Center for Discrete Mathematics and Theoretical Computer Science

About: Center for Discrete Mathematics and Theoretical Computer Science is a facility organization based out in Piscataway, New Jersey, United States. It is known for research contribution in the topics: Local search (optimization) & Optimization problem. The organization has 140 authors who have published 175 publications receiving 2345 citations.

An Effective Hybrid Memetic Algorithm for the Minimum Weight Dominating Set Problem

Abstract: The minimum weight-dominating set (MWDS) problem is NP-hard and has a lot of applications in the real world. Several metaheuristic methods have been developed for solving the problem effectively, but suffering from high CPU time on large-scale instances. In this paper, we design an effective hybrid memetic algorithm (HMA) for the MWDS problem. First, the MWDS problem is formulated as a constrained 0–1 programming problem and is converted to an equivalent unconstrained 0–1 problem using an adaptive penalty function. Then, we develop a memetic algorithm for the resulting problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, a crossover operator, a population-updating method, and a path-relinking procedure. These strategies make a good tradeoff between intensification and diversification. A number of experiments were carried out on three types of instances from the literature. Compared with existing algorithms, HMA is able to find high-quality solutions in much less CPU time. Specifically, HMA is at least six times faster than existing algorithms on the tested instances. With increasing instance size, the CPU time required by HMA increases much more slowly than required by existing algorithms.

Beck's Three Permutations Conjecture: A Counterexample and Some Consequences

Abstract: Given three permutations on the integers 1 through $n$, consider the set system consisting of each interval in each of the three permutations. In 1982, Beck conjectured that the discrepancy of this set system is $O(1)$. In other words, the conjecture says that each integer from 1 through $n$ can be colored either red or blue so that the number of red and blue integers in each interval of each permutations differs only by a constant. (The discrepancy of a set system based on two permutations is at most two.) Our main result is a counterexample to this conjecture: for any positive integer $n = 3^k$, we construct three permutations whose corresponding set system has discrepancy $\Omega(\log{n})$. Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This construction also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to $\ell$ permutations has discrepancy $O(\sqrt{\ell})$. Our work was inspired by an intriguing paper from SODA 2011 by Eisenbrand, P{\'a}lv{\"o}lgyi and Rothvo\ss, who show a surprising connection between the discrepancy of three permutations and the bin packing problem: They show that Beck's conjecture implies a constant worst-case bound on the additive integrality gap for the Gilmore-Gomory LP relaxation for bin packing in the special case when all items have sizes strictly between $1/4$ and $1/2$, also known as the three partition problem. Our counterexample shows that this approach to bounding the additive integrality gap for bin packing will not work. We can, however, prove an interesting implication of our construction in the reverse direction: there are instances of bin packing and corresponding optimal basic feasible solutions for the Gilmore-Gomory LP relaxation such that any packing that contains only patterns from the support of these solutions requires at least ${\rm opt} + \Omega(\log{m})$ bins, where $m$ is the number of items. Finally, we discuss some implications that our construction has for other areas of discrepancy theory.