90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
Refine
Document Type
- ZIB-Report (7)
Language
- English (7)
Has Fulltext
- yes (7)
Is part of the Bibliography
- no (7)
Keywords
- polyhedral combinatorics (7) (remove)
Institute
Our main result is that every $n$-dimensional polytope can be described by at most $2n-1$ polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an $n$-dimensional pointed polyhedral cone we prove the bound $2n-2$ and for arbitrary polyhedra we get a constructible representation by $2n$ polynomial inequalities.
Point-to-Multipoint systems are one kind of radio systems supplying wireless access to voice/data communication networks. Such systems have to be run using a certain frequency spectrum, which typically causes capacity problems. Hence it is, on the one hand, necessary to reuse frequencies but, on the other hand, no interference must be caused thereby. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there exist no polynomial time approximation algorithms with a guaranteed quality. One kind of algorithms which turned out to be successful for many other combinatorial optimization problems uses cutting plane methods. In order to apply such methods, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring basic properties of chromatic scheduling polytopes and several classes of facet-defining inequalities.
Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound 2n-2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities.
Stable multi-sets are an evident generalization of the well-known stable sets. As integer programs, they constitute a general structure which allows for a wide applicability of the results. Moreover, the study of stable multi-sets provides new insights to well-known properties of stable sets. In this paper, we continue our investigations started in [{\sl Koster and Zymolka 2002}] and present results of three types: on the relation to other combinatorial problems, on the polyhedral structure of the stable multi-set polytope, and on the computational impact of the polyhedral results. First of all, we embed stable multi-sets in a framework of generalized set packing problems and point out several relations. The second part discusses properties of the stable multi-set polytope. We show that the vertices of the linear relaxation are half integer and have a special structure. Moreover, we strengthen the conditions for cycle inequalities to be facet defining, show that the separation problem for these inequalities is polynomial time solvable, and discuss the impact of chords in cycles. The last result allows to interpret cliques as cycles with many chords. The paper is completed with a computational study to the practical importance of the cycle inequalities. The computations show that the performance of state-of-the-art integer programming solvers can be improved significantly by including these inequalities.
A subset ${\cal C}$ of the power set of a finite set $E$ is called cardinality homogeneous if, whenever ${\cal C}$ contains some set $F$, ${\cal C}$ contains all subsets of $E$ of cardinality $|F|$. Examples of such set systems ${\cal C}$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of $E$. With each cardinality homogeneous set system ${\cal C}$, we associate the polytope $P({\cal C})$, the convex hull of the incidence vectors of all sets in ${\cal C}$, and provide a complete and nonredundant linear description of $P({\cal C})$. We show that a greedy algorithm optimizes any linear function over $P({\cal C})$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type $P({\cal C})$.
A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.
Stable Multi-Sets
(2000)
In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined. The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks.