Let $P$ be an optimization problem, and let $A$ be an approximation algorithm for $P$. The domination ratio $\domr(A,n)$ is the maximum real $q$ such that the solution $x(I)$ obtained by $A$ for any instance $I$ of $P$ of size $n$ is not worse than at least a fraction $q$ of the feasible solutions of … Read more
In this paper, we describe a methodology for the analysis of greedy randomized adaptive search procedures (GRASP). GRASP is a metaheuristic for combinatorial optimization. It usually consists of a construction procedure based on a greedy randomized algorithm and a local search. Hybrid approaches of GRASP with path-relinking developed for the 3-index assignment problem (AP3) and … Read more
This paper addresses recent advances and application of hybridizations of greedy randomized adaptive search procedures (GRASP) and path-relinking. We present a template for implementing path-relinking as an intensification procedure for GRASP. Enhancements to the procedure, recently described in the literature, are reviewed. The effectiveness of the procedure is illustrated experimentally. CitationAT&T Labs Research Technical Report, … Read more
A nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is said to be {\em $n$-realizable\/} if the set $I_n=\{ 1,2,\cdots,n\}$ can be partitioned into $k$ mutually disjoint subsets $S_1,S_2,\cdots, S_k$ such that $\sum\limits_{x\in S_i}x=m_i$ for each $1\le i\le k$. In this paper, we will prove that a nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k\rangle$ is … Read more
Given a directed graph G(V,A), the p-Median problem consists of determining p nodes (the median nodes) minimizing the total distance from the other nodes of the graph. We present a Branch-and-Cut algorithm yielding provably good solutions for instances up to 3795 nodes (14,402,025 variables). Key ingredients of our approach are: lagrangian relaxation, a simple procedure … Read more
We provide a characterization of the cases when the greedy algorithm may produce the unique worst possible solution for the problem of finding a minimum weight base in a uniform independence system when the weights are taken from a finite range. We apply this theorem to TSP and the minimum bisection problem. The practical message … Read more
We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of $\frac{1}{2}\log m_T+1$, where … Read more
Given a convex rational polytope $\Omega(b):=\{x\in\R^n_+\,|\,Ax=b\}$, we consider the function $b\mapsto f(b)$, which counts the nonnegative integral points of $\Omega(b)$. A closed form expression of its $\Z$-transform $z\mapsto \mathcal{F}(z)$ is easily obtained so that $f(b)$ can be computed as the inverse $\Z$-transform of $\mathcal{F}$. We then provide two variants of an inversion algorithm. As a … Read more
An operation on trivalent graphs leads from the truncated cube to buckminsterfullerene, and C60 is the only fullerene with disjoint pentagons which can be obtained by this method. The construction and the proof emphasize maximal independent sets that contain two fifths of the vertices of trivalent graphs. In the case of C60, these sets define … Read more
In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the max-cut problem studied by Goemans and Williamson. Since the problem is NP-hard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. … Read more