On cycles and the stable multi-set polytope

AbstractStable multi-sets are an integer extension of stable sets in graphs. In this paper, we continue our investigations started by Koster and Zymolka [Stable multi-sets, Math. Methods Oper. Res. 56(1) (2002) 45–65]. We present further results on the stable multi-set polytope and discuss their computational impact.The polyhedral investigations focus on the cycle inequalities. We strengthen their facet characterization and show that chords need not weaken the cycle inequality strength in the multi-set case. This also helps to derive a valid right hand side for clique inequalities.The practical importance of the cycle inequalities is evaluated in a computational study. For this, we revisit existing polynomial time separation algorithms. The results show that the performance of state-of-the-art integer programming solvers can be improved by exploiting this general structure

Treewidth: Computational Experiments.

Many NP-complete graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for diverse optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed k, linear time algorithms exist to solve the decision problem ``treewidth at most k, their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.operations research and management science;

The partial constraint satisfaction problem : facets and lifting theorems

In this paper the partial constraint satisfaction problem (PCSP) is introduced and formulated as a {0,1}-programming problem. We define the partial constraint satisfaction polytope as the convex hull of feasible solutions for this programming problem. As examples of the class of problems studied we mention the frequency assignment problem and the maximum satisfiability problem. Lifting theorems are presented and some classes of facet-defining valid inequalities for PCSP are given. Computational results show that these valid inequalities reduce the gap between LP-value and IP-value substantially.mathematical applications;