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SIAM J. on Discrete Mathematics

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1989

Volume 2, Issue 4 (partial)


The Asymmetric Assignment Problem and Some New Facets of the Traveling Salesman Polytope on a Directed Graph

Egon Balas

SIAM J. Discrete Math. 2, pp. 425-451 (27 pages) | Cited 11 times

Online Publication Date: August 08, 2006

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An assignment (spanning union of node-disjoint dicycles) in a directed graph is called asymmetric if it contains at most one arc of each pair $(i, j)$, $( j,i)$. The asymmetric assignment polytope is the convex hull of the incidence vectors of all asymmetric assignments. A class of facets is described for this polytope defined on a complete digraph, associated with certain odd-length closed alternating trails. The inequalities defining these facets are also facet inducing for the traveling salesman polytope defined on the same digraph. Furthermore, this class of facets is distinct from each of the classes identified earlier.

Induced Subgraphs of the Power of a Cycle

Jean-Claude Bermond and Claudine Peyrat

SIAM J. Discrete Math. 2, pp. 452-455 (4 pages) | Cited 1 time

Online Publication Date: August 08, 2006

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In this article, it is shown that if $G$ is an induced subgraph of the $d$th power of a cycle of length $n$, and $G$ has minimum degree $d + k$, then $G$ has at least $[ (d + k)/2d ]n$ vertices. This answers a problem of Kézdy.

A Matter of Degree

G. Chartrand, H. Hevia, O. Oellermann, F. Saba, and A. Schwenk

SIAM J. Discrete Math. 2, pp. 456-466 (11 pages)

Online Publication Date: August 08, 2006

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The concepts of $n$th degrees and $n$th-order odd vertices in graphs are introduced. The first degree of a vertex $v$ in a graph $G$ is the degree of $v$, while the $n$th degree ($n\geqq 2$) of $v $ is the sum of the $(n - 1)$st degrees of the vertices adjacent to $v $ in $G$. By a first-order odd vertex in a graph $G$ is meant an (ordinary) odd vertex in $G$, while for $n\geqq 2$, an $n$th-order odd vertex of $G$ is a vertex adjacent to an odd number of $(n - 1)$st-order odd vertices. The number of $n$th-order odd vertices, $n = 1,2, \cdots $, is investigated. A sequence $s_{1}, s_{2}, \cdots ,s_n , \cdots $ of integers is defined to be a generalized odd vertex sequence if there exists a graph $G$ containing exactly $s_{n}$$n$th-order odd vertices for every positive integer $n$. Generalized odd vertex sequences are characterized. Relationships between the $n$th degrees of the vertices of a graph $G$ and the walks of length $n$ in $G$ are described. The analogous problem for digraphs is also discussed.

Pebbling in Hypercubes

Fan R. K. Chung

SIAM J. Discrete Math. 2, pp. 467-472 (6 pages) | Cited 8 times

Online Publication Date: August 08, 2006

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This paper considers the following game on a hypercube, first suggested by Lagarias and Saks. Suppose $2^n$ pebbles are distributed onto vertices of an $n$-cube (with $2^n$ vertices). A pebbling step is to remove two pebbles from some vertex and then place one pebble at an adjacent vertex. The question of interest is to determine if it is possible to get one pebble to a specified vertex by repeatedly using the pebbling steps from any starting distribution of $2^n$ pebbles. This question is answered affirmatively by proving several stronger and more general results.

Complexity of Scheduling Parallel Task Systems

Jianzhong Du and Joseph Y.-T. Leung

SIAM J. Discrete Math. 2, pp. 473-487 (15 pages) | Cited 20 times

Online Publication Date: August 08, 2006

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One of the assumptions made in classical scheduling theory is that a task is always executed by one processor at a time. With the advances in parallel algorithms, this assumption may not be valid for future task systems. In this paper, a new model of task systems is studied, the so-called Parallel Task System, in which a task can be executed by one or more processors at the same time. The complexity of scheduling Parallel Task Systems to minimize the schedule length is examined. For nonpreemptive scheduling, it is shown that the problem is strongly NP-hard even for two processors when the precedence constraints consist of a set of chains. For independent tasks, the problem is strongly NP-hard for five processors, but solvable in pseudo-polynomial time for two and three processors. For preemptive scheduling, it is shown that the problem is strongly NP-hard for arbitrary number of processors for a set of independent tasks. Furthermore, it is shown that it is NP-hard, but solvable in pseudo-polynomial time, for a fixed number of processors.

On Two Classical Ramsey Numbers of the Form $R(3,n)$

Geoffrey Exoo

SIAM J. Discrete Math. 2, pp. 488-490 (3 pages) | Cited 1 time

Online Publication Date: August 08, 2006

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New lower bounds are given for the classical Ramsey numbers $R (3,10)$ and $R (3,12)$. Both constructions were made using a variant of the Metropolis Algorithm and were built on smaller cyclic constructions.

Pair Labellings with Given Distance

Z. Furedi, J. R. Griggs, and D. J. Kleitman

SIAM J. Discrete Math. 2, pp. 491-499 (9 pages) | Cited 2 times

Online Publication Date: August 08, 2006

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Given a graph $G$ and $d \in \mathbb{Z}^+$, the pair labelling number, $r(G,d)$, is defined to be the minimum $n$ such that each vertex in $G$ can be assigned a pair of numbers from $\{ 1, \cdots ,n\} $ in such a way that any two numbers used at adjacent vertices differ by at least $d$. A question of Roberts’ is answered by determining all possible values of $r(G,d)$ given the chromatic number of $G$. The answer follows by determining the chromatic number of the graph that has pairs of integers as vertices and edges joining pairs that are distance at least $d$ apart. For general $t \in \mathbb{Z}^+$, the analogous questions for $t$-sets instead of pairs are considered. A solution for general $t$ is conjectured which, for $d = 1$, reduces to Lovász's theorem on Kneser graphs.

Cutoff Point and Monotonicity Properties for Multinomial Group Testing

F. K. Hwang and Y. C. Yao

SIAM J. Discrete Math. 2, pp. 500-507 (8 pages)

Online Publication Date: August 08, 2006

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The classical binomial group testing problem studies plans to sort defectives from good items using a minimum number of group tests where a group test is a test applicable to any subset of items with a yes and no answer to the question of whether the subset contains any defectives. In a multinomial group testing problem, each item can be in one of $k$ ordered states and a group test on a subset always reveals the highest state of any item in the subset. Two properties known for binomial group testing are the cutoff point, which characterizes when individual testing is optimal, and the monotonicity, which states that the minimum expected number of tests increases in the probability of an item being defective. This paper studies the multinomial group testing problem and gives a new sufficient condition that guarantees individual testing is optimal. (A sufficient condition in Kumar [SIAM J. Appl. Math., 19 (1970), pp. 340–350] is shown to be incorrect.) It is also shown that the monotonicity does not hold for the multinomial case.

Average Performance of Heuristics for Satisfiability

Rajeev Kohli and Ramesh Krishnamurti

SIAM J. Discrete Math. 2, pp. 508-523 (16 pages) | Cited 4 times

Online Publication Date: August 08, 2006

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Distribution-free tight lower bounds on the average performance ratio for random search, for a greedy heuristic and for a probabilistic greedy heuristic are derived for an optimization version of satisfiability. On average, the random solution is never worse than $\frac{1}{2}$ of the optimal, regardless of the data-generating distribution. The lower bound on the average greedy solution is at least $\frac{1}{2}$ of the optimal, and this bound increases with the probability of the greedy heuristic selecting the optimal at each step. In the probabilistic greedy heuristic, probabilities are introduced into the search strategy so that a decrease in the probability of finding the optimal solution occurs only if the nonoptimal solution becomes closer to the optimal. Across problem instances, and regardless of the distribution giving rise to data, the minimum average value of the solutions identified by the robabilistic greedy heuristic is no less than $\frac{2}{3}$ of the optimal.

Partitions of $A^w $

H. Lefmann and B. VOIGT

SIAM J. Discrete Math. 2, pp. 524-529 (6 pages)

Online Publication Date: August 08, 2006

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A canonizing Ramsey type theorem for Baire mappings $\Delta : A^w \to \mathcal{Y}$, where $\mathcal{Y}$ is a metric space is established.

The Average Number of Stable Matchings

Boris Pittel

SIAM J. Discrete Math. 2, pp. 530-549 (20 pages) | Cited 8 times

Online Publication Date: August 08, 2006

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The probable behavior of an instance of size $n$ of the stable marriage problem, chosen uniformly at random, is studied. The expected number of stable matchings is shown to be asymptotic to $e^{ - 1} n \ln n$ for $n \to \infty $. The total rank of women by men in the male optimal (pessimal) matching is proved to be close to $n$ In $n$ (respectively, $n^2 $/$\ln n$, with high probability.

A Mathematical Model for Periodic Scheduling Problems

Paolo Serafini and Walter Ukovich

SIAM J. Discrete Math. 2, pp. 550-581 (32 pages) | Cited 50 times

Online Publication Date: August 08, 2006

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A mathematical model is proposed for scheduling activities of periodic type. First a model is proposed for scheduling periodic events with particular time constraints. This problem, which could be considered the extension to periodic phenomena of ordinary scheduling with precedence constraints, is shown to be NP-complete. An algorithm for it of implicit enumeration type is designed based on network flow results, and its average complexity is discussed. Some extensions of the model are considered. The results of this first part serve as a basis in modelling periodic activities using resources. Several cases are considered. Finally some applications are presented for which the proposed model can be a useful tool.

Birigidity in the Plane

Brigitte Servatius

SIAM J. Discrete Math. 2, pp. 582-589 (8 pages) | Cited 3 times

Online Publication Date: August 08, 2006

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The two-dimensional generic rigidity matroid $R(G)$ of a graph $G$ is considered. The notions of vertex and edge birigidity are introduced. It is proved that vertex birigidity of $G$ implies the connectivity of $R(G)$ and that the connectivity of $R(G)$ implies the edge birigidity of $G$. These implications are not equivalences.
A class of minimal vertex birigid graphs is exhibited and used to show that $R(G)$ is not representable over any finite field.

Prime Testing for the Split Decomposition of a Graph

Jeremy Spinrad

SIAM J. Discrete Math. 2, pp. 590-599 (10 pages) | Cited 2 times

Online Publication Date: August 08, 2006

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This paper develops an $O(n^2 )$ algorithm for testing whether a graph is decomposable with respect to the split decomposition. The fastest previous algorithm required $w (n^3 )$ time for this problem. This leads to an $O(n^2 )$ expected time algorithm for computing the split decomposition of a graph.
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