On the acyclic subgraph polytope

The acyclic subgraph problem can be formulated as follows. Given a digraph with arc weights, find a set of arcs containing no directed cycle and having maximum total weight. We investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope. We also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time. This implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs. This generalizes a result of Lucchesi for planar digraphs.     

Facets of the balanced (acyclic) induced subgraph polytope

A signed graph is a graph whose edges are labelled positive or negative. A signed graph is said to be balanced if the set of negative edges form a cut. The balanced induced subgraph polytopeP(G) of a graphG is the convex hull of the incidence vectors of all node sets that induce balanced subgraphs ofG. In this paper we exhibit various (rank) facet defining inequalities. We describe several methods with which new facet defining inequalities ofP(G) can be constructed from known ones. Finding a maximum weighted balanced induced subgraph of a series parallel graph is a polynomial problem. We show that for this class of graphsP(G) may have complicated facet defining inequalities. We derive analogous results for the polytope of acyclic induced subgraphs.Research supported in part by the Natural Sciences and Engineering Research Council of Canada; the second author has also been supported by C.P. Rail.     

The generalized assignment problem: Valid inequalities and facets

Three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem. General properties of the facet defining inequalities are discussed and, for a special case, the convex hull is completely characterized. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint.Partial financial support under NSF grant #CCR-8812736.Partial financial support under NSF grant #DMS-8606188.     

Generating facets for the cut polytope of a graph by triangular elimination

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier–Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.      

Polyhedral study of the connected subgraph problem

In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.     

On the partial order polytope of a digraph

We introduce the partial order polytope of a digraphD, defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofD. For this polytope we prove some classes of inequalities to be facet-defining and show that there is a polynomial separation algorithm for each of these classes. The results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two-chorded odd cycle inequalities. The polyhedral results concerning the partial order polytope are of interest since a cutting plane based algorithm to solve the maximum weighted transitive acyclic subdigraph problem can be used to solve the maximum weighted acyclic subdigraph problem, the maximum weighted linear ordering problem and a flexible manufacturing problem. For the acyclic subdigraph polytope we show that the separation of simplet-reinforcedk-fence-inequalities is -complete.     

The partition problem

In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.     

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.     

Orderly algorithms for generating restricted classes of graphs

Orderly algorithms for the generation of exhaustive lists of nonisomorphic graphs are discussed. The existence of orderly methods to generate the graphs with a given subgraph and without a given subgraph is established. This method can be used to list all the nonisomorphic subgraphs of a given graph, as well as to produce catalogs of Hamiltonian graphs, pancyclic graphs, degree-constrained graphs, and other classes. A generalization of this method is given that can be used to generate lists of graphs with given girth, planar graphs, k-colorable graphs, and k-connected graphs, for example. Finally, these observations are employed to generate restricted classes of digraphs, notably acyclic digraphs and poset digraphs. The generation of poset digraphs is shown to supply a practical orderly method for producing a catalog of lattices. Similar observations concerning vertex addition generation methods allow one to improve on existing methods for the generation of catalog of interval and circle graphs.     

On the monotonization of polyhedra

In polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra. The presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure. It is therefore customary to approach the study of such a polytopeP through the study of one of its (full dimensional) relaxations (monotonizations) known as the submissive and the dominant ofP. Finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem. Our paper goes a long way towards solving this problem. We address the problem in the framework of a generalized monotonization of a polyhedronP, g-mon(P), that subsumes both the submissive and the dominant, and give a sufficient condition for an inequality that defines a facet of g-mon(P) to define a facet ofP. For the important cases of the traveling salesman (TS) polytope in both its symmetric and asymmetric variants, and of the linear ordering polytope, we give sufficient conditions trivially easy to verify, for a facet of the monotone completion to define a facet of the original polytope itself. Research supported by grant DMI-9201340 of the National Science Foundation and contract N00014-89-J-1063 of the Office of Naval Research. Research supported by MURST, Italy.     

(1,k)-configuration facets for the generalized assignment problem

class of facet defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from (1,k)-configuration inequalities.Partial financial support under NSF grant #CCR-8812736.Partial financial support under NSF grant #DMS-8606188.     

On Facets of Knapsack Equality Polytopes

The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope, where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the right-hand side, is considered. Equality constraints of this form arose in a real-world application of integer programming to a truck dispatching scheduling problem. Families of facet defining inequalities for this polytope are identified, and complete linear inequality representations are obtained for some classes of polytopes.     

The domino inequalities: facets for the symmetric traveling salesman polytope

Adam Letchford defines in [4] the Domino Parity inequalities for the Symmetric Traveling Salesman Polytope (STSP) and gives a polynomial algorithm for the separation of such constraints when the support graph is planar, generalizing a result of Fleischer and Tardos [2] for maximally violated comb inequalities. Naddef in [5] gives a set of necessary conditions for such inequalities to be facet defining for the STSP. These conditions lead to the Domino inequalities and it is shown in [5] that one does not lose any facet inducing inequality restricting the Domino Parity inequalities to Domino inequalities, except maybe for some very particular case. We prove here that all the domino inequalities are facet inducing for the STSP, settling the conjecture given in [5]. As a by product we will also have a complete proof that the comb inequalities are facet inducing. Mathematics Subject Classification (2000):Main 90C57, secondary 90C27     

Two-edge connected spanning subgraphs and polyhedra

This paper studies the problem of finding a two-edge connected spanning subgraph of minimum weight. This problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks. We discuss the polytope associated with the solutions to this problem. We show that when the graph is series-parallel, the polytope is completely described by the trivial constraints and the so-called cut constraints. We also give some classes of facet defining inequalities of this polytope when the graph is general.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and CP Rail and the German Research Association (Deutsche Forschungsgemeinschaft SFR 303).     

环面图上的一个Lebesgue型定理及其在线性染色中的应用

设c是图G的一个顶点染色, 如果c的任意两个色类都导出一个最大度至多为2的无圈子图,则称c为G的一个无圈染色. 我们首先证明了环面图上的一个Lebesgue 型定理, 作为其应用证明了对任一个围长不小于5 的环面图G, 除非△(G) = 4 而且G有一个子图H使得H的每一个面都是与三个3度点和二个4度点相关的5度面, H一定是(「(△(G))/2」+ 4)- 线性列表可染色的. 这一结果推广和改进了一些已知结论.     

Acrylic improper colorings of graphs

In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class V_i satisfies some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999