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.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

Box-inequalities for quadratic assignment polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope. Received: April 17, 2000 / Accepted: July 3, 2001?Published online September 3, 2001     

A note on Faigle and Kern’s dual greedy polyhedra

U. Faigle and W. Kern have recently extended the work of their earlier paper and of M. Queyranne, F. Spieksma and F. Tardella and have shown that a dual greedy algorithm works for a system of linear inequalities with {:0,1}-coefficients defined in terms of antichains of an underlying poset and a submodular function on the set of ideals of the poset under some additional condition on the submodular function.?In this note we show that Faigle and Kern’s dual greedy polyhedra belong to a class of submodular flow polyhedra, i.e., Faigle and Kern’s problem is a special case of the submodular flow problem that can easily be solved by their greedy algorithm. Received: February 1999 / Accepted: December 1999?Published online February 23, 2000     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

New classes of fast lower bounds for bin packing problems

The bin packing problem is one of the classical NP-hard optimization problems. In this paper, we present a simple generic approach for obtaining new fast lower bounds, based on dual feasible functions. Worst-case analysis as well as computational results show that one of our classes clearly outperforms the previous best “economical” lower bound for the bin packing problem by Martello and Toth, which can be understood as a special case. In particular, we prove an asymptotic worst-case performance of 3/4 for a bound that can be computed in linear time for items sorted by size. In addition, our approach provides a general framework for establishing new bounds. Received: August 11, 1998 / Accepted: February 1, 2001?Published online September 17, 2001     

A polyhedral approach to single-machine scheduling problems

We report new results for a time-indexed formulation of nonpreemptive single-machine scheduling problems. We give complete characterizations of all facet inducing inequalities with integral coefficients and right-hand side 1 or 2 for the convex hull of the set of feasible partial schedules, i.e., schedules in which not all jobs have to be started. Furthermore, we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope, which is the convex hull of the set of feasible complete schedules, i.e., schedules in which all jobs have to be started. To obtain insight in the effectiveness of these classes of facet-inducing inequalities, we develop a branch-and-cut algorithm based on them. We evaluate its performance on the strongly NP-hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates. Received March 24, 1994 / Revised version received February 13, 1997 Published online June 28, 1999     

A branch and cut approach to the cardinality constrained circuit problem

The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits, and a branch-and-cut solution approach using these inequalities. Received: April 1998 / Accepted: October 2000?Published online October 26, 2001     

On the rank of mixed 0,1 polyhedra

For a polytope in the [0,1] ⁿ cube, Eisenbrand and Schulz showed recently that the maximum Chvátal rank is bounded above by O(n ²logn) and bounded below by (1+ε)n for some ε>0. Chvátal cuts are equivalent to Gomory fractional cuts, which are themselves dominated by Gomory mixed integer cuts. What do these upper and lower bounds become when the rank is defined relative to Gomory mixed integer cuts? An upper bound of n follows from existing results in the literature. In this note, we show that the lower bound is also equal to n. This result still holds for mixed 0,1 polyhedra with n binary variables. Received: March 15, 2001 / Accepted: July 18, 2001?Published online September 17, 2001     

A branch-and-cut algorithm for capacitated network design problems

We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called “knapsack branching”. We also report on our computational experience using real-life data. Received April 29, 1997 / Revised version received January 9, 1999? Published online June 28, 1999     

The stable admissions polytope

The stable admissions polytope– the convex hull of the stable assignments of the university admissions problem – is described by a set of linear inequalities. It depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been fruitful in the analysis of the stable marriage problem. Received: April 10, 1998 / Accepted: June 3, 1999?Published online January 27, 2000     

Alternative formulations for the Set Packing Problem and their application to the Winner Determination Problem

An alternative formulation for the set packing problem in a higher dimension is presented. The addition of a new family of binary variables allows us to find new valid inequalities, some of which are shown to be facets of the polytope in the higher dimension. We also consider the Winner Determination Problem, which is equivalent to the set packing problem and whose special structure allows us to easily implement these valid inequalities in a very easy way. The computational experiments illustrate the performance of the valid inequalities and obtain good results.     

A cutting plane algorithm for the General Routing Problem

The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for the GRP based on facet-inducing inequalities and show that it is capable of providing very strong lower bounds and, in most cases, optimal solutions. Received: November 1998 / Accepted: September 2000?Published online March 22, 2001     

Set packing relaxations of some integer programs

This paper is about set packing relaxations of combinatorial optimization problems associated with acyclic digraphs and linear orderings, cuts and multicuts, and set packings themselves. Families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation. Received: September 1997 / Accepted: November 1999?Published online June 8, 2000     

The Steiner tree problem I: Formulations,compositions and extension of facets

In this paper we give some integer programming formulations for the Steiner tree problem on undirected and directed graphs and study the associated polyhedra. We give some families of facets for the undirected case along with some compositions and extensions. We also give a projection that relates the Steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph. This is used to show that the LP-relaxation of the directed formulation is superior to the LP-relaxation of the undirected one.Corresponding author.     

The 0-1 Knapsack problem with a single continuous variable

Specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable, called the mixed 0-1 knapsack problem. First different classes of facet-defining inequalities are derived based on restriction and lifting. The order of lifting, particularly of the continuous variable, plays an important role. Secondly we show that the flow cover inequalities derived for the single node flow set, consisting of arc flows into and out of a single node with binary variable lower and upper bounds on each arc, can be obtained from valid inequalities for the mixed 0-1 knapsack problem. Thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0-1 constraints. Initial computational results on a variety of problems are presented. Received May 22, 1997 / Revised version received December 22, 1997 Published online November 24, 1998     

Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover, the total cost must not exceed a given budget. This problem has been introduced by Kleinberg [7] and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing, and virtual-circuit routing. Kolliopoulos and Stein [9] and Dinitz, Garg, and Goemans [4] developed algorithms improving the first approximation results of Kleinberg for the problem of minimizing the violation of edge capacities and for other variants. However, known techniques do not seem to be capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover, all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem. Received: August 23, 2000 / Accepted: April 20, 2001?Published online October 2, 2001     

On the core of ordered submodular cost games

A general ordertheoretic linear programming model for the study of matroid-type greedy algorithms is introduced. The primal restrictions are given by so-called weakly increasing submodular functions on antichains. The LP-dual is solved by a Monge-type greedy algorithm. The model offers a direct combinatorial explanation for many integrality results in discrete optimization. In particular, the submodular intersection theorem of Edmonds and Giles is seen to extend to the case with a rooted forest as underlying structure. The core of associated polyhedra is introduced and applications to the existence of the core in cooperative game theory are discussed. Received: November 2, 1995 / Accepted: September 15, 1999?Published online February 23, 2000     

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001