Linear programming bounds for cliques in Paley graphs

07/12/2019 ∙ by Mark Magsino, et al. ∙ The Ohio State University 0

The Lovász theta number is a semidefinite programming bound on the clique number of (the complement of) a given graph. Given a vertex-transitive graph, every vertex belongs to a maximal clique, and so one can instead apply this semidefinite programming bound to the local graph. In the case of the Paley graph, the local graph is circulant, and so this bound reduces to a linear programming bound, allowing for fast computations. Impressively, the value of this program with Schrijver's nonnegativity constraint rivals the state-of-the-art closed-form bound recently proved by Hanson and Petridis. We conjecture that this linear programming bound improves on the Hanson-Petridis bound infinitely often, and we derive the dual program to facilitate proving this conjecture.

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1 Introduction

The Paley graph is defined for every prime with vertex set , the finite field of elements, and an edge between if and only if , where

is the multiplicative subgroup of quadratic residues modulo . The Paley graphs provide a family of quasi-random graphs (see Chung, Graham and Wilson[chung89]) with several nice properties (see §13.2 in Bollobas[bollobas01]). For instance, the Paley graph is a so-called strongly regular graph in which every vertex has neighbors, every pair of adjacent vertices share common neighbors, and every pair of non-adjacent vertices share common neighbors. The Paley graph of order can be used to construct an optimal packing of lines through the origin of , known as the corresponding Paley equiangular tight frame [strohmer03, renes07, waldron09], and these packings have received some attention in the context of compressed sensing [bandeira13, bandeira17].

For a simple, undirected graph , we say is a clique if every pair of vertices in is adjacent, and we define the clique number of , denoted by , to be the size of the largest clique in . It is a famously difficult open problem to determine the order of magnitude of as . The best known closed-form bounds that are valid for all primes are given by

(1)

The lower bound in (1) is due to Cohen [cohen88]. The same lower bound, with a weaker term, is actually valid for any self-complementary graph. Recall that the Ramsey number is the least integer such that every graph on at least vertices contains either a clique of size  or a set of pairwise non-adjacent vertices. Then for any self-complementary graph on at least vertices, it holds that . Together with the classical upper bound of by Erdős and Szekeres [erdos35], it is straightforward to establish the lower bound in (1). The work of Graham and Ringrose [graham90] on least quadratic non-residues shows that there exists such that for infinitely many primes , and so the lower bound in (1) is not sharp in general.

The upper bound in (1) was proved very recently by Hanson and Petridis [hanson19] using a clever application of Stepanov’s polynomial method. This improved upon the previously best known closed-form upper bounds of , proved by Maistrelli and Penman [maistrelli06], and , proved to hold for a majority of primes by Bachoc, Matolcsi and Ruzsa[bachoc13]. Numerical data for primes by Shearer [shearer86] and Exoo [exooData] suggests that there should be a polylogarithmic upper bound on , but it remains an open problem to determine whether there exists such that infinitely often.

This open problem bears some significance in the field of compressed sensing. In particular, Tao [taoBlog] posed the problem of finding an explicit family of matrices with and for which there exists such that for every , it holds that

for every with at most nonzero entries. Such matrices are known as restricted isometries. Families of restricted isometries are known to exist for every by an application of the probabilistic method, and yet to date, the best known explicit construction [bourgain11, mixon15] takes . It is conjectured [bandeira17] that the Paley equiangular tight frame behaves as a restricted isometry for a larger choice of , but proving this is difficult, as it would imply the existence of such that for all sufficiently large

. As partial progress along these lines, the authors recently established that the singular values of random subensembles of the Paley equiangular tight frame obey a Kesten–McKay law 

[magsino19].

The goal of this paper is to describe a promising approach to find new upper bounds on the clique numbers of Paley graphs. In Section 2, we recall a semidefinite programming approach of Lovász [lovasz79] that yields bounds on the clique numbers of arbitrary graphs. By passing to an appropriate subgraph of , we show that this produces numerical bounds on that usually coincide with the Hanson–Petridis bound and sometimes improve upon it. In Section 3, we show how to compute these bounds by linear programming, extending the range in which we are able to produce computational evidence. In Section 4, we derive the relevant dual program and summarize how one might use weak duality to prove a new bound on by constructing appropriate number-theoretic functions; see Proposition 5. We then conclude with suggestions for future work in this direction.

2 Semidefinite Programming Bounds

For a graph , its complement is the graph on vertices with edges . An isomorphism between graphs and is a bijection such that contains an edge between if and only if contains an edge between , and an automorphism is an isomorphism between and itself. When is isomorphic to , we say that is self-complementary. We say is vertex-transitive if, for every pair of vertices , there exists an automorphism of with .

In the sequel, we label the vertices of every graph on vertices by and similarly index the rows and columns of matrices by with addition considered modulo . The Lovász theta number for a graph on vertices is defined by the semidefinite program

Lovász [lovasz79] proved the following.

Proposition 1.

Let be any graph on vertices.

  • .

  • If is vertex-transitive, then .

Proof.

For (i), suppose is a maximal clique in , consider the indicator function

as a column vector in

indexed by , and put . Then is feasible in the program . Counting the nonzero entries of then gives

The proof of (ii) is more involved; see Theorem 8 in Lovász [lovasz79]. ∎

As a consequence of Proposition 1, every self-complementary vertex-transitive graph on vertices satisfies . This is enough to recover the well-known bound of . Indeed, to show that is self-complementary, fix any nonzero quadratic non-residue and consider the bijection defined by . Then for all , we have if and only if . That is, is an isomorphism between and . To show that is vertex-transitive, let and consider the map defined by . Clearly . To see that is an automorphism of , it suffices to observe that for any two vertices , . Hence, follows from Proposition 1.

We can improve upon this bound by focusing our attention to the neighborhood of in , namely, the set of quadratic residues. Let denote the subgraph of induced by . Since is vertex-transitive, there exists a maximal clique of containing the vertex . In particular, . By Proposition 1(i), we conclude that

(2)

For a graph on vertices, Schrijver [schrijver79] proposed strengthening to

where denotes entrywise nonnegativity. Clearly , and the proof of Proposition 1(i) further establishes establishes . This strengthening leads to the bound

(3)

To compare the bounds (2) and (3) to the Hanson–Petridis bound (1), we set

and we compare these in the range in Figure 1 and Table 1. We observe for most primes in this range, providing an equivalent upper bound on . Interestingly, for 17 values of .

Gvozdenović, Laurent and Vallentin[gvoz09] used semidefinite programming to compute several values of , which in their notation is . For instance, they compute in 4.5 hours on a 3GHz processor with 1GB of RAM. They further introduced the so-called block-diagonal hierarchy of semidefinite programs, which allowed them to compute sharper bounds than somewhat more efficiently in the range . In order to compute numerical values of and efficiently, we leverage the symmetry of to reformulate both and as linear programs in the next section. Using this approach on a 3.4GHz processor with 8GB of RAM, we compute in under 20 seconds.

Figure 1: Comparison of and for the 211 primes with . For 60 such primes, . For 17 such primes, . (left) The blue points appear to concentrate around the red curve . (right) If a point lies below the red curve , then , in which case we plot the point as a circle.

3 Reduction to Linear Programming

Recall that a matrix is circulant if for all . A graph is said to be circulant if there exists a labeling of its vertices such that its adjacency matrix is circulant. We note that for every prime , the graph is circulant. Indeed, select a generator of the mulitplicative subgroup , and order the elements of as . Then is circulant since if and only if . Since the complement of a circulant graph is also circulant, it holds that is circulant as well.

Schrijver [schrijver79] showed that the semidefinite programming formulations of both and

can be reduced to linear programs for certain classes of graphs. In order to state one such linear programming formulation, we take the Fourier transform of

to be the function defined by

Proposition 2.

Let be any circulant graph with vertex set . Then

(4)
Proof.

Let denote the right-hand side of . First, we show that . Take any feasible in the program and consider the circulant matrix defined by . Then follows from , the edge constraints on follow from the edge constraints on

, and since the eigenvalues of

are the Fourier coefficients , we see that follows from . Furthermore, . Since every feasible point in can be mapped to a feasible point in with the same value, we conclude that .

For the other direction, fix any that is feasible in , and for each , consider the matrix defined by . Then is also feasible in with the same value:

Averaging over this orbit produces a circulant matrix that, by convexity, is also feasible in , and that, by linearity, has the same value. Take defined by to obtain a feasible point in with the same value. This implies the reverse inequality . ∎

Arguing similarly establishes the following.

Proposition 3.

Let be any circulant graph with vertex set . Then

(5)

We used the linear program formulations in Propositions 3 and 2 to compute the values of and reported in Figure 1 and Table 1.

4 Dual Certificates

In this section, we derive the dual program of for arbitrary circulant graphs . Since every feasible point of the dual program of gives an upper bound on , this section might allow one to prove a new closed-form upper bound on . Recall that for a closed convex cone , its dual cone is given by

Given closed convex cones , the primal program

(6)

has the corresponding dual program

We will use the above formulation to derive a relatively clean expression for the dual program of (5).

Proposition 4.

Let be any circulant graph with vertex set . Then

Proof.

By strong duality, it suffices to show that the right-hand side is the dual program of (5). To this end, we first write the linear program (5) in the form (6). We identify functions with column vectors in indexed by . Let denote the projection operator defined by

Then and for every if and only if . Next, let denote the reversal operator defined by , and let denote the cosine transform defined by

Then satisfies if and only if and . Overall, (5) is equivalent to

where denotes the set of nonnegative real numbers. Since , the dual program is given by following, written in terms of dual variables :

(7)

Since is a circulant graph, we see that precisely when , and so . Also, . We apply these facts to observe that is feasible in (7) if and only if is feasible in (7), and with the same value. Indeed, maps to itself, and

By averaging these two feasible points, we obtain the following equivalent program:

(8)

At this point, we may relax the constraint since is even. Also, satisfies if and only if . Changing variables to and then gives the result. ∎

As such, given a circulant graph , any that is feasible in the corresponding linear program in Proposition 4 yields an upper bound on . Recalling (3), we now specialize to the case of Paley graphs:

Proposition 5.

Given a prime , let denote a generator of the multiplicative group of quadratic residues modulo , and set . Suppose that together satisfy

  • for every with ,

  • , and

  • .

Then .

Arguing similarly to Proposition 4 gives a comparable dual program for . In fact, the resulting program corresponds to adding the constraint to the program in Proposition 4. Considering the numerical data in Table 1, we expect these bounds to match frequently.

5 Future Work

In this paper, we used linear programming to find numerical upper bounds on that usually match and sometimes improve on the Hanson–Petridis bound . Our experiments suggest the following.

Conjecture 6.

For infinitely many primes , it holds that .

With appropriate number-theoretic functions, one might use Proposition 5 to prove creftype 6. This pursuit of “magic functions” bears some resemblance to recent progress in sphere packing; see Cohn [cohn17] for a survey. We note that Gvozdenović, Laurent and Vallentin[gvoz09] introduced a semidefinite programming hierarchy that gives numerical bounds on that are sharper than in the range . However, these semidefinite programs are still rather slow. For our linear programming computations, we used GLPK within SageMath [sage]. We believe that our code could be sped up significantly by incorporating the fast Fourier transform[vanderbei12], possibly giving new bounds on for significantly larger primes .

Acknowledgements.

MM and DGM were partially supported by AFOSR FA9550-18-1-0107. DGM was also supported by NSF DMS 1829955 and the Simons Institute of the Theory of Computing.

References