Approximate Graph Coloring by Semidefinite Programming

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page

Graphs with tiny vector chromatic numbers and huge chromatic numbers

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^(1 - 2/k) colors. Here Δ is the maximum degree in the graph and is assumed to be of the order of n^5 for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^(1- 2/k) (and hence cannot be colored with significantly fewer than Δ^(1-2/k) colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem

The Lovasz number of random graphs

We study the Lovasz number theta along with two further SDP relaxations theta1, theta1/2 of the independence number and the corresponding relaxations of the chromatic number on random graphs G(n,p). We prove that these relaxations are concentrated about their means Moreover, extending a result of Juhasz, we compute the asymptotic value of the relaxations for essentially the entire range of edge probabilities p. As an application, we give an improved algorithm for approximating the independence number in polynomial expected time, thereby extending a result of Krivelevich and Vu. We also improve on the analysis of an algorithm of Krivelevich for deciding whether G(n,p) is k-colorable

Systematic construction of quantum contextuality scenarios with rank advantage

A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.Comment: 8+8 page

Hidden Variables for Pauli Measurements

The Pauli measurements (the measurements that can be performed with Clifford operators followed by measurement in the computational basis) are a fundamental object in quantum information. It is well-known that there is no assignment of outcomes to all Pauli measurements that is both complete and consistent. We define two classes of hidden variable assignments based on relaxing either condition. Partial hidden variable assignments retain the consistency condition, but forfeit completeness. Contextual hidden variable assignments retain completeness but forfeit consistency. We use techniques from spectral graph theory to characterize the incompleteness and inconsistency of the respective hidden variable assignments. As an application, we interpret our incompleteness result as a statement of contextuality and our inconsistency result as a statement of nonlocality. Our results show that we can obtain large amounts of contextuality and nonlocality using Clifford gates and measurements

Quantum Computation, Markov Chains and Combinatorial Optimisation

This thesis addresses two questions related to the title, Quantum Computation, Markov Chains and Combinatorial Optimisation. The first question involves an algorithmic primitive of quantum computation, quantum walks on graphs, and its relation to Markov Chains. Quantum walks have been shown in certain cases to mix faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. We design a lifted Markov chain that in some sense simulates any quantum walk. Concretely, we construct a lifted Markov chain on a connected graph G with n vertices that mixes exactly to the average mixing distribution of a quantum walk on G. Moreover, the mixing time of this chain is the diameter of G. We then consider practical consequences of this result. In the second part of this thesis we address a classic unsolved problem in combinatorial optimisation, graph isomorphism. A theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product of the two graphs. Furthermore, a straightforward corollary of this theorem and Lovász’s sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in n. We enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. Interesting cases include complete multipartite graphs and disjoint unions of cliques. We find that all perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices. Open questions and further research directions are discussed

Geometric Ramifications of the Lovász Theta Function and Their Interplay with Duality

The Lovasz theta function and the associated convex sets known as theta bodies are fundamental objects in combinatorial and semidefinite optimization. They are accompanied by a rich duality theory and deep connections to the geometric concept of orthonormal representations of graphs. In this thesis, we investigate several ramifications of the theory underlying these objects, including those arising from the illuminating viewpoint of duality. We study some optimization problems over unit-distance representations of graphs, which are intimately related to the Lovasz theta function and orthonormal representations. We also strengthen some known results about dual descriptions of theta bodies and their variants. Our main goal throughout the thesis is to lay some of the foundations for using semidefinite optimization and convex analysis in a way analogous to how polyhedral combinatorics has been using linear optimization to prove min-max theorems. A unit-distance representation of a graph $G$ maps its nodes to some Euclidean space so that adjacent nodes are sent to pairs of points at distance one. The hypersphere number of $G$, denoted by $t(G)$, is the (square of the) minimum radius of a hypersphere that contains a unit-distance representation of $G$. Lovasz proved a min-max relation describing $t(G)$ as a function of $\vartheta(\overline{G})$, the theta number of the complement of $G$. This relation provides a dictionary between unit-distance representations in hyperspheres and orthonormal representations, which we exploit in a number of ways: we develop a weighted generalization of $t(G)$, parallel to the weighted version of $\vartheta$; we prove that $t(G)$ is equal to the (square of the) minimum radius of an Euclidean ball that contains a unit-distance representation of $G$; we abstract some properties of $\vartheta$ that yield the famous Sandwich Theorem and use them to define another weighted generalization of $t(G)$, called ellipsoidal number of $G$, where the unit-distance representation of $G$ is required to be in an ellipsoid of a given shape with minimum volume. We determine an analytic formula for the ellipsoidal number of the complete graph on $n$ nodes whenever there exists a Hadamard matrix of order $n$. We then study several duality aspects of the description of the theta body $\operatorname{TH}(G)$. For a graph $G$, the convex corner $\operatorname{TH}(G)$ is known to be the projection of a certain convex set, denoted by $\widehat{\operatorname{TH}}(G)$, which lies in a much higher-dimensional matrix space. We prove that the vertices of $\widehat{\operatorname{TH}}(G)$ are precisely the symmetric tensors of incidence vectors of stable sets in $G$, thus broadly generalizing previous results about vertices of the elliptope due to Laurent and Poljak from 1995. Along the way, we also identify all the vertices of several variants of $\widehat{\operatorname{TH}}(G)$ and of the elliptope. Next we introduce an axiomatic framework for studying generalized theta bodies, based on the concept of diagonally scaling invariant cones, which allows us to prove in a unified way several characterizations of $\vartheta$ and the variants $\vartheta'$ and $\vartheta^+$, introduced independently by Schrijver, and by McEliece, Rodemich, and Rumsey in the late 1970's, and by Szegedy in 1994. The beautiful duality equation which states that the antiblocker of $\operatorname{TH}(G)$ is $\operatorname{TH}(\overline{G})$ is extended to this setting. The framework allows us to treat the stable set polytope and its classical polyhedral relaxations as generalized theta bodies, using the completely positive cone and its dual, and it allows us to derive a (weighted generalization of a) copositive formulation for the fractional chromatic number due to Dukanovic and Rendl in 2010 from a completely positive formulation for the stability number due to de Klerk and Pasechnik in 2002. Finally, we study a non-convex constraint for semidefinite programs (SDPs) that may be regarded as analogous to the usual integrality constraint for linear programs. When applied to certain classical SDPs, it specializes to the standard rank-one constraint. More importantly, the non-convex constraint also applies to the dual SDP, and for a certain SDP formulation of $\vartheta$, the modified dual yields precisely the clique covering number. This opens the way to study some exactness properties of SDP relaxations for combinatorial optimization problems akin to the corresponding classical notions from polyhedral combinatorics, as well as approximation algorithms based on SDP relaxations