Convex Hull Of Integer Points Research Articles

Enumerating Integer Points in Polytopes with Bounded Subdeterminants

We show that one can enumerate the vertices of the convex hull of integer points in polytopes whose constraint matrices have bounded and nonzero subdeterminants, in time polynomial in the dimension and encoding size of the polytope. This extends a previous result by Artmann et al. who showed that integer linear optimization in such polytopes can be done in polynomial time.

Stage- and scenario-wise Fenchel decomposition for stochastic mixed 0-1 programs with special structure

Solving stochastic integer programs (SIPs) is generally difficult. This paper considers a comparative study of stage- and scenario-wise Fenchel decomposition (FD) for two-stage SIPs with special structure. The standard FD approach is based on stage-wise or Benders’ decomposition. This work derives a scenario FD method based on decomposing the SIP problem by scenario and performs a computational study of the two approaches. In particular, two algorithms are studied, stage-wise FD (ST-FD) and scenario-wise FD (SC-FD) algorithms. The algorithms use FD cuts generated based on the scenario subproblem under each decomposition setting to iteratively recover (partially) the convex hull of integer points in the neighborhood of the optimal solution. The L-shaped method is used to solve the LP relaxation of the SIP problem in the ST-FD algorithm, while the progressive hedging algorithm (PHA) is used in the SC-FD algorithm. Computational results on knapsack test instances demonstrate the viability of both approaches towards solving large instances in reasonable amount of time and outperforming a direct solver in most cases. Overall, the ST-FD algorithm provides the best performance in our experiments.

Sherali–Adams relaxations of graph isomorphism polytopes

We investigate the Sherali–Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali–Adams relaxations characterize a new vertex classification algorithm for graph isomorphism, which we call the generalized vertex classification algorithm. This algorithm generalizes the classic vertex classification algorithm and generalizes the work of Tinhofer on polyhedral methods for graph automorphism testing. We establish that the Sherali–Adams lift & project hierarchy when applied to a graph isomorphism polytope of a graph with n vertices needs Ω(n) iterations in the worst case before converging to the convex hull of integer points. We also show that this generalized vertex classification algorithm is also strongly related to the well-known Weisfeiler–Lehman algorithm, which we show can also be characterized in terms of the Sherali–Adams relaxations of a semi-algebraic set whose integer points encode graph isomorphisms.

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.

On the k-dominating set polytope of web graphs

Abstract In this work we present some results on the polyhedral structure of the convex hull of integer points in polyhedra of the form { x ⩾ 0 : M x ⩾ k 1 } , for a 0, 1 matrix M and a positive integer number k. In particular, we consider the k-dominating set problem in a graph. Given a graph G = ( V , E ) , a set D ⊆ V is a k-dominating set if every vertex in V is adjacent to at least k vertices of D. The k-dominating set problem consists in finding a k-dominating set of minimum cardinality. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G and it is a natural generalization of the well-known dominating set polytope of a graph. We apply our results for general problems to the k-dominating set polytope of some particular families of web graphs.

Upper and lower bound results on the convex hull of integer points in polyhedra

In typical linear programming problems, we are concerned with finding non-negative integers {x1,…, xn} that maximize a linear form c1x1 + … + cnxn, subject to a number of linear inequalities, for The maximum is necessarily attained at one of the vertices of the convex hull of integer points defined by the inequalities, so we have an interest in estimating the number M of these vertices. We give two results; one improving an upper bound result for M of Hayes and Larman concerning the Knapsack polytope, the other an example showing that, in 3-dimensions, it is possible to choose the coefficients aij to obtain a lower bound for M.

Arbitrarily Complex Corner Polyhedra are Dense in $R^n $

A cone determined by n independent linear inequalities in n-space is specified, for example by the constraint set of a mathematical programming problem. We show that under certain weak assumptions, the closed convex hull of the integer points in the cone has infinitely many extreme points. This result is used to prove the existence of small perturbations of the coefficients of any given cone, producing inequalities with integer coefficients, for which the orresponding closed convex hull of integer points has an arbitrarily large number of extreme points. This property may explain the volatility of convergence times of cutting plane algorithms.