Zero-one IP problems: Polyhedral descriptions & cutting plane procedures

A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie

The Traveling Salesman Problem

This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances

On cardinality constrained cycle and path polytopes

Given a directed graph D = (N, A) and a sequence of positive integers 1 <= c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality c_p for some p, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.Comment: 24 page

The complexity of lifted inequalities for the knapsack problem

AbstractIt is well known that one can obtain facets and valid inequalities for the knapsack polytope by lifting simple inequalities associated with minimal covers. We study the complexity of lifting. We show that recognizing integral lifted facets or valid inequalities can be done in O(n2) time, even if the minimal cover from which they are lifted is not given. We show that the complexities of recognizing nonintegral lifted facets and valid inequalities are similar, respectively, to those of recognizing general (not necessarily lifted) facets and valid inequalities. Finally, we show that recognizing valid inequalities is in co- NPC while recognizing facets is in Dn. The question of whether recognizing facets is complete for Dn is open

Revlex-Initial 0/1-Polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.Comment: Accepted for publication in J. Comb. Theory Ser. A; 24 pages; simplified proof of Theorem 1; corrected and improved version of Theorem 4 (the average degree is now bounded by d+4 instead of d+8); several minor corrections suggested by the referee