the invariance principle applied to polynomial we get,

Using the Lipschitz condition satisfied by the payoff,

             (By Cauchy Schwartz Inequality)

  (by definition of  )

           (Equation 2)

Along with Equation 4, the above inequality implies Equation 4. This finishes the proof of Claim 2.  

5. Dictatorship Tests and Rounding Schemes

The soundness analysis can be translated in to an efficient rounding scheme. Specifically, given a cut of the graph , let denote the extension of the cut to the Gaussian graph . The idea is to sample a random copy of the graph inside the Gaussian graph , and output the cut induced by on the copy. The details of the rounding scheme so obtained are as follows:

Input: SDP solution for the GW SDP relaxation of the graph .

• Sample random Gaussian vectors of the same dimension as the SDP solution .
• Project the SDP vectors along directions . Let

  

• Compute for all as    where
• Assign vertex , the value with probability and with the remaining probability.

Let denote the expected value of the cut returned by the above rounding scheme on an SDP solution . Then,

The following is an immediate consequence of Claim 2,

Theorem 7 For a -quasirandom function ,

The above theorem exposes an interesting duality between rounding schemes and dictatorship tests.

This post is about a beautiful twist on flows that arises when studying (the dual) of the Sparsest Cut SDP.  These objects, which I’m going to call “PSD flows,” are rather poorly understood, and there are some very accessible open problems surrounding them.  Let’s begin with the definition of a normal flow:

Let be a finite, undirected graph, and for every pair , let be the set of all paths between and in .  Let .  A flow in G is simply a mapping .  We define, for every edge , the congestion on as

which is the total amount of flow going through .  Finally, for every , we define

as the total amount of flow sent from u to v.

Now, in the standard (all-pairs) maximum concurrent flow problem, the goal is to find a flow F which simultaneously sends units of flow from every vertex to every other, while not putting more than one unit of flow through any edge, i.e.

In order to define a PSD flow, it helps to write this in a slightly different way.  If we define the symmetric matrix

then we have

Claim 1: .

To see that this is true, we can take a matrix with for all and fix it one entry at a time so that and , without decreasing the total demand satisfied by the flow.

For instance, if and , then it must be that , so we can reroute units of flow going through the edge to go along one of the extraneous flow paths which gives the excess .  Similar arguments hold for the other cases (Exercise!).

PSD flows

So those are normal flows.  To define a PSD flow, we define for any symmetric matrix A, the Laplacian of A, which has diagonal entries and off-diagonal entries .  It is easy to check that

Hence if for all , then certainly (i.e. L(A) is positive semi-definite).  The PSD flow problem is precisely

where is defined as above.  Of course, now we are allowing to have negative entries, which makes this optimization trickier to understand.  We allow the flow to undersatisfy some demand, and to overcongest some edges, but now the “error” matrix has to induce a PSD Laplacian.

Scaling down the capacities

Now, consider some , and write

Requiring for every simply induces a standard flow problem where each edge now has capacity .  In the case of normal flows, because we can decouple the demand/congestion constraints as in Claim 1, we can easily relate to (the first is exactly times the second, because we can just scale a normal flow down by and now it satisfies the reduced edge capacities).

Question: Can we relate and ?  More specifically, do they differ by some multiplicative constant depending only on ?

This is a basic question whose answer is actually of fundamental importance in understanding the Sparsest Cut SDP.  I asked this question in its primal form almost 4 years ago (see question 3.2 here).

Note that the answer is affirmative if we can decouple the demand/congestion constraints in the case of PSD flows.  In other words, let and let .

Question: Can we relate to ?

In the next post, I’ll discuss consequences of this question for constructing integrality gaps for the Sparsest Cut SDP.

Before we can apply more advanced fixed point theorems to the Evasiveness Conjecture, we need a little background on simplicial complexes, and everything starts with simplices.

Simplices

It’s most intuitive to start with the geometric viewpoint, in which case an -simplex is defined to be the convex hull of affinely independent points in .  These points are called the vertices of the simplex.  Here are examples for

A simplicial complex is then a collection of simplices glued together along lower-dimensional simplices.  More formally, if is a (geometric) simplex, then a face of is a subset formed by taking the convex hull of a subset of the vertices of .

Finally, a (geometric) simplicial complex is a collection of simplices such that

1. If and is a face of , then , and
2. If and , then is a face of both and .

Property (1) gives us downward closure, and property (2) specifies how simplices can be glued together (only along faces).  For instance, the first picture depicts a simplicial complex.  The second does not.

There is also an abstract way to define a simplicial complex.  An abstract simplicial complex is a ground set of vertices, together with a collection of subsets of satisfying the axioms

1. 
2. If and , then .

For instance, a standard (undirected) graph can be thought of as a 1-dimensional abstract simplicial complex.  We can always realize an abstract complex as a geometric complex by taking affinely independent points in and adding in the appropriate convex hulls.  (Exercise:  Every 1-dimensional complex can be realized in , but not .)  In general, we will switch freely between the two notions.  (Here is an interesting paper of Matousek, Tancer, and Wagner on the computational complexity of realizability.)

Contractibility and collapsibility

Say that a set is contractible if there exists a continuous mapping such that

1. is the identity on .
2. for some fixed .

In other words, can be contracted to a point in place. For instance, the closed unit ball in is contractible while the unit sphere is not (think about it!)  As another example, consider that a geometric realization of a graph (a 1-dimensional simplicial complex) is contractible if and only if the graph is a connected tree.

We now define two basic operations on simplicial complexes.  The first involves removing a vertex

The second, called the link of in is

For example, consider the following complex , in which a distinguished simplex is marked.

Then we form the two simplices and (respectively) as follows.

The link of separates from .  The following lemma generalizes the natural strategy for showing that a connected tree is contractible.

Lemma: If is a geometric simplicial complex and both and are contractible, then so is .

Instead of proving this lemma, we will work with the more combinatorial notion of collapsibility.

Let be an abstract simplicial complex.  A free face of is a non-maximal face which is contained in a unique maximal face.  We can collapse to a subcomplex by removing a free face , along with all faces containing .

After collapsing the free face A.

This yeilds an inductive notion of collapsibility: is collapsible if there exists a sequence of collapses that leads from to the empty set.  It is straightforward to verify that a geometric simplicial complex is contractible whenever it is collapsible, but the reverse direction does not hold.

Now we state a version of the preceding lemma for collapsibility.

Collapsibility Lemma: If and are collapsible, then so is .

Proof: The key observation is that the sequence of moves used to collapse can be used to collapse to (verify using the definition of ), at which point we can collapse to the empty set by assumption.

Collapsibility and Evasiveness

Before ending the lecture, we should say why collapsibility is relevant to the evasiveness conjecture.  The connection is relatively straightforward:  Let be a non-trivial, monotone boolean function.  For a subset , let represent the characteristic vector of .  Then associated to , we have the simplicial complex

Let and be the two functions on bits corresponding to fixing the value of the th bit.  Then we have:

A simple induction using the Collapsibility Lemma (which we will do in the next lecture) now yields our main topological connection:

If is non-evasive, then is collapsible (hence also contractible).

[Credits: Some pictures taken from Wikipedia.]

whew.

Grant deadline coming soon.  More posts when the madness ends.

Continuing our look at some toplogical methods, today we’ll see the evasiveness conjecture in decision tree complexity.  In the next lecture, we’ll see how we can sometimes analyze the complexity using fixed point theorems, and their generalizations (like the Hopf index formula), following the work of Kahn, Saks, and Sturtevant.  These two lectures are co-blogged with Elisa Celis, with a lot of input from Lovasz’s lecture notes.

Decision tree complexity and evasiveness

Consider a boolean function on n bits.  We define the decision tree complexity of f as follows.  Given an unknown input , you are allowed to ask about the values of various bits of x, e.g. .  Your goal is to compute using as few questions as possible, and your questions can be adaptive, depending on answers to previous questions.  The complexity of such a strategy is the maximum number of questions asked for any .  The decision tree complexity, written , is the minimum complexity of any strategy that computes f.  (There are many other interesting models of decision complexity, see e.g. this survey.)

Clearly , because we can trivially query all the bits of x, and then output f(x).  A function f is called evasive if this upper bound is met, i.e. .  As an example, consider the parity function , where is addition modulo 2.  Clearly is evasive because after bits of x are asked about, the setting of the final bit determines the value of f.

For a more general example, consider any such that is odd.  In this case, for every , exactly one of or has the same property that the number of inputs resulting in a 1 is odd.  (These two functions are the natural restriction of f to functions on n-1 bits, which results from fixing the value of the ith bit.)  Thus an adversary could keep answering questions “?” so that the restricted function retains this property.  Since the number of inputs yielding a 1 is always odd, the restricted function always takes both possible values, implying that f is evasive–the advesary ensures that the value cannot be determined until all n possible questions are asked.

For an example of a non-evasive property, think of a point as speciying a directed graph on vertices, where there is exactly one directed edges connecting every pair of vertices, and x specifies the direction of this edge (this is called a tournament).  Thinking of the vertices as players, a directed edge from u to v means that u defeats v.  Now if the digraph specified by x has one vertex that defeats everyone else.  What is ?

Well, first we can conduct a single elimination tournament, where vertex 1 plays vertex 2, and the winner players vertex 3, and the winner of that players vertex 4, etc.  At the end, there is only one remaining vertex that remains undefeated.  Now asking more questions, we can determine whether indeeds defeats everyone else.  The total number of questions was , hence , implying that f is not evasive.

Montone graph properties and the evasiveness conjecture

Let .  In general, we can encode an arbitrary undirected N-vertex graph as an element .  A function is called a graph property if relabeling the vertices of doesn’t affect the value of .  The function f is monotone if the value of the function can never change from 1 to 0 when flipping one of the input bits from 0 to 1.  In the setting of graph properties, this corresponds to those which are maintained under addition of edges to the graph, e.g. “is G connected?” or “does G have a k-clique?”

Evasiveness Conjecture (Aanderaa-Karp-Rosenberg): Every non-trivial monotone graph property is evasive.

Here, non-trivial means that , where and denote the all-zeros and all-ones strings, respectively.

For example, consider the example “is G connected?”  The adversary is simple:  When asked about a possible edge , she answers NO unless this answer would imply that the graph is disconnected.  In other words, she answers NO unless she has answered NO already for all edges across a cut except for , in which case she has to answer YES.

Now, suppose there is a strategy which figures out the connectivity of without asking a question about some edge .  In this case, the conclusion must be that because the adversary always maintains that by answering everything in the future YES, she could force the graph to be connected.  In this case, the edges answered YES have to form a spanning tree of G (otherwise by answering all unasked questions NO, the graph would become disconnected).  Consider a path P from i to j in this YES spanning tree.  Let be the edge of P which was asked about last.  Clearly the adversary answered YES for , but this contradicts the advesary’s strategy.  Since has not been asked yet, the adversary is safe to answer NO for , and still later by answering YES on , she could force the graph to be connected.  Thus no such strategy exists, and connectivity is evasive.

A natural generalization of the evasiveness conjecture is to general monotone functions which are invariant under a group acting transitively on the coordinates.  For a permutation , let act on via .  We say that is invariant under if for every .  Let be the group of all permutations under which f is invariant.  We will say that f is weakly symmetric if acts transitively on .

Generalized Evasiveness Conjecture: If f is a non-trivial, monotone, weakly symmetric function, then f is evasive.

Clearly this generalizes the previous conjecture because every graph property is invariant under permutations of the vertices (which induces a transitive action on the edges).

A proof when is prime

We’ll end this lecture with a proof of the generalized conjecture which holds when n is prime.  The proof will hint at the topological connections coming up in the next lecture.

First, we generalize the proof that f is evasive when is odd.  For , let denote the hamming weight of x, i.e. the number of 1’s in x, and for , define

Topologically-minded readers will recognize this as some kind of Euler characteristic, and this connection will bear out in the next lecture.  We claim that if , then f is evasive.  To see this, note that .  Hence if , then one of or is non-zero, so an adversary can keep answering so that the restriction has .  Finally, notice that if , then must take both values 0 and 1, so the adversary can maintain this until all n questions are asked.

Theorem: Suppose n is prime. If is monotone, non-trivial, and weakly symmetric, then f is evasive.

Proof: We’ll show that .  First, we argue that contains a cyclic permutation.  Write , where .  By transitivity, we have , hence divides .  Now since n is prime, by Cauchy’s theorem there is an element or order (i.e. an n-cycle).

Since f is non-trivial, we know that and .  For any with , it is clear that all n elements are distinct, since n is prime.  But f is invariant under , thus the set partitions into equivalence classes , where for every i.  But this immediately implies that , which gives .

Now we’ll move away from spectral methods, and into a few lectures on topological methods.  Today we’ll look at the Borsuk-Ulam theorem, and see a stunning application to combinatorics, given by Lovász in the late 70’s.  A great reference for this material is Matousek’s book, from which I borrow heavily.  I’ll also discuss why the Lovász-Kneser theorem arises in theoretical CS.

The Borsuk-Ulam Theorem

We begin with a statement of the theorem.  We will think of the n-dimensional sphere as the subset of given by

Borsuk-Ulam Theorem: For every continuous mapping , there exists a point with .

Pairs of points are called antipodal.

There are a couple of common illustrative examples for the case .  The theorem says that if you take the air out of a basketball, crumple it (no tearing), and flatten it out, then there are two points directly on top of each other which were antipodal before.  Another common example states that at every point in time, there must be two points on the earth which both have exactly the same temperature and barometric pressure (assuming, of course, that these two parameters vary continuously over the surface of the eath).

The n=1 case is completely elementary.  For the rest of the lecture, let’s use and to denote the north and south poles (the dimension will be obvious from context).  To prove the n=1 case, simply trace out the path in starting at and going clockwise around .  Simultaneously, trace out the path starting at and going counter-clockwise at the same speed.  It is easy to see that eventually these two paths have to collide:  At the point of collision, .

We will give the sketch of a proof for   Let , and note that our goal is to prove that for some .  Note that is antipodal in the sense that for all .  Now, if for every , then by compactness there exists an such that for all .  Because of this, we may approximate arbitrarily well by a smooth map, and prove that the approximation has a 0.  So we will assume that itself is smooth.

Now, define by , i.e. the north/south projection map.  Let be a hollow cylinder, and let be given by so that linearly interpolates between and .

Also, let’s define an antipodality on by .  Note that is antipodal with respect to , i.e. , because both and are antipodal.  For the sake of contradiction, assume that for all .

Now let’s consider the structure of the zero set .  Certainly since , and these are h’s only zeros.  Here comes the sketchy part:  Since is smooth, is also smooth, and thus locally can be approximated by an affine mapping .  It follows that if is not empty, then it should be a subspace of dimension at least one.  By an arbitrarily small perturbation of the initial , ensuring that is sufficiently generic, we can ensure that is either empty or a subspace of dimension one.  Thus locally, should look like a two-sided curve, except at the boundaries and , where (if non-empty) would look like a one-sided curve.  But, for instance, cannot contain any Y-shaped branches.

It follows that is a union of closed cycles and paths whose endpoints must lie at the boundaries and .  ( is represented by red lines in the cylinder above.)  But since there are only two zeros on the sphere, and none on the sphere, must contain a path from to   Since is antipodal with respect to , must also satisfy this symmetry, making it impossible for the segment initiating at N to ever meet up with the segment initiating at S.  Thus we arrive at a contradiction, implying that must take the value 0.

Notice that the only important property we used about (other than its smoothness) is that is has a number of zeros which is twice an odd number.  If had, e.g. four zeros, then we could have two -symmetric paths emanating from and returning to the bottom.  But if has six zeros, then we would again reach a contradiction.

The Kneser conjecture

While the Borsuk-Ulam theorem seems initially far removed from any combinatorial applications, the following result of Lyusternik and Shnirel’man starts to hint at the combinatorial significance.

LS Lemma: Let be a cover of by closed sets.  Then some contains an antipodal pair.

Proof: Consider the continuous map given by , where denotes e.g. the geodesic distance on .  By the Borsuk-Ulam theorem, there exists a with .  Now if for some , then .  Otherwise, if this holds for no , then .

In fact, it is not difficult to show that the LS Lemma is equivalent to Borsuk-Ulam.  For our application to the Kneser conjecture, we will need to consider both open and closed sets.

Corollary 1: Let be a cover of by open sets.  Then some contains an antipodal pair.

Proof: This follows from the LS Lemma by choosing a family of closed sets such that forms a cover of .  The existence of such sets follows from applying compactness to the following open cover of :  For every , choose an open set whose closure lies completely within some .  By compactness, there is a finite subcover.  By piecing together elements of its closure, we can form the sets .

Finally, we address the case of both open and closed sets.

Corollary 2: Let be a cover of with each open or closed.  Then some contains an antipodal pair.

Proof: Suppose that are closed and are open.  Consider the open cover , where denotes the -neighborhood of .  Taking and applying Corollary 1 to each cover results in a sequence .  If for some , we are done.  Thus we may pass to a subsequence for which for some fixed .  Choose a convergent subsequence , and observe that since is closed, we have .  We conclude that there exists an with .

The Kneser graphs.

Now, we introduce the Kneser graphs .  The vertex set is , i.e. the vertices are k-element subsets of .  There is an edge between two sets if and only if and are disjoint.  In 1955, Kneser posed the following conjecture.

Conjecture: For every and , , where denotes the chromatic number.

First, we note that the upper bound is easy:  Construct a coloring by defining

Clearly if , then and have the same minimum element, and thus they are not disjoint.  On the other hand, if , then , and the latter set contains only elements.  Thus again .

The conjecture stood open until Lovász resolved it, using the Borsuk-Ulam theorem.  Recently, Greene (an undergraduate at the time) gave a particularly simple proof which we reproduce here.

Proof: Consider and let .  Let be a set of points in general position, so that no -dimensional hyperplane through the origin contains more than points of .  Clearly we may assume that actually has vertex set .  For every , let

denote the open hemisphere centered at .

Now, suppose we have a proper coloring of , and define sets as follows:  is the set of all points such that contains a k-set colored .  Finally, .

By Corollary 2 above, there must exist an and a such that .  If , then since and are disjoint, there must exist two disjoint k-sets colored , which means we did not start with a proper coloring.

If, instead , it implies that both .  But then the -dimensional hyperplane contains points, contradicting the fact that was chosen in general position.

Appearances in TCS

One of the most useful properties of the Kneser graphs is that there is a large gap between their chromatic number and their fractional chromatic number .  The fractional chromatic number of graph is the minimum value such that can be covered by independent sets, where each vertex occurs in at least of them.

It is easy to see that by choosing the n independent sets

The Kneser graphs are a particularly natural family to have a large gap between and because we can define for an arbtirary G as the minimum ratio such that G admits a homomorphism into .  One can also see that using the Erdős–Ko–Rado theorem.

See e.g. this FOCS’08 paper of Alon, et. al. for a recent application which uses this gap between fractional and actual chromatic numbers.  The Kneser graphs also arise very naturally in PCP constructions, as one can see in this paper of Dinur, Regev, and Smyth about the computational hardness of coloring hypergraphs.

In the previous lecture, we gave an upper bound on the second eigenvalue of the Laplacian of (bounded degree) planar graphs in order to analyze a simple spectral partitioning algorithm.  A natural question is whether these bounds extend to more general families of graphs.  Well-known generalizations of planar graphs are those which can be embedded on a surface of fixed genus, and, more generally, families of graphs that arise by forbidding minors.  In fact, Spielman and Teng conjectured that for any graph excluding as a minor, one should have .   Of course planar graphs have genus 0, and by Wagner’s theorem, are precisely the graphs which exclude and as minors.  In this lecture, we will follow an intrinsic approach of Biswal, myself, and Rao which, in particular, is able to resolve the conjecture of Spielman and Teng.  First, we see why even pushing the conformal approach to bounded genus graphs is difficult.

Bounded genus graphs

For graphs of bounded genus, there is hope to use an approach based on conformal mappings.  In 1980, Yang and Yau proved that

for any compact Riemannian surface of genus .  (Note that for the Laplace-Beltrami operator, one usually writes as the first non-zero eigenvalue, rather than .)  In analog with Hersch’s proof of the genus 0 case, they use Riemann-Roch to obtain a degree- conformal mapping to the Riemann sphere, then try to pull back a second eigenfunction.  A factor of the degree is lost in the Rayleigh quotient (hence the factor in the preceding bound), and Hersch’s Möbius trick is still required.

genus 0

genus 1

genus 2

genus 3

An analogous proof for graphs of bounded genus would proceed by constructing a circle packing of on the sphere , but instead of the circles having disjoint interiors, we would be assured that every point of is contained in at most circles.  Unfortunately, such a result is impossible (this has to do with the handling of branch points in the discrete setting).  Kelner has to take a different approach in his proof that for graphs of genus at most .

He starts with a circle packing of on a compact surface of genus (whose existence follows from results of Beardon and Stephenon and He and Schramm).  Then Kelner randomly subdivides repeatedly, and these subdivisions give progressively better approximations to some sequence of surfaces .  Once the approximation is of high enough quality, one applies Riemann-Roch to , and infers something about a subdivision of .  The final element is to track how the second eigenvalue of changes (in expectation) under random subdivision.

Needless to say, this approach is already quite delicate, and for graphs that can’t be equipped with some kind of conformal structure, we seem to have reached a dead end.  In this lecture, we’ll see how to use intrinsic deformations of the geometry of in order to bound its eigenvalues.  Eventually, this will reduce to the study of certain kinds of multi-commodity flows.

Metrics on graphs

Let be an arbitrary n-vertex graph with maximum degree .  Recall that we can write

where .  (Also recall that we can replace by any Hilbert space, and the same formula holds.)  The first step is to prepare this equality for “non-linearization” by getting rid of the linear condition and the sum .  (This is a popular sort of passage in the non-linear geometry of Banach spaces, which also plays a rather important role in applications to the theoretical CS.)  The goal is to get only terms that look like .  Fortunately, there is a well-known way to do this:

which follows easily from the equality when .

Thus if we want to bound , we need to find an for which the latter ratio (without the ) is .  Now, for someone who works a lot with linear programming relaxations, it’s very natural to consider a “relaxation”

where the minimization is over all pseudo-metrics d, i.e. symmetric non-negative functions which satisfy the triangle inequality, but might have even for .  Certainly , but Bourgain’s embedding theorem (which states that every n-point metric space embeds into a Hilbert space with distortion at most ) also assures us that .  Since we are trying to show that , this term is morally negligible.  One can see the paper for a more advanced embedding argument that doesn’t lose this factor, but for now we concentrate on proving that .  The embedding theorems allow us to concentrate on finding an intrinsic metric on the graph with small “Rayleigh quotient,” without having to worry about an eventual geometric representation.

As a brief preview… we are going to find a good metric by taking a certain kind of all-pairs multi-commodity flow at optimality, and weighting the edges by their congestion in the optimal flow.  Thus as the flow spreads out on the graph, it has the effect of “uniformizing” its geometry.

Discrete Riemannian metrics, convexification, and duality

Let’s now assume that is planar.  We want to show that .  First, let’s restrict ourselves to vertex weighted metrics on .  Given any non-negative weight function , we can define the length of a path in by summing the weights of vertices along it:  .  Then we can define a vertex-weighted shortest-path pseudo-metric on in the natural way

where is the set of all u-v paths in .  We also have the nice relationship

since .

So if we define

then by (1), we have .

Examples. Let’s try to exhibit weights for two well-known examples:  the grid, and the complete binary tree.

For the grid, we can simply take for all .  Clearly .  On the other hand, a random pair of points in the grid is apart, hence .  It follows that , as desired.

For the complete binary tree with root , we can simply put and for .  (Astute readers will guess the geometrically decreasing weights are actually the optimal choice.)  In this case, , while all the pairs on opposite sides of the root have .  It again follows that .  Our goal is to provide such a weight for any planar graph.

A natural way to study the optimal weight is via duality; unfortunately, cannot be written as a convex program.  Thus we will pass to a “convexified” objetive:

The key here is that we have replaced the denominator by a linear form, which allows us to write as a convex program.  To relate it to , note that by Cauchy-Schwarz, for every

so our task reduces to finding an for which .  To see why we are not going lose much in the Cauchy-Schwarz bound, note that in the two “extremal” examples above, was concentrated near its maximum value (the tight case for Cauchy-Schwarz).

Now that we have a convex objective, we can find its dual.  If you are used to dealing with programs where the triangle inequality occurs in the primal, you already know that some kind of flow is going to rear its head in the dual.

Multi-flows and -congestion.

Let be the set of all paths in .  A flow in G will simply be a non-negative mapping which assigns values to paths.  We will define the flow from u to v as the quantity .  A complete flow in G is a flow F such that for all .  Finally, we define the congestion of the flow at a vertex v by , i.e. the amount of flow going through .  The -congestion is now the quantity

Working out the Lagrangian multipliers and using strong duality (Slater’s condition) yields the following.

Theorem (Duality): For any graph , we have

where the minimums are over all vertex weights and over all complete flows .

Thus in order to show the existence of a weight with , we need to show that for every complete flow F in G, .

Congestion, drawings, and crossings

First, let’s revisit our examples.  In the grid, a complete flow has total “length” about since about pairs need to send flow down a path of length at least .  To minimize the -congestion, we would spread this out evenly, putting congestion at every node, yelding for any complete flow .  In the complete binary tree, the argument is even easier since clearly flow has to go through the root.

Randomized rounding.

First, let’s round to an integral flow, i.e. such that the u-v flow is sent along a single u-v path for every pair of vertices.  Let be the random integral flow that arises in the following way:  For every pair of vertices u and v, let choose a path with probability .  It is easy to check that

Notice that is the amount of -congestion incurred by any complete flow congesting against itself (every vertex sends out flow n, so even that incurs congestion).

Thus we can assume that our flow is integral.  But now since our graph is planar, it has a drawing in the plane.  Now, I claim that every complete integral flow induces a drawing of the complete graph on n vertices in the plane, via the drawing of .  Furthermore, edges of cross only at vertices of .

In the picture, the graph is represented by black edges, while some of the edges of are drawn in colors.

How many edge crossings occur in the drawing of ?  Well, we have no idea how many crossings there are “inside” a vertex of , but if edges go in, we can make sure there are at most crossings (in the worst case, every edge crosses every other).  But this immediately implies that we have a drawing of in the plane with at most crossings.  Now, by the well-known crossing number inequality (found independently by Leighton, and also Ajtai, Chvatal, Newborn, and Szemeredi), the number of such crossings is always at least , implying that for any complete flow in .  This completes our alternate argument that for planar graphs.

In fact, it is trivial to prove the crossing inequality for the complete graph:  Every copy of must induce one crossing, now double count (it is easy to check that every crossing involving less than four points can be removed, so every crossing involves four points, hence we overcount by a factor of n, representing the one free vertex).  This kind of argument extends immediately to bounded genus graphs via similar crossing arguments.  For more advanced arguments, including the excluded-minor case conjectured by Spielman and Teng, one has to use not only complete graphs, but dense graphs, in which case the full power of the crossing number inequality is needed.  The beautiful probabilistic proof of Chazelle, Sharir, and Welzl (see the exposition on Terry Tao’s blog) can be appropriately generalized to excluded-minor graphs, and even to families of higher-dimensional graphs.

]]> http://tcsmath.wordpress.com/2008/10/13/lecture-5-uniformizing-graphs-multi-flows-and-eigenvalues/feed/ 1 James grid cbt routing http://tcsmath.wordpress.com/2008/10/08/lecture-4-conformal-mappings-circle-packings-and-spectral-geometry/ http://tcsmath.wordpress.com/2008/10/08/lecture-4-conformal-mappings-circle-packings-and-spectral-geometry/#comments Wed, 08 Oct 2008 10:40:52 +0000 James Lee http://tcsmath.wordpress.com/?p=343 ]]>

In Lecture 2, we used spectral partitioning to rule out the existence of a strong parallel repetition theorem for unique games.  In practice, spectral methods are a very successful heuristic for graph partitioning, and in the present lecture we’ll see how to analyze these partitioning algorithms for some common families of graphs.

Balanced separators, eigenvalues, and Cheeger’s inequality

Lipton and Tarjan proved that every planar graph has a negligibly small set of nodes whose remval splits the graph into two roughly equal pieces.  More specifically, every n-node planar graph can be partitioned into three disjoint sets such that there are no edges from to , the separator has at most nodes, and .  This allows one to do all sorts of things, e.g. a simple divide-and-conquer algorithm gives a linear time -approximation for the maximum independent set problem in such graphs, for any .

So there is a natural question of how well spectral methods do, for example, on planar graphs.  Spielman and Teng showed that for bounded-degree planar graphs, a simple recursive spectral algorithm recovers a partition of the vertex set so that .  In other words, for bounded-degree planar graphs, spectral methods recover the Lipton-Tarjan separator theorem!  This is proved by combining Cheeger’s inequality with their main theorem.

Theorem [Spielman-Teng]: Every n-node planar graph with maximum degree has , where is the second eigenvalue of the combinatorial Laplacian on .

Recall that we introduced the combinatorial Laplacian in Lecture 2.  If is an arbitrary finite graph, in this lecture it will make more sense to think about the Laplacian as an operator on functions given by

If we define the standard inner product , then one can easily check that for any such , we have .  In particular, this implies that is a positive semi-definite operator.  If we denote its eigenvalues by , then it is also easy to check that , with corresponding eigenfunction for every .

Thus by standard variational principles, we have

Let us also define the Cheeger constant .  For an arbitrary subset , let

,

note that this definition varies from the we defined in Lecture 2, because we will be discussing eigenfunctions without boundary conditions.  Now one defines .

Finally, we have the version of Cheeger’s inequality (proved by Alon and Milman in the discrete setting) for graphs without boundary.

Cheeger’s inequality: If is any graph with maximum degree , then

This follows fairly easy from the Dirichlet version of Cheeger’s inequality presented in Lecture 2.  Here’s a sketch:  Let satisfy , and suppose, without loss of generality, that has .  Define for and otherwise.  Then for , so we can plug into the Dirichlet version of Cheeger’s inequality with boundary conditions on .  For the full analysis, see this note which essentially follows this approach.  By examining the proof, note that one can find a subset with by a simple “sweep” algorithm:  Arrange the vertices so that , and output the best of the cuts .

So using the eigenvalue theorem of Spielman and Teng, along with Cheeger’s inequality, we can find a set with .  While this cut has the right Cheeger constant, it is not necessarily balanced (i.e. could be very small).  But one can apply this algorithm recursively, perhaps continually cutting small chunks off of the graph until a balanced cut is collected.  Refer to the Spielman-Teng paper for details.  A great open question is how one might use spectral information about to recover a balanced cut immediately, without the need for recursion.

Conformal mappings and circle packings

Now we focus on proving the bound for any planar graph .  A natural analog is to look at what happens for the Laplace-Beltrami operator for a Riemannian metric on the 2-sphere.  In fact, Hersch considered this problem almost 40 years ago and proved that , for any such Riemannian manifold .  His approach was to first use the uniformization theorem to get a conformal mapping from onto , and then try to pull-back the standard second eigenfunctions on (which are just the three coordinate projections).  Since the Dirichlet energy is conformally invariant in dimension 2, this almost works, except that the pulled-back map might not be orthogonal to the constant function.  To fix this, he has to post-process the initial conformal mapping with an appropriate Möbius transformation.

Unaware of Hersch’s work, Spielman and Teng derived eigenvalue bounds for planar graphs using the discrete analog of this approach:  Circle packings replace conformal mappings, and one still has to show the existence of an appropriate post-processing Möbius transformation.

Circle packings and eigenvalue bounds

First, let’s consider mappings and write

where is the Euclidean norm on   To see that this holds, note that the minimum will always be achieve for some coordinate projection , , or of , by the elementary inequality for any non-negative numbers and .

So how do we get a geometric representation of our graph?  A very natural representation comes from the circle packing theorem of Koebe (rediscovered later by Thurston, following from work of Andreev).  Look here for a historical account, and the relationship with conformal mappings.

Circle-packing theorem: For any planar graph , there exists a set of interior-disjoint circles in such that and are tangent if and only if .

Dan Spielman’s notes contain a nice proof of the theorem using convex programming.

Now, suppose we take such a circle packing, and compose it with a stereographic projection (which takes circles to circles) so that we get a circle-packing on the unit sphere.  Let be the mapping which takes a vertex to the center of its circle on the unit sphere .  If we are somehow lucky, and it happens that , then we could use , along with (1) to get an eigenvalue bound as follows.

stereographic projection

Clearly we have since for every , so it suffices to bound the numerator in (1).  For each vertex , there is a cap on the sphere associated to it.  Let denote the Euclidean length from to the boundary of .  In that case, for any edge , we have

Thus the numerator in (1) is at most .  On the other hand, the area enclosed by is at least , and the caps are disjoint, hence

where is the surface area of .  It follows that the numerator is at most , and therefore .

Of course, this all depended on a statement of the following form, which would guarantee that we can take with .

Given any collection of disjoint caps on the sphere, there exists a circle-preserving map such that in the image, the center of mass of the centers of the caps lies at the origin.

Möbius transformation and moving the center of mass

In fact, we prove the following related, but slightly easier statement, and the interested reader can see Spielman and Teng for the full details.

Transform lemma: Let be points on the unit sphere.  Then there exists a circle-preserving map such that .

This lemma doesn’t manage to prove quite what we need, since the centers of circles do not have to map to the centers of circles in the image, but it captures the essence.

To prove the transform lemma, let’s first introduce some circle-preserving maps.

Let be the stereographic projection with as the pole.

Furthermore, let be the scaling map , with .  Finally, define .  Observe that as , we have for all .  In other words, this map pushes everything on the sphere toward .

Now consider the map given by

where is the open unit ball in .  For , maps and .  Also, as approaches , we have for every .

Define by .  Now, as , we have and , as long as we do not have for any .  But this can be handled by spreading the point mass at each out uniformly in a cap of radius , and taking where is one unit of measure distributed uniformly in an -cap around each .

Supposing we do this, we will have as .  This implies that can be extended to a continuous function , and furthermore its restriction to is the identity.  But now basic topology shows that there must exist an for which .  Finally, as , one can check that is eventually uniformly bounded below 1, which means we can take and conclude the existence of an with .

The basic topological fact we needed is the following.

Lemma: If is a continuous map and is the identity, then there exists an with .

Proof: Let be defined by .  Note that is continuous.  Now, suppose that 0 is not in the image of , and define a map from to itself by .  By assumption, is continuous, however it has no fixed point, contradicting Brouwer’s theorem.

[Some picture credits go to Wikipedia, Oded Schramm, and Dan Spielman's lecture notes.]