Semidefinite Programming in Combinatorial and Polynomial Optimization

In the recent years semidefinite programming has become a widely used tool for designing better efficient algorithms for approximating hard combinatorial optimization problems and, more generally, polynomial optimization problems, which deal with optimizing a polynomial objective function over a basic closed semialgebraic set. The underlying paradigm is that, while testing nonnegativity of a polynomial is a hard problem, one can test efficiently whether it can be written as a sum of squares of polynomials, using semidefinite programming. In this note we sketch some of the main mathematical tools that underlie this approach and illustrate its application to some graph problems dealing with maximum cuts, stable sets and graph coloring

On the sparsity order of a graph and its deficiency in chordality

Given a graph $G$ on $n$ nodes, let {cal P_G denote the cone consisting of the positive semidefinite $ntimes n$ matrices (with real or complex entries) having a zero entry at every position corresponding to a non edge of $G$. Then, the order of $G$ is defined as the maximum rank of a matrix lying on an extreme ray of the cone {cal P_G. It is shown in [AHMR88] that the graphs of order 1 are precisely the chordal graphs and a characterization of the graphs having order $2$ is conjectured there in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with order 2 in the complex case and we give a decomposition result for the graphs having order $le 2$ in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also establish an inequality relating the order {rm ord_{oF(G) of a graph $G$ ($oF=oR$ or $oC$) and the parameter {rm fill(G) defined as the minimum number of edges needed to be added to $G$ in order to obtain a chordal graph. Namely, we show that {rm ord_{oF(G)le 1 +epsilon_oF cdot {rm fill(G) where $epsilon_oR=1$ and $epsilon _oC =2$; this settles a conjecture posed in [HPR89]

Renal Handling of Urate and Gout

We study the facial structure of the set {cal E_{ntimes n of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face of {cal E_{ntimes n. It turns out that the spectrum of face dimensions is lacunary and that {cal E_{ntimes n has polyhedral faces of dimension up to approx sqrt {2n. As an application, we describe in detail the faces of {cal E_{4times 4. We also discuss results related to optimization over {cal E_{ntimes n

A Lex-BFS-based recognition algorithm for Robinsonian matrices

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be re- ordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a non- negative symmetric n × n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n + m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A

Handelman 's hierarchy for the maximum stable set problem.

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman's hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)

A characterization of box 1/d1/d-integral binary clutters

Let Q6 denote the port of the dual Fano matroid F*7 and let Q7 denote the clutter consisting of the circuits of the Fano matroid F7 that contain a given element. Let be a binary clutter on E and let d = 2 be an integer. We prove that all the vertices of the polytope {x E+ | x(C) = 1 for C } n {x | a = x = b} are -integral, for any -integral a, b, if and only if does not have Q6 or Q7 as a minor. This includes the class of ports of regular matroids. Applications to graphs are presented, extending a result from Laurent and Pojiak [7]

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs