Lovasz Theta Number Research Articles

The theta number of simplicial complexes

We introduce a generalization of the celebrated Lovasz theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman’s ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended to a hierarchy of semidefinite programming upper bounds reaching the independence number. We analyze the value of the theta number and of the hierarchy for dense random simplicial complexes.

The Density of Sets Avoiding Distance 1 in Euclidean Space

We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analog of the Lovasz theta number and of a combinatorial argument involving finite subgraphs of the unit distance graph. In turn, we straightforwardly obtain an asymptotic improvement for the measurable chromatic number of Euclidean space. We also tighten previous results for dimensions between 4 and 24.

Ellipsoidal Relaxations of the Stable Set Problem: Theory and Algorithms

A new exact approach to the stable set problem is presented, which attempts to avoid the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimizing over it is equal to the Lovasz theta number. This ellipsoid can then be used to construct useful convex relaxations of the stable set problem, which can be embedded within a branch-and-bound framework. Extensive computational results are given, which indicate the potential of the approach.

On the theta number of powers of cycle graphs

We give a closed formula for Lovasz's theta number of the powers of cycle graphs C k d?1 and of their complements, the circular complete graphs K k/d . As a consequence, we establish that the circular chromatic number of a circular perfect graph is computable in polynomial time. We also derive an asymptotic estimate for the theta number of C k d .

Approximating the Lovász θ Function with the Subgradient Method

Abstract The famous Lovasz theta number θ ( G ) is expressed as the optimal solution of a semidefinite program. As such, it can be computed in polynomial time to an arbitrary precision. Nevertheless, computing it in practice yields some difficulties as the size of the graph gets larger and larger, despite recent significant advances of semidefinite programming (SDP) solvers. We present a way around SDP which exploits a well-known equivalence between SDP and lagrangian relaxations of non-convex quadratic programs. This allows us to design a subgradient algorithm which is shown to be competitive with SDP algorithms in terms of efficiency, while being preferable as far as memory requirements, flexibility and stability are concerned.

New heuristics for the vertex coloring problem based on semidefinite programming

The Lovasz theta number Lovasz (IEEE Trans Inf Theory 25:1–7, 1979) is a well-known lower bound on the chromatic number of a graph \(G\), and \(\varPsi _K(G)\) is its impressive strengthening Gvozdenovic and Laurent (SIAM J Optim 19(2):592–615, 2008). The bound \(\varPsi _K(G)\) was introduced in very specific and abstract setting which is tough to translate into usual mathematical programming framework. In the first part of this paper we unify the motivations and approaches to both bounds and rewrite them in a very similar settings which are easy to understand and straightforward to implement. In the second part of the paper we provide explanations how to solve efficiently the resulting semidefinite programs and how to use optimal solutions to get good coloring heuristics. We propose two vertex coloring heuristics based on \(\varPsi _K(G)\) and present numerical results on medium sized graphs.

Copositive programming motivated bounds on the stability and the chromatic numbers

The Lovasz theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovasz theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovasz theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs.

Computing Semidefinite Programming Lower Bounds for the (Fractional) Chromatic Number Via Block-Diagonalization

Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572-591] hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds: the “$\psi$”-hierarchy converging to the fractional chromatic number and the “$\Psi$”-hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lovasz theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed in [SIAM J. Optim., 19 (2008), pp. 572-591]. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs, and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits us to compute the bounds by using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit block-diagonalization of the Terwilliger algebra given by Schrijver [IEEE Trans. Inform. Theory, 51 (2005), pp. 2859-2866]. Our numerical results indicate that the new bounds can be much stronger than the Lovasz theta number. For some of the DIMACS instances we improve the best known lower bounds significantly.

The Operator $\Psi$ for the Chromatic Number of a Graph

We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi(\overline G)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\overline G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if $\beta(G)$ is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number $\chi^*(\cdot)$ and $\chi(\cdot)$ unless $\text{P} = \text{NP}$. Moreover, based on the Motzkin-Straus formulation for $\alpha(G)$, we give (quadratically constrained) quadratic and copositive programming formulations for $\chi(G)$. Under some mild assumptions, $n/\beta(G)\le \Psi_\beta(G)$, but, while $n/\beta(G)$ remains below $\chi^*(G)$, $\Psi_\beta(G)$ can reach $\chi(G)$ (e.g., for $\beta(\cdot)=\alpha(\cdot)$). We also define new polynomial time computable lower bounds for $\chi(G)$, improving the classic Lovasz theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. Gvozdenovic and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592-615].

Semidefinite programming relaxations for graph coloring and maximal clique problems

The semidefinite programming formulation of the Lovasz theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number ź(G) or the clique number ź(G) of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either ź(G) or ź(G) by adding several types of cutting planes. We explore several such strengthenings, and show that some of them can be computed with the same effort as the theta number. We also investigate computational simplifications for graphs with rich automorphism groups.

A Convex Quadratic Characterization of the Lovász Theta Number

In previous works an upper bound on the stability number $\alpha(G)$ of a graph G based on convex quadratic programming was introduced and several of its properties were established. The aim for this investigation is to relate theoretically this bound (usually represented by $\upsilon(G)$) with the well-known Lovasz $\vartheta(G)$ number. First, a new set of convex quadratic bounds on $\alpha(G)$ that generalize and improve the bound $\upsilon(G)$ is proposed. Then it is proved that $\vartheta(G)$ is never worse than any bound belonging to this set of new bounds. The main result of this note states that one of these new bounds equals $\vartheta(G)$, a fact that leads to a new characterization of the Lovasz theta number.

Maximum stable set formulations and heuristics based on continuous optimization

The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovasz theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovasz semidefinite program. In this paper, we show that restricting the matrix variable in the Lovasz semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics.