Svatopluk Poljak

Checking robust nonsingularity is NP-hard

We consider the following problem: givenk+1 square matrices with rational entries,A0,A1,...,Ak, decide ifA0+r1A1+···+rkAk is nonsingular for all possible choices of real numbersr1, ...,rk in the interval [0, 1]. We show that this question, which is closely related to the robust stability problem, is NP-hard. The proof relies on the new concept ofradius of nonsingularity of a square matrix and on the relationship between computing this radius and a graph-theoretic problem.

A recipe for semidefinite relaxation for (0,1)-quadratic programming: In memory of Svata Poljak

We review various relaxations of (0,1)-quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partition Problem and the Max-Clique Problem. Finally we show our relaxation to be best possible among all quadratic majorants with zero trace.

Laplacian eigenvalues and the maximum cut problem

We introduce and study an eigenvalue upper boundϕ(G) on the maximum cut mc (G) of a weighted graph. The functionϕ(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented.We show thatϕ is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove thatϕ(G) is never worse that 1.131 mc(G) for a planar, or more generally, a weakly bipartite graph with nonnegative edge weights. We give a dual characterization ofϕ(G), and show thatϕ(G) is computable in polynomial time with an arbitrary precision.

Eigenvalues in Combinatorial Optimization

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

Nonpolyhedral Relaxations of Graph-Bisection Problems

We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node- and edge- incidence vectors of cu...

On a positive semidefinite relaxation of the cut polytope

Abstract We study the convex set L n defined by L n Z≔ {X|X = (xij) a positive semidefinite n × n matrix, xii = 1 for all i}. We describe several geometric properties of L n. In particular, we show that L n has 2n − 1 vertices, which are its rank one matrices, corresponding to all bipartitions of the set {1, 2, …, n}. Our main motivation for investigating the convex set L n comes from combinatorial optimization, namely from approximating the max-cut problem. An important property of L n is that, due to the positive semidefinite constraints, one can optimize over it in polynomial time. On the other hand, L n still inherits the difficult structure of the underlying combinatorial problem. In particular, it is NP-hard to decide whether the optimum of the problem min Tr(CX), X ∈ Ln is reached at a vertex. This result follows from the complete characterization of the matrices C of the form C = bbt for some vector b, for which the optimum of the above program is reached at a vertex.

On the maximum number of qualitatively independent partitions

Two partitions P = (P1, …, Pt) and P = (P′1, …, P′1) of a given underlying set are called qualitatively independent if Pi ∩ P′j ≠ O for all i, j (1 ⩽ i, j ⩽ t). It is shown there exist at most 12([nt][2nt]) pairwise qualitatively independent t-partitions on n elements; and constructions of size O(t(n log t)/t2) are given. Also, collections of r-wise independent partitions are considered.

Combinatorial properties and complexity of a max-cut approximation

Abstract We study various properties of an eigenvalue upper bound on the max-cut problem. We show that the bound behaves in a manner similar to the max-cut for the operations of switching, vertex splitting, contraction and decomposition. It can also be adjusted for branch and bound techniques. We introduce a Gram representation of a weighted graph, in order to construct weighted graphs with pre-given eigenvalue properties. As a corollary, we prove that the decision problem as to whether the upper bound coincides with the actual value of the max-cut is NP-complete. We study the mutual relation between the max-cut and the bound on the line graphs, which allow a good approximation. We show that the ratio between the upper bound and the actual size of the max-cut is close to 9/8 for the studied classes, and for several other graphs.

Convex relaxations of (0, 1)-quadratric programming

We consider three parametric relaxations of the 0, l-quadratic programming problem. These relaxations are to: quadratic maximization over simple box constraints, quadratic maximization over the sphere, and the maximum eigenvalue of a bordered matrix. When minimized over the parameter, each of the relaxations provides an upper bound on the original discrete problem. Moreover, these bounds are efficiently computable. Our main result is that, surprisingly, all three bounds are equal.

Solving the max-cut problem using eigenvalues

Abstract We present computational experiments for solving the max-cut problem using an eigenvalue relaxation. Our motivation is twofold — we are interested both in the quality of the bound, and in developing an efficient code to compute it. We describe the theoretical background of the method, an implementation of the algorithm, and its practical performance. The experiments have been done for various data sets, including random graphs of different densities, clustering problems, problems arising from quadratic 0–1 optimization, and some graphs taken from the literature. The basic algorithm is used to compute an upper and lower bound on the max-cut. The relative gap between these bounds is typically much less than 10%. We also present results where the basic algorithm is used in a “branch and bound” setting to find the exact value of the max-cut. The largest problems solved to optimality are dense geometric graphs with up to 100 nodes.