On Smooth Lattice Polytopes with Small Degree

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this article, we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco, and Piene. We follow their approach of interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then apply techniques from Adjunction Theory and Mori Theory.     

Tight bounds for periodicity theorems on the unbounded Knapsack problem

Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.     

Facets of the independent set polytope

Theoretical results pertaining to the independent set polytope P^ISP=conv{x{0,1}ⁿ:Axb} are presented. A conflict hypergraph is constructed based on the set of dependent sets which facilitates the examination of the facial structure of P^ISP. Necessary and sufficient conditions are provided for every nontrivial 0-1 facet-defining inequalities of P^ISP in terms of hypercliques. The relationship of hypercliques and some classes of knapsack facet-defining inequalities are briefly discussed. The notion of lifting is extended to the conflict hypergraph setting to obtain strong valid inequalities, and back-lifting is introduced to strengthen cut coefficients. Preliminary computational results are presented to illustrate the usefulness of the theoretical findings.Mathematics Subject Classification (2000): 90C11, 90C57, 90C35     

Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (2000): 90C11, 90C27     

Facets of the Graph Coloring Polytope

In this paper we present a study of the polytope associated to a classic linear integer programming formulation of the graph coloring problem. We determine some families of facets. This is the initial step for the development of a branch-and-cut algorithm to solve large instances of the graph coloring problem.     

Stability Properties of Neighbourly Random Polytopes

We introduce a quantitative parameter measuring m-neighbourliness of symmetric convex polytopes in ℝ ^k . We discuss this parameter for random polytopes generated by subgaussian vectors and show its stability properties. Research of P. Mankiewicz was partially supported by KBN Grant no. 1 P03A 015 27. N. Tomczak-Jaegermann holds the Canada Research Chair in Geometric Analysis.     

On the two-dimensional Knapsack Problem

We address the two-dimensional Knapsack Problem (2KP), aimed at packing a maximum-profit subset of rectangles selected from a given set into another rectangle. We consider the natural relaxation of 2KP given by the one-dimensional KP with item weights equal to the rectangle areas, proving the worst-case performance of the associated upper bound, and present and compare computationally four exact algorithms based on the above relaxation, showing their effectiveness.     

Knapsack feasibility as an absolute value equation solvable by successive linear programming

We formulate the NP-hard n-dimensional knapsack feasibility problem as an equivalent absolute value equation (AVE) in an n-dimensional noninteger real variable space and propose a finite succession of linear programs for solving the AVE. Exact solutions are obtained for 1,880 out of 2,000 randomly generated consecutive knapsack feasibility problems with dimensions between 500 and one million. For the 120 approximately solved problems the error consists of exactly one noninteger component with value in (0, 1), which when replaced by 0, results in a relative error of less than 0.04%. We also give a necessary and sufficient condition for the solvability of the knapsack feasibility problem in terms of minimizing a concave quadratic function on a polyhedral set. Average time for solving exactly a million-variable knapsack feasibility problem was less than 14 s on a 4 GB machine.     

Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables, through the lifting of continuous variables fixed at their upper bounds. We introduce the concept of a superlinear inequality and show that, in this case, lifting is significantly simpler than for general inequalities. We use the superlinearity theory, together with the traditional lifting of 0-1 variables, to describe families of facets of the mixed 0-1 knapsack polytope. Finally, we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (1991): 90C11, 90C27     

On the set covering polytope: I. All the facets with coefficients in {0, 1, 2}

While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. We characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1 or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. We show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure. We point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation.Research supported by the National Science Foundation through grant ECS-8503198 and the Office of Naval Research through contract N0001485-K-0198.     

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (Ann Probab 4:667–673, 1976) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (Order 15:279–295, 1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F _n?+?1, where F _n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of a flippable pair introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.     

Combinatorial optimization - Polyhedra and efficiency: A book review

We review the recent three-volume monograph authored by Alexander Schrijver, Combinatorial Optimization - Polyhedra and Efficiency, Springer-Verlag, 2003, ISBN 3-540-44389-4, 1881 pages (in a slip-case), price: € 89,95 .Received: November 2003, Revised: January 2004, AMS classification: 90C57, 68R10, 05C99     

Gainfree Leontief substitution flow problems

Leontief substitution systems have been studied by economists and operations researchers for many years. We show how such linear systems are naturally viewed asLeontief substitution flow problems on directed hypergraphs, and that important solution properties follow from structural characteristics of the hypergraphs. We give a strongly polynomial, non-simplex algorithm for Leontief substitution flow problems that satisfy againfree property leading to acyclic extreme solutions. Integrality conditions follow easily from this algorithm. Another structural property,support disjoint reachability, leads to necessary and sufficient conditions for extreme solutions to be binary. In a survey of applications, we show how the Leontief flow paradigm links polyhedral combinatorics, expert systems, mixed integer model formulation, and some problems in graph optimization.Dedicated to the memory of Robert G. JeroslowSee Acknowledgement section.Research supported in part by the ONR (Office of Naval Research) under URI Grant number N00014-86-K-0689, and Center for Operations Research and Econometrics, Universite Catholique de Louvain.     

Stochastic on-line knapsack problems

Different classes of on-line algorithms are developed and analyzed for the solution of {0, 1} and relaxed stochastic knapsack problems, in which both profit and size coefficients are random variables. In particular, a linear time on-line algorithm is proposed for which the expected difference between the optimum and the approximate solution value isO(log^3/2 n). An(1) lower bound on the expected difference between the optimum and the solution found by any on-line algorithm is also shown to hold.Corresponding author.Partially supported by the Basic Research Action of the European Communities under Contract 3075 (Alcom).Partially supported by research project Models and Algorithms for Optimization of the Italian Ministry of University and Scientific and Technological Research (MURST 40%).     

Facets of two Steiner arborescence polyhedra

The Steiner arborescence (or Steiner directed tree) problem concerns the connection of a set of target vertices of a digraph to a given root vertex. This problem is known to be NP-hard. In the present paper we study the facial structure of two polyhedra associated with the problem. Several classes of valid inequalities are considered, and a new class with arbitrarily large coefficients is introduced. All these inequalities are shown to define distinct facets of both the Steiner polyhedra considered. This is achieved by exploiting two lifting theorems which also allow generalization of the new inequalities. Composition theorems are finally given and used to derive large families of new facet-inducing inequalities with exponentially large coefficients.     

On the facets of the mixed–integer knapsack polyhedron

We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.Supported, in part, by NSF grants DMII–0070127 and DMII–0218265.Mathematics Subject Classification (2000): 90C10, 90C11, 90C57     

Ehrhart Polynomials of 3-Dimensional Simple Integral Convex Polytopes

The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry.