Some formulations for the group steiner tree problem

The group Steiner tree problem consists of, given a graph G, a collection R of subsets of V(G) and a cost c(e) for each edge of G, finding a minimum-cost subtree that connects at least one vertex from each R∈R. It is a generalization of the well-known Steiner tree problem that arises naturally in the design of VLSI chips. In this paper, we study a polyhedron associated with this problem and some extended formulations. We give facet defining inequalities and explore the relationship between the group Steiner tree problem and other combinatorial optimization problems.     

The node-edge weighted 2-edge connected subgraph problem: Linear relaxation, facets and separation

Let G=(V,E) be a undirected k-edge connected graph with weights c_e on edges and w_v on nodes. The minimum 2-edge connected subgraph problem, 2ECSP for short, is to find a 2-edge connected subgraph of G, of minimum total weight. The 2ECSP generalizes the well-known Steiner 2-edge connected subgraph problem. In this paper we study the convex hull of the incidence vectors corresponding to feasible solutions of 2ECSP. First, a natural integer programming formulation is given and it is shown that its linear relaxation is not sufficient to describe the polytope associated with 2ECSP even when G is series-parallel. Then, we introduce two families of new valid inequalities and we give sufficient conditions for them to be facet-defining. Later, we concentrate on the separation problem. We find polynomial time algorithms to solve the separation of important subclasses of the introduced inequalities, concluding that the separation of the new inequalities, when G is series-parallel, is polynomially solvable.     

A graph isomorphism algorithm using pseudoinverses

For a graph ofm nodes andn edges, an algorithm for testing the isomorphism of graphs is given. The complexity of the algorithm is a maximum ofO(mn ²) in almost all cases, with a considerable reduction if sparsity is exploited. If isomorphism is present, the pseudoinverses of the Laplace matrices of the graphs will be row and column permutations of each other. Advantage can be taken of certain features of the incidence matrices or of properties of the graphs to reduce computation time.     

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of t-(v,k,λ) designs. For this class of highly regular graphs, we obtain a worst-case running time of O(vlogv+O(1)) for bounded parameters t, k, λ.In a first step, our approach makes use of the Babai-Luks algorithm to compute canonical forms of t-designs. In a second step, we show that t-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.     

George Dantzig’s impact on the theory of computation

George Dantzig created the simplex algorithm for linear programming, perhaps the most important algorithm developed in the 20th century. This paper traces a single historical thread: Dantzig’s work on linear programming and its application and extension to combinatorial optimization, and the investigations it has stimulated about the performance of the simplex algorithm and the intrinsic complexity of linear programming and combinatorial optimization.     

Size-constrained graph partitioning polytopes

We consider the problem of clustering a set of items into subsets whose sizes are bounded from above and below. We formulate the problem as a graph partitioning problem and propose an integer programming model for solving it. This formulation generalizes several well-known graph partitioning problems from the literature like the clique partitioning problem, the equi-partition problem and the k-way equi-partition problem. In this paper, we analyze the structure of the corresponding polytope and prove several results concerning the facial structure. Our analysis yields important results for the closely related equi-partition and k-way equi-partition polytopes as well.     

Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

Facet-inducing web and antiweb inequalities for the graph coloring polytope

For a graph G and its complement , we define the graph coloring polytope P(G) to be the convex hull of the incidence vectors of star partitions of . We examine inequalities whose support graphs are webs and antiwebs appearing as induced subgraphs in G. We show that for an antiweb in G the corresponding inequality is facet-inducing for P(G) if and only if is critical with respect to vertex colorings. An analogous result is also proved for the web inequalities.     

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0–1linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.     

Multigraph realizations of degree sequences: Maximization is easy, minimization is hard

The following minimization problem is shown to be NP-hard: Given a graphic degree sequence, find a realization of this degree sequence as loopless multigraph that minimizes the number of edges in the underlying support graph. The corresponding maximization problem is known to be solvable in polynomial time.     

Stochastic lot-sizing problem with deterministic demands and Wagner-Whitin costs

In this paper, we consider a two-stage stochastic uncapacitated lot-sizing problem with deterministic demands and Wagner-Whitin costs. We develop an extended formulation in the higher dimensional space that provides integral solutions by showing that its constraint matrix is totally unimodular. We also provide the integral polyhedron of the problem in the original space by projecting the extended formulation to the original space.     

On the combinatorial structure of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, supplying wireless access to voice/data communication networks for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee.As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.     

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2K₂-partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2K₂-partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2K₂-hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.     

Strengthened clique-family inequalities for the stable set polytope

The stable set polytope is a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument.     

FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. A conference version of this article, containing a part of the results presented here, appeared in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL, January 22–24, 2006, pp. 743–748. The first author gratefully acknowledges support from NSF grant DMS-0608785, a 2003 UC-Davis Chancellor’s fellow award, the Alexander von Humboldt foundation, and IMO Magdeburg. The remaining authors were supported by the European TMR network ADONET 504438.     

Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems

Point-to-Multipoint systems are a 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 a combinatorial optimization problem, the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard and it is known that, for these problems, there exist no polynomial time algorithms with a fixed approximation ratio. Algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems. 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. J. L. Marenco: This work supported by UBACYT Grant X036, CONICET Grant 644/98 and ANPCYT Grant 11-09112. A. K. Wagler: This work supported by the Deutsche Forschungsgemeinschaft (Gr 883/9–1).     

Simultaneously lifting sets of binary variables into cover inequalities for knapsack polytopes

Cover inequalities are commonly used cutting planes for the 0–1 knapsack problem. This paper describes a linear-time algorithm (assuming the knapsack is sorted) to simultaneously lift a set of variables into a cover inequality. Conditions for this process to result in valid and facet-defining inequalities are presented. In many instances, the resulting simultaneously lifted cover inequality cannot be obtained by sequentially lifting over any cover inequality. Some computational results demonstrate that simultaneously lifted cover inequalities are plentiful, easy to find and can be computationally beneficial.