The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) … Read more
The quadratic traveling salesman problem asks for a tour of minimal costs where the costs are associated with each two arcs that are traversed in succession. This structure arises, e. g., if the succession of two arcs represents the costs of loading processes in transport networks or a switch between different technologies in communication networks. … Read more
An instance of the quadratic assignment problem (QAP) with cost matrix Q is said to be linearizable if there exists an instance of the linear assignment problem (LAP) with cost matrix C such that for each assignment, the QAP and LAP objective function values are identical. The QAP linearization problem can be solved in O(n4) … Read more
We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of … Read more
We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of … Read more
A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. GRASP heuristics are multistart procedures which apply local search to a set of starting solutions generated with a randomized greedy algorithm or semi-greedy method. The best local optimum found over the iterations is returned as the heuristic solution. Path-relinking is a search … Read more
We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b \in (0,1/2], and a parameter \gamma, either finds an \Omega(b)-balanced cut of conductance O(\sqrt{\gamma}) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least \gamma, and runs in … Read more
Tressler’s Theorem states that the long-standing Hadamard conjecture (concerning the existence of n by n orthogonal matrices with elements of the same absolute value, for n=4k, k=1,2,…) will be settled if we find n-2 pairwise orthogonal words in a hyperplane of words. In this paper we will prove the counterpart of Tressler’s Theorem: the existence … Read more
We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with diff erent basis vector sets. We discuss how several well-known TSP lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between … Read more
The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations … Read more