A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous … Read more
We propose in this work 50 new test instances for the Steiner problem in graphs. These instances are characterized by large integrality gaps and symmetry aspects which make them harder to both exact methods and heuristics than the test problems currently in use for the evaluation and comparison of existing and newly developed algorithms. Our … Read more
GRASP is a multi-start metaheuristic for combinatorial problems, in which each iteration consists basically of two phases: construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum is found during the local search phase. The best overall solution is kept as the result. In this chapter, … Read more
This paper provides a simple approach for solving a semidefinite program, SDP\@. As is common with many other approaches, we apply a primal-dual method that uses the perturbed optimality equations for SDP, $F_\mu(X,y,Z)=0$, where $X,Z$ are $n \times n$ symmetric matrices and $y \in \Re^n$. However, we look at this as an overdetermined system of … Read more
We introduce an anti-matroid as a family $\cal F$ of subsets of a ground set $E$ for which there exists an assignment of weights to the elements of $E$ such that the greedy algorithm to compute a maximal set (with respect to inclusion) in $\cal F$ of minimum weight finds, instead, the unique maximal set … Read more
The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most $k$ edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits. ArticleDownload View PDF
We present new fixed parameter algorithms for feedback vertex set and disjoint cycles on planar graphs. We give an $O(c^{\sqrt{k}} + n)$ algorithm for $k$-feedback vertex set and an $O(c^{\sqrt{k}} n)$ algorithm for $k$-disjoint cycles on planar graphs. CitationManuscript 4 July, 2001.ArticleDownload View PDF
Sherali and Adams \cite{SA90}, Lov\’asz and Schrijver \cite{LS91} and, recently, Lasserre \cite{Las01b} have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a $0-1$ polytope $P\subseteq \oR^n$ converging to $P$ in $n$ steps. Lasserre’s approach uses results about representations of positive polynomials as sums of squares and the dual … Read more
A frame relay service offers virtual private networks to customers by provisioning a set of long-term private virtual circuits (PVCs) between customer endpoints on a large backbone network. During the provisioning of a PVC, routing decisions are made without any knowledge of future requests. Over time, these decisions can cause inefficiencies in the network and … Read more
Semidefinite programming (SDP) relaxations are proving to be a powerful tool for finding tight bounds for hard discrete optimization problems. This is especially true for one of the easier NP-hard problems, the Max-Cut problem (MC). The well-known SDP relaxation for Max-Cut, here denoted SDP1, can be derived by a first lifting into matrix space and … Read more