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SIAM J. on Discrete Mathematics

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1996

Volume 9, Issue 4, pp. 511-681


Graph Algorithms for Conformance Testing Using the Rural Chinese Postman Tour

Yinan N. Shen and Fabrizio Lombardi

SIAM J. Discrete Math. 9, pp. 511-528 (18 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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This paper presents new results and graph algorithms for the automatic testing of protocols using “unique input/output” (UIO) sequences. UIO sequences can be efficiently employed in checking conformance of protocols to their specifications by using transition testing. The optimization of the test sequence is based on finding the rural Chinese postman tour, of the state transition diagram of a finite state machine (FSM).
The process of conformance test generation using a touring algorithm is valid provided that certain connectivity properties of the graph are present. This implies that a weakly connected graph must be constructed. It is possible that this connectivity condition may not be met when multiple UIO sequences are used even if the reset capability and/or the self-loop properties are present. The “weakly connected graph problem” consists of finding an edge-induced subgraph of the FSM which is still weakly connected when multiple UIO sequences are used. The “multiple UIO tour minimization problem” addresses the assignment of edges to UIO sequences for minimizing the degree of the directed UIO graph. This process may not also minimize the length of the tour. The above two problems, left open in previous papers, are solved in this paper. It is proved that by appropriately changing the original assignment graph and using network flow techniques with a new UIO generation process referred to as chaining, efficient solutions can be provided. The theoretical approaches behind the solution to these problems are fully characterized.

Multipartition Series

David G. Wagner

SIAM J. Discrete Math. 9, pp. 529-544 (16 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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We investigate a class of generating series which enumerate multi-analogues of set partitions with very general weights and constraints imposed, and develop some of the relevant theory. The weights and constraints we consider are embodied in the definition of a “system,” which includes weighted multiset systems as a simple special case. Three topics are discussed. First, we derive a composition formula valid for all systems, which specializes to composition formulas for familiar combinatorial structures in many cases. Second, we extend the Heilmann–Lieb theorem on matching polynomials to a similar statement valid for more general factors of multigraphs. Finally, we introduce a multi-analogue of the order polynomial of a labelled poset, and by applying our general composition theorem give a formula for the effect of composition of labelled posets on their $E$-polynomials.

Valuated Matroid Intersection I: Optimality Criteria

Kazuo Murota

SIAM J. Discrete Math. 9, pp. 545-561 (17 pages) | Cited 9 times

Online Publication Date: July 12, 2006

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The independent assignment problem (or the weighted matroid intersection problem) is extended using Dress and Wenzel’s matroid valuations, which are attached to the vertex set of the underlying bipartite graph as an additional weighting. Specifically, the problem considered is as follows: given a bipartite graph $G = (V^ + ,V^ - ;A)$ with arc weight $w:A \to \mathbf{R}$ and matroid valuations $\omega ^ + $ and $\omega^ - $ on $V^ + $ and $V^ - $ respectively, find a matching $M( \subseteq A)$ that maximizes $\sum \{ \omega (a) \mid a \in M\} + \omega^ + (\partial ^ + M) + \omega^ - (\partial ^ - M)$, where $\partial ^ + M$ and $\partial ^ - M$ denote the sets of vertices in $V^ + $ and $V^ - $ incident to $M$. As natural extensions of the previous results for the independent assignment problem, two optimality criteria are established: one in terms of potentials and the other in terms of negative cycles in an auxiliary graph.

Valuated Matroid Intersection II: Algorithms

Kazuo Murota

SIAM J. Discrete Math. 9, pp. 562-576 (15 pages) | Cited 13 times

Online Publication Date: July 12, 2006

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Based on the optimality criteria established in part I [SIAM J. Discrete Math., 9 (1996), pp. 545–561] we show a primal-type cycle-canceling algorithm and a primal–dual-type augmenting algorithm for the valuated independent assignment problem: given a bipartite graph $G = (V^ + ,V^ - ;A)$ with arc weight $w:A \to \mathbf{R}$ and matroid valuations $\omega^ + $ and $\omega ^ - $ on $V^ + $ and $V^ - $, respectively; find a matching $M( \subseteq A)$ that maximizes $\sum \{ w(a)\mid a \in M\} + \omega^ + (\partial ^ + M) + \omega^ - (\partial ^ - M)$, where $\partial ^ + M$ and $\partial ^ - M$ denote the sets of vertices in $V^ + $ and $V^ - $ incident to $M$. The proposed algorithms generalize the previous algorithms for the independent assignment problem as well as for the weighted matroid intersection problem, including those due to Lawler [Math. Prog., 9 (1975), pp. 31–56], Ini and Tomizawa [J. Oper. Res. Soc. Japan, 19 (1976), pp. 32–57], Fujishige [J. Oper. Res. Soc. Japan, 20 (1977), pp. 1–15], Frank [J. Algorithms, 2 (1981), pp. 328–336], and Zimmermann [Discrete Appl. Math., 36 (1992), pp. 179–189].

Plane Embeddings of $2$-Trees and Biconnected Partial $2$-Trees

Andrzej Proskurowski, Maciej M. Sysło, and Paweł Winter

SIAM J. Discrete Math. 9, pp. 577-596 (20 pages)

Online Publication Date: July 12, 2006

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We consider different plane embeddings of partial $2$-trees and give an efficient algorithm constructing a minimum cardinality cover of faces, where each face is covered by exactly one vertex. These tasks are facilitated by a unique tree representation of plane embeddings of $2$-trees.

On the Trellis Complexity of the Densest Lattice Packings in $\mathbb{R}^n $

Ian F. Blake and Vahid Tarokh

SIAM J. Discrete Math. 9, pp. 597-601 (5 pages) | Cited 3 times

Online Publication Date: July 12, 2006

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An inequality relating the trellis complexity of lattices to their dimension and Hermite parameter is established. Using this inequality, a conjecture of Forney is proved indicating that the trellis complexity of the densest lattice packings in $\mathbb{R}^n $ grows exponentially as a function of their coding gain.

The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities

Sunil Chopra and Giovanni Rinaldi

SIAM J. Discrete Math. 9, pp. 602-624 (23 pages) | Cited 8 times

Online Publication Date: July 12, 2006

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A present trend in the study of the symmetric traveling salesman polytope is to use, as a relaxation of the polytope, the graphical traveling salesman polyhedron (GTSP). Following a parallel approach for the asymmetric traveling salesman polytope, we define the graphical asymmetric traveling salesman problem on a general digraph $D$ and its associated polyhedron GATSP($D$). We give basic polyhedral results and lifting theorems for GATSP($D$) and we give a general condition for a facet-defining inequality for GTSP to yield a symmetric facet-defining inequality for GATSP. Using this approach we show that all known major families of facet-defining inequalities of GTSP define facets of GATSP. Finally, we discuss possible extension of these results to the asymmetric traveling salesman polytope.

Cayley Graphs with Neighbor Connectivity One

L. L. Doty, R. J. Goldstone, and C. L. Suffel

SIAM J. Discrete Math. 9, pp. 625-642 (18 pages)

Online Publication Date: July 12, 2006

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We give an algebraic characterization of the generating sets of Cayley graphs with neighbor connectivity equal to one for a class of Cayley graphs that includes the Cayley graphs of all abelian groups. We also show that the determination of the neighbor connectivity of a graph is NP-hard.

IIS-Hypergraphs

Jennifer Ryan

SIAM J. Discrete Math. 9, pp. 643-653 (11 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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Given an inconsistent set of inequalities $Ax \leq b$, the irreducibly inconsistent subsystems (IIS’s) designate subsets of the inequalities such that at least one member of each subset must be deleted in order to achieve a feasible system. Each IIS can be considered the edge of a hypergraph. The purpose of this paper is to present several properties of this special class of hypergraphs (IIS-hypergraphs). IIS-hypergraphs are bicolourable, and their placement in Berge’s hierarchy of “hypergraphs generalizing bipartite graphs” is discussed. The greedy algorithm finds the minimum transversal for 2-uniform IIS-hypergraphs. It is shown that the greedy algorithm does not work for general IIS-hypergraphs.However, if the IIS-hypergraph is “nondegenerate” (implying uniform), the transversal number always can be found in time polynomial in the size of the hypergraph. An interesting intermediate result arises regarding blocking pairs of polyhedra arising from subspaces in $\Re^{n} $.

A Gray Code for Necklaces of Fixed Density

Terry Min Yih Wang and Carla D. Savage

SIAM J. Discrete Math. 9, pp. 654-673 (20 pages) | Cited 4 times

Online Publication Date: July 12, 2006

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A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all $n$-bit necklaces with $d$ ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the last and the first in the list, differ only by the transposition of two bits. The total time required is ${\text{O}}(nN(n,d))$, where $N(n,d)$ denotes the number of $n$-bit binary necklaces with $d$ ones. This is the first algorithm for generating necklaces of fixed density which is known to achieve this time bound.

Finding Independent Sets in Triangle-Free Graphs

Kathryn Fraughnaugh and Stephen C. Locke

SIAM J. Discrete Math. 9, pp. 674-681 (8 pages)

Online Publication Date: July 12, 2006

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Finding a maximum independent set in a graph is well known to be an NP-complete problem. Here an $O(n^2 )$-time algorithm that finds an independent set of order at least $(6n - m)/13$ in a triangle-free graph with $n$ vertices and $m$ edges is presented. A tight lower bound on independence in 4-regular triangle-free graphs is $4n/13$, so the bound is sharp for this class.
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