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2007

Volume 21, Issue 4, pp. 823-1092


Neighborhood Broadcasting in Hypercubes

Jean-Claude Bermond, Afonso Ferreira, Stéphane Pérennes, and Joseph G. Peters

SIAM J. Discrete Math. 21, pp. 823-843 (21 pages)

Online Publication Date: November 07, 2007

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In the broadcasting problem, one node needs to broadcast a message to all other nodes in a network. If nodes can only communicate with one neighbor at a time, broadcasting takes at least $\lceil \log_2 N \rceil$ rounds in a network of $N$ nodes. In the neighborhood broadcasting problem, the node that is broadcasting needs to inform only its neighbors. In a binary hypercube with $N$ nodes, each node has $\log_2 N$ neighbors, so neighborhood broadcasting takes at least $\lceil \log_2 \log_2 (N+1) \rceil$ rounds. In this paper, we present asymptotically optimal neighborhood broadcast protocols for binary hypercubes.

Mod (2p + 1)-Orientations and $K_{1,2p+1}$-Decompositions

Hong-Jian Lai

SIAM J. Discrete Math. 21, pp. 844-850 (7 pages)

Online Publication Date: November 07, 2007

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In this paper, we establish an equivalence between the contractible graphs with respect to the mod $(2p+1)$-orientability and the graphs with $K_{1, 2p+1}$-decompositions. This is applied to disprove a conjecture proposed by Barat and Thomassen that every 4-edge-connected simple planar graph $G$ with $|E(G)|\equiv 0$ (mod 3) has a claw decomposition.

On the Existence of $(K_5 \setminuse)$-Designs with Application to Optical Networks

Gennian Ge and Alan C. H. Ling

SIAM J. Discrete Math. 21, pp. 851-864 (14 pages)

Online Publication Date: November 16, 2007

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Motivated by the connection between graph decompositions and traffic grooming in optical networks, we continue the investigation of the existence problem for $(K_5 \setminus e)$-designs of order $n$. It is proved that the necessary conditions for the existence of such designs are also sufficient with 3 definite exceptions $(n=9,10,18)$ and 12 possible exceptions with $n=234$ being the largest. This gives a near solution for the long standing problem posed by Bermond et al. in [Ars Combin., 10 (1980), pp. 211–254]. As a consequence, we also give an optimal grooming on $n$ nodes with $C=9$ when such a $(K_5 \setminus e)$-design of order $n$ exists.

Improved Asymptotic Bounds for Codes Using Distinguished Divisors of Global Function Fields

Harald Niederreiter and Ferruh Özbudak

SIAM J. Discrete Math. 21, pp. 865-899 (35 pages)

Online Publication Date: November 16, 2007

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For a prime power $q$, let $\alpha_q$ be the standard function in the asymptotic theory of codes, that is, $\alpha_q(\delta)$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $\delta$ of $q$-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on $\alpha_q(\delta)$ was improved by Elkies, Xing, Niederreiter and Özbudak, and Maharaj. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function $\alpha_q^{\rm lin}$ for linear codes.

The Complexity of the List Partition Problem for Graphs

Kathie Cameron, Elaine M. Eschen, Chính T. Hoàng, and R. Sritharan

SIAM J. Discrete Math. 21, pp. 900-929 (30 pages)

Online Publication Date: December 07, 2007

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The $k$-partition problem is as follows: Given a graph $G$ and a positive integer $k$, partition the vertices of $G$ into at most $k$ parts $A_1, A_2, \ldots , A_k$, where it may be specified that $A_i$ induces a stable set, a clique, or an arbitrary subgraph, and pairs $A_i, A_j (i \neq j)$ be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list $k$-partition problem generalizes the $k$-partition problem by specifying for each vertex $x$, a list $L(x)$ of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list $k$-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.

Lower Bounds on Locality Sensitive Hashing

Rajeev Motwani, Assaf Naor, and Rina Panigrahy

SIAM J. Discrete Math. 21, pp. 930-935 (6 pages)

Online Publication Date: December 07, 2007

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Given a metric space $(X,d_X)$, $c \ge 1$, $r > 0$, and $p,q \in [0,1]$, a distribution over mappings $\mathscr{H} : X \to \mathbb{N}$ is called a $(r,cr,p,q)$-sensitive hash family if any two points in $X$ at distance at most $r$ are mapped by $\mathscr{H}$ to the same value with probability at least $p$, and any two points at distance greater than $cr$ are mapped by $\mathscr{H}$ to the same value with probability at most $q$. This notion was introduced by Indyk and Motwani in 1998 as the basis for an efficient approximate nearest neighbor search algorithm and has since been used extensively for this purpose. The performance of these algorithms is governed by the parameter $\rho = \frac{\log(1/p)}{\log(1/q)}$, and constructing hash families with small $\rho$ automatically yields improved nearest neighbor algorithms. Here we show that for $X = \ell_1$ it is impossible to achieve $\rho \le \frac{1}{2c}$. This almost matches the construction of Indyk and Motwani which achieves $\rho \le \frac{1}{c}$.

A Worst-Case Analysis of the Sequential Method to List the Minimal Hitting Sets of a Hypergraph

Ken Takata

SIAM J. Discrete Math. 21, pp. 936-946 (11 pages)

Online Publication Date: December 07, 2007

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It is open whether the minimal hitting sets of a hypergraph can be listed in time polynomial in the input and output size. We show that a well-known sequential approach described by Berge and studied since the 1950s is not polynomial in the above sense, even if we allow an optimal ordering of the edges. This answers a question posed by H. Hirsh. The proof uses hypergraphs based on read-once formulas. We also offer a generalization of this sequential approach.

On the Number of Fixed Pairs in a Random Instance of the Stable Marriage Problem

B. Pittel, L. Shepp, and E. Veklerov

SIAM J. Discrete Math. 21, pp. 947-958 (12 pages)

Online Publication Date: December 07, 2007

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Consider a group of $n$ men and $n$ women, each ranking the members of the opposite sex as a potential marriage partner. A matching (marriage) of men and women is called stable if there is no pair (man, woman) who are not matched but prefer each other to their partners in the matching. It is known that, for every instance of the rankings, there is at least one stable matching and that there are instances with exponentially many stable matchings. Assume that the instance is chosen uniformly at random among all $(n!)^{2n}$ possibilities. In this case the likely number of stable matchings is known to be $n^{1/2-o(1)}$, with high probability, and of order $n\ln n$, with probability $0.84$ at least. In this paper we show that the average number of fixed pairs (man, woman), i.e., pairs common to all stable matchings, is asymptotic to $\ln^2 n$. More generally, the average number of women (men) with $k$ stable husbands (wives) is asymptotic to $(\ln ^{k+1} n)/(k-1)!$.

Constant Weight Conflict-Avoiding Codes

Koji Momihara, Meinard Müller, Junya Satoh, and Masakazu Jimbo

SIAM J. Discrete Math. 21, pp. 959-979 (21 pages) | Cited 1 time

Online Publication Date: December 12, 2007

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A conflict-avoiding code (CAC) $C$ of length $n$ with weight $k$ is a family of binary sequences of length $n$ and weight $k$ satisfying $\sum_{0\le t\le n-1}x_{it}x_{j,t+s}\le \lambda$ for any distinct codewords $x_i=(x_{i0},x_{i1},\ldots,x_{i,n-1})$ and $x_j=(x_{j0},x_{j1},\ldots,x_{j,n-1})$ in $C$ and for any integer $s$, where the subscripts are taken modulo $n$. A CAC with maximum code size for given $n$ and $k$ is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of potential users to transmit information efficiently and reliably. In this paper, the case $\lambda=1$ is treated, and various direct and recursive constructions of optimal CACs for weight $k=4$ and $5$ are obtained by providing constructions of CACs for general weight $k$. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.

Ramsey Numbers and the Size of Graphs

Benny Sudakov

SIAM J. Discrete Math. 21, pp. 980-986 (7 pages)

Online Publication Date: December 12, 2007

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For two graphs $H$ and $G$, the Ramsey number $r(H, G)$ is the smallest positive integer $n$ such that every red-blue edge coloring of the complete graph $K_n$ on $n$ vertices contains either a red copy of $H$ or a blue copy of $G$. Motivated by questions posed by Erdős and Harary, in this note we study how the Ramsey number $r(K_s, G)$ depends on the size of the graph $G$. For $s \geq 3$, we prove that for every $G$ with $m$ edges, $r(K_s,G) \geq c(m/\log m)^{(s+1)/(s+3)}$ for some positive constant $c$ depending only on $s$. This lower bound improves an earlier result of Erdős, Faudree, Rousseau, and Schelp, and it is tight up to a polylogarithmic factor when $s=3$. We also study the maximum value of $r(K_s,G)$ as a function of $m$.

On $k$-Term DNF with the Largest Number of Prime Implicants

Robert H. Sloan, Balázs Szörényi, and György Turán

SIAM J. Discrete Math. 21, pp. 987-998 (12 pages)

Online Publication Date: January 09, 2008

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It is known that a $k$-term DNF can have at most $2^k - 1$ prime implicants and that this bound is sharp. We determine all $k$-term DNF having the maximal number of prime implicants. It is shown that a DNF is maximal if and only if it corresponds to a nonrepeating decision tree with literals assigned to the leaves in a certain way. We also mention some related results and open problems.

Coloring Bull-Free Perfectly Contractile Graphs

Benjamin Lévêque and Frédéric Maffray

SIAM J. Discrete Math. 21, pp. 999-1018 (20 pages)

Online Publication Date: January 09, 2008

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We consider the class of graphs that contain no bull, no odd hole, and no antihole of length at least five. We present a new algorithm that colors optimally the vertices of every graph in this class. This algorithm is based on the existence in every such graph of an ordering of the vertices with a special property. More generally we prove, using a variant of lexicographic breadth-first search, that in every graph that contains no bull and no hole of length at least five there is a vertex that is not the middle of a chordless path on five vertices. This latter fact also generalizes known results about chordal bipartite graphs, totally balanced matrices, and strongly chordal graphs.

Combinatorial Optimization with Explicit Delineation of the Ground Set by a Collection of Subsets

Moshe Dror and James B. Orlin

SIAM J. Discrete Math. 21, pp. 1019-1034 (16 pages)

Online Publication Date: January 09, 2008

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We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements $N$ has additional characteristics. For each problem in this paper, the set $N$ is expressed explicitly by subsets of $N$ either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.

Average Distance and Edge-Connectivity II

Peter Dankelmann, Simon Mukwembi, and Henda C. Swart

SIAM J. Discrete Math. 21, pp. 1035-1052 (18 pages)

Online Publication Date: January 22, 2008

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The average distance $\mu(G)$ of a connected graph $G$ of order $n$ is the average of the distances between all pairs of vertices of $G$. We prove that for a 3-edge-connected graph $G$ of order $n$ the inequality $\mu(G)\le{n}/{6}+24$ on the average distance holds. Our bound is shown to be best possible even if $G$ is 4-edge-connected, and our results answer, in part, a question of Plesník [J. Graph Theory, 8 (1984), pp. 1–24].

Explicit Construction of Small Folkman Graphs

Linyuan Lu

SIAM J. Discrete Math. 21, pp. 1053-1060 (8 pages)

Online Publication Date: January 22, 2008

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A Folkman graph is a $K_4$-free graph $G$ such that if the edges of $G$ are 2-colored, then there exists a monochromatic triangle. Erdős offered a prize for proving the existence of a Folkman graph with at most 1 million vertices. In this paper, we construct several “small” Folkman graphs within this limit. In particular, there exists a Folkman graph on 9697 vertices.

Nonseparating Induced Cycles Consisting of Contractible Edges in $k$-Connected Graphs

Yoshimi Egawa, Katsumi Inoue, and Ken-ichi Kawarabayashi

SIAM J. Discrete Math. 21, pp. 1061-1070 (10 pages)

Online Publication Date: January 25, 2008

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Egawa and Saito proved that every $k$-connected graph with girth at least 4 has an induced cycle $C$ such that $G-V(C)$ is $(k-3)$-connected, and every edge of $C$ is contractible. This means that we can find not only a nonseparating cycle $C$ but also one that consists of contractible edges. Motivated by this result, we prove that if $G$ is a $k$-connected graph which does not contain $K_4^{-}$, then $G$ has an induced cycle $C$ such that $G - V(C)$ is $(k-2)$-connected and either every edge of $C$ is $k$-contractible or $C$ is a triangle. As a corollary of this result, we get the following result: Every $k$-connected graph with girth at least 4 has an induced cycle $C$ such that $G-V(C)$ is $(k-2)$-connected, and every edge of $C$ is contractible. This theorem is a generalization of some known theorems. In particular, this generalizes the above-mentioned result proved by Egawa and Saito and the result of Egawa which says that a $k$-connected graph with girth at least 4 has an induced cycle $C$ such that $G-V(C)$ is $(k-2)$-connected.

Matched-Factor $d$-Domatic Coloring of Graphs

K. S. Sudeep and Sundar Vishwanathan

SIAM J. Discrete Math. 21, pp. 1071-1082 (12 pages)

Online Publication Date: January 25, 2008

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Consider a graph $G$ and a collection of connected spanning subgraphs $G_1, G_2, \ldots, G_k$, not necessarily edge-disjoint. A subset $U_i$ of the vertex set is said to $d$-$dominate$ $G_i$ if in $G_i$, all the vertices are at distance at most $d$ from some vertex in $U_i$. Alon et al. [Discrete Math., 262 (2003), pp. 17–25] introduced and studied a function $\mu(k)$, which is defined as the minimum radius of domination $d$ such that the vertex set of every graph with a collection of $k$ spanning subgraphs can be partitioned into $U_1, U_2, \ldots, U_k$ such that $U_i$ $d$-dominates $G_i$. They proved that $\mu(k) < \frac{3}{2}k$, and the proof yields a polynomial time algorithm for the same. We prove that the problem is $\cal NP$-complete, and we also answer a question from their paper by improving their bound to $(\frac{3}{2}-\epsilon)k$. We also present an algorithm which finds such a coloring in polynomial time.

Cycle Systems in the Complete Bipartite Graph Plus a One-Factor

Liqun Pu, Hao Shen, Jun Ma, and San Ling

SIAM J. Discrete Math. 21, pp. 1083-1092 (10 pages)

Online Publication Date: January 25, 2008

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Let $K_{n,n}$ denote the complete bipartite graph with $n$ vertices in each partite set and $K_{n,n}+I$ denote $K_{n,n}$ with a one-factor added. It is proved in this paper that there exists an $m$-cycle system of $K_{n,n}+I$ if and only if $n \equiv 1 (\rm{mod} 2)$, $m \equiv 0 (\rm{mod} 2)$, $4 \leq m \leq 2n$, and $n(n+1) \equiv$ 0 (mod $m$).
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