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SIAM J. on Discrete Mathematics

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2009

Volume 23, Issue 4, pp. 1655-2213


Paths with No Small Angles

Imre Bárány, Attila Pór, and Pavel Valtr

SIAM J. Discrete Math. 23, pp. 1655-1666 (12 pages)

Online Publication Date: November 13, 2009

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Giving a (partial) solution to a problem of Fekete [Geometry and the Traveling Salesman Problem, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada, 1992] and Fekete and Woeginger [Comput. Geom., 8 (1997), pp. 195–218], we show that given a finite set $X$ of points in the plane, it is possible to find a polygonal path with $|X|-1$ segments and with vertex set $X$ so that every angle on the polygonal path is at least $\pi/9$. According to a conjecture of Fekete and Woeginger, $\pi/9$ can be replaced by $\pi/6$. Previously, the result has not been known with any positive constant. We show further that the same result holds, with an angle smaller than $\pi/9$, in higher dimensions.

Approximate Nonlinear Optimization over Weighted Independence Systems

Jon Lee, Shmuel Onn, and Robert Weismantel

SIAM J. Discrete Math. 23, pp. 1667-1681 (15 pages)

Online Publication Date: November 13, 2009

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We consider optimizing a nonlinear objective function over a weighted independence system presented by a linear-optimization oracle. We provide an efficient algorithm that determines an $r$-best solution for nonlinear functions of the total weight of an independent set, where $r$ depends only on certain Frobenius numbers of the individual weights and is independent of the size of the ground set. In contrast, we show that finding an optimal (0-best) solution requires exponential time.

Finding Planted Partitions in Random Graphs with General Degree Distributions

Amin Coja-Oghlan and André Lanka

SIAM J. Discrete Math. 23, pp. 1682-1714 (33 pages)

Online Publication Date: November 18, 2009

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We consider the problem of recovering a planted partition such as a coloring, a small bisection, or a large cut in an (apart from that) random graph. In the last 30 years many algorithms for this problem have been developed that work provably well on various random graph models resembling the Erdős–Rényi model $G_{n,m}$. In these random graph models edges are distributed uniformly, and thus the degree distribution is very regular. By contrast, the recent theory of large networks shows that real-world networks frequently have a significantly different distribution of the edges and hence also a different degree distribution. Therefore, a variety of new types of random graphs have been introduced to capture these specific properties. One of the most popular models is characterized by a prescribed expected degree sequence. We study a natural variant of this model that features a planted partition. Our main result is that there is a polynomial time algorithm for recovering (a large part of) the planted partition in this model even in the sparse case, where the average degree is constant. In contrast to prior work, the input of the algorithm consists only of the graph, i.e., no further parameters of the model (such as the expected degree sequence) are revealed to the algorithm.

Probabilistic Analysis of a Motif Discovery Algorithm for Multiple Sequences

Bin Fu, Ming-Yang Kao, and Lusheng Wang

SIAM J. Discrete Math. 23, pp. 1715-1737 (23 pages)

Online Publication Date: November 18, 2009

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We study a natural probabilistic model for motif discovery that has been used to experimentally test the quality of motif discovery programs. In this model, there are $k$ background sequences, and each character in a background sequence is a random character from an alphabet $\Sigma$. A motif $G=g_1g_2\cdots g_m$ is a string of $m$ characters. Each background sequence is implanted into a probabilistically generated approximate copy of $G$. For an approximate copy $b_1b_2\cdots b_m$ of $G$, every character $b_i$ is probabilistically generated such that the probability for $b_i\neq g_i$ is at most $\alpha$. In this paper, we give the first analytical proof that multiple background sequences do help with finding subtle and faint motifs. This work is a theoretical approach with a rigorous probabilistic analysis. We develop an algorithm that under the probabilistic model can find the implanted motif with high probability when the number of background sequences is reasonably large. Specifically, we prove that for $\alpha<0.1771$ and any constant $x\geq8$, there exist constants $t_0,\delta_0,\delta_1>0$ such that if the length of the motif is at least $\delta_0\log n$, the alphabet has at least $t_0$ characters, and there are at least $\delta_1\log n_0$ input sequences, then in $O(n^3)$ time our algorithm finds the motif with probability at least $1-\frac{1}{2^x}$, where $n$ is the longest length of any input sequence and $n_0\leq n$ is an upper bound for the length of the motif.

Multiple Random Walks in Random Regular Graphs

Colin Cooper, Alan Frieze, and Tomasz Radzik

SIAM J. Discrete Math. 23, pp. 1738-1761 (24 pages)

Online Publication Date: November 25, 2009

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We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random $r$-regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728–740], where it was shown with high probability (whp) that for $r\geq3$ the cover time is asymptotic to $\theta_r n\ln n$, where $\theta_r=(r-1)/(r-2)$. In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph $G$. For $k$ independent walks on $G$, the cover time $C_G(k)$ is asymptotic to $C_G/k$, where $C_G$ is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is $\theta_r n/\binom{k}{2}$. If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for $k$ walks to broadcast a single piece of information to each other is asymptotic to $\frac{2\ln k}{k}\theta_r n$ as $k,n\rightarrow\infty$. We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of $k$ explosive particles ($k$ even) tends to $(2\ln2)\theta_r n$ as $k\rightarrow\infty$. The case of $n$ coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to $2\theta_r n$. Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with $k$ predators and $\ell$ prey the expected detection time is $\theta_r H_{\ell}n/k$, where $H_{\ell}$ is the $\ell$th harmonic number.

The Fractional Chromatic Number of Graphs of Maximum Degree at Most Three

Hamed Hatami and Xuding Zhu

SIAM J. Discrete Math. 23, pp. 1762-1775 (14 pages) | Cited 1 time

Online Publication Date: November 25, 2009

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This paper studies the fractional chromatic number of graphs with maximum degree at most 3. It is proved that if $G$ is triangle free and has maximum degree at most 3, then $\chi_f(G)\leq3-\frac{3}{64}$. If $G$ has girth at least $k$ and maximum degree at most 3, then $\chi_f(G)\leq c_k$, where $c_k$ is a decreasing sequence converging to $8/3$, and $c_{15}\approx2.66681$.

Locating Errors Using ELAs, Covering Arrays, and Adaptive Testing Algorithms

Conrado Martínez, Lucia Moura, Daniel Panario, and Brett Stevens

SIAM J. Discrete Math. 23, pp. 1776-1799 (24 pages)

Online Publication Date: December 04, 2009

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In this paper, we define and study error locating arrays (ELAs), which can be used in software testing for locating faulty interactions among parameters or components in a system. We give constructions of ELAs for arbitrary strength $t$, based on covering arrays. We show that the number of tests given by ELAs grows as $O(\log k)$, where $k$ is the number of parameters/components in the system, assuming other quantities (the number $g$ of values per parameter, the strength $t$ of faulty interactions, and the number $d$ of faulty interactions) are bounded by a constant. We then give a series of results for the case of pairwise interactions ($t=2$). We study the computational complexity of deciding whether a graph describing the faulty pairwise interactions is “locatable.” We characterize the locatable graphs for the binary case ($g=2$). We design and analyze efficient algorithms that locate errors under certain assumptions on the structure of the faulty pairwise interactions. Under the assumption of known “safe values,” our algorithm performs a number of tests that is polynomial in $\log k$ and $d$, where $k$ is the number of parameters in the system and $d$ is an upper bound on the number of faulty pairwise interactions. For the binary alphabet case, we provide an algorithm that does not require safe values and runs in expected polynomial time in $\log k$ whenever $d\in O(\log\log k)$.

A New Intersection Model and Improved Algorithms for Tolerance Graphs

George B. Mertzios, Ignasi Sau, and Shmuel Zaks

SIAM J. Discrete Math. 23, pp. 1800-1813 (14 pages) | Cited 1 time

Online Publication Date: December 04, 2009

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Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs, which generalizes in a natural way both interval and permutation graphs, has attracted many research efforts since their introduction in [M. C. Golumbic and C. L. Monma, Congr. Numer., 35 (1982), pp. 321–331], as it finds many important applications in constraint-based temporal reasoning, resource allocation, and scheduling problems, among others. In this article we propose the first non-trivial intersection model for general tolerance graphs, given by three-dimensional parallelepipeds, which extends the widely known intersection model of parallelograms in the plane that characterizes the class of bounded tolerance graphs. Apart from being important on its own, this new representation also enables us to improve the time complexity of three problems on tolerance graphs. Namely, we present optimal $\mathcal{O}(n\log n)$ algorithms for computing a minimum coloring and a maximum clique and an $\mathcal{O}(n^{2})$ algorithm for computing a maximum weight independent set in a tolerance graph with $n$ vertices, thus improving the best known running times $\mathcal{O}(n^{2})$ and $\mathcal{O}(n^{3})$ for these problems, respectively.

The Surviving Rate of a Graph for the Firefighter Problem

Cai Leizhen and Wang Weifan

SIAM J. Discrete Math. 23, pp. 1814-1826 (13 pages) | Cited 1 time

Online Publication Date: December 09, 2009

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We consider the following firefighter problem on a graph $G=(V,E)$. Initially, a fire breaks out at a vertex $v$ of $G$. In each subsequent time unit, a firefighter protects one vertex, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. Let $\mathrm{sn}(v)$ denote the maximum number of vertices the firefighter can save when a fire breaks out at vertex $v$ of $G$. We define the surviving rate $\rho(G)$ of $G$ to be the average percentage of vertices that can be saved when a fire randomly breaks out at a vertex of $G$, i.e., $\rho(G)=\sum_{v\in V}\mathrm{sn}(v)/n^2$. In this paper, we prove that for every tree $T$ on $n$ vertices, $\rho(T)>1-\sqrt{2/n}$. Furthermore, we show that $\rho(G)>1/6$ for every outerplanar graph $G$, and $\rho(H)>3/10$ for every Halin graph $H$ with at least 5 vertices.

A Tight Upper Bound on the Probabilistic Embedding of Series-Parallel Graphs

Yuval Emek and David Peleg

SIAM J. Discrete Math. 23, pp. 1827-1841 (15 pages)

Online Publication Date: December 09, 2009

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We prove that every unweighted series-parallel graph can be probabilistically embedded into its spanning trees with logarithmic distortion. This is tight due to an $\Omega(\log n)$ lower bound established by Gupta, Newman, Rabinovich, and Sinclair on the distortion required to probabilistically embed the $n$-vertex diamond graph into a collection of dominating trees. Our upper bound is gained by presenting a polynomial time probabilistic algorithm that constructs spanning trees with low expected stretch. This probabilistic algorithm can be derandomized to yield a deterministic polynomial time algorithm for constructing a spanning tree of a given (unweighted) series-parallel graph $G$, whose communication cost is at most $O(\log n)$ times larger than that of $G$.

Upward Spirality and Upward Planarity Testing

Walter Didimo, Francesco Giordano, and Giuseppe Liotta

SIAM J. Discrete Math. 23, pp. 1842-1899 (58 pages)

Online Publication Date: December 09, 2009

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A digraph is upward planar if it admits a planar drawing where all edges are monotone in the upward direction. It is known that the problem of testing a digraph for upward planarity is NP-complete in general. This paper describes an $O(n^4)$-time upward planarity testing algorithm for all digraphs that have a series-parallel structure, where $n$ is the number of vertices of the input. This significantly enlarges the family of digraphs for which a polynomial-time testing algorithm is known. Furthermore, the study is extended to general digraphs, and a fixed parameter tractable algorithm for upward planarity testing is described, whose time complexity is $O(d^t \cdot t \cdot n^3 + d \cdot t^2 \cdot n + d^2 \cdot n^2)$ where $t$ is the number of triconnected components of the digraph and $d$ is an upper bound on the diameter of any split component of the digraph. Our results use the new notion of upward spirality that, informally speaking, is a measure of the “level of winding” that a triconnected component of a digraph $G$ can have in an upward planar drawing of $G$.

The Ultimate Categorical Independence Ratio of Complete Multipartite Graphs

Ágnes Tóth

SIAM J. Discrete Math. 23, pp. 1900-1904 (5 pages)

Online Publication Date: December 11, 2009

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The independence ratio $i(G)$ of a graph $G$ is the ratio of its independence number and the number of vertices. The ultimate categorical independence ratio of a graph $G$ is defined as $\lim_{k\to\infty}i(G^{\times k})$, where $G^{\times k}$ denotes the $k$th categorical power of $G$. This parameter was introduced by Brown, Nowakowski, and Rall, who asked about its value for complete multipartite graphs. In this paper we determine the ultimate categorical independence ratio of complete multipartite graphs.

The LBFS Structure and Recognition of Interval Graphs

Derek G. Corneil, Stephan Olariu, and Lorna Stewart

SIAM J. Discrete Math. 23, pp. 1905-1953 (49 pages)

Online Publication Date: December 11, 2009

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A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple–free graphs, two families where the well-known lexicographic breadth first search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and that by exploiting this structure one can design a linear time, easily implementable, interval graph recognition algorithm.

On-line Ramsey Numbers

David Conlon

SIAM J. Discrete Math. 23, pp. 1954-1963 (10 pages)

Online Publication Date: December 11, 2009

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Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colors them in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph $G$. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number $\tilde{r}(G)$ of $G$. Our main result, relating to the conjecture that $\tilde{r}(K_t)=o(({r(t)\atop2}))$, is that there exists a constant $c>1$ such that $\tilde{r}(K_t)\leq c^{-t}({r(t)\atop2})$ for infinitely many values of $t$. We also prove a more specific upper bound for this number, showing that there exists a positive constant $c$ such that $\tilde{r}(K_t)\leq t^{-c\frac{\log t}{\log \log t}}4^t$. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph $K_{t,t}$.

Quasirandom Rumor Spreading on the Complete Graph Is as Fast as Randomized Rumor Spreading

Nikolaos Fountoulakis and Anna Huber

SIAM J. Discrete Math. 23, pp. 1964-1991 (28 pages)

Online Publication Date: December 11, 2009

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In this paper, we provide a detailed comparison between a fully randomized protocol for rumor spreading on a complete graph and a quasirandom protocol introduced by Doerr, Friedrich, and Sauerwald [Quasirandom rumor spreading, in Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 773–781]. In the former, initially there is one vertex which holds a piece of information, and during each round every one of the informed vertices chooses uniformly at random and independently one of its neighbors and informs it. In the quasirandom version of this method (cf. Doerr, Friedrich, and Sauerwald) each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor, and at each subsequent round it picks the next neighbor from its list and informs it. We give a precise analysis of the evolution of the quasirandom protocol on the complete graph with $n$ vertices and show that it evolves essentially in the same way as the randomized protocol. In particular, if $S(n)$ denotes the number of rounds that are needed until all vertices are informed, we show that for any slowly growing function $\omega(n)$, we have $\log_2 n + \ln n - 4 \ln \ln n \leq S(n) \leq \log_2 n + \ln n + \omega(n)$, with probability $1-o(1)$.

First-Fit Algorithm for the On-Line Chain Partitioning Problem

BartŁomiej Bosek, Tomasz Krawczyk, and Edward Szczypka

SIAM J. Discrete Math. 23, pp. 1992-1999 (8 pages)

Online Publication Date: January 06, 2010

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We consider a problem of partitioning a partially ordered set into chains by first-fit algorithm. In general this algorithm uses arbitrarily many chains on a class of bounded width posets. In this paper we prove that First-Fit uses at most $3tw^2$ chains to partition any poset of width $w$ which does not induce two incomparable chains of height $t$. In this way we get a wide class of posets with polynomial bound for the on-line chain partitioning problem. We also discuss some consequences of our result for coloring graphs by First-Fit.

An Efficient Sparse Regularity Concept

Amin Coja-Oghlan, Colin Cooper, and Alan Frieze

SIAM J. Discrete Math. 23, pp. 2000-2034 (35 pages)

Online Publication Date: January 06, 2010

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Let ${\bf A}$ be a $0/1$ matrix of size $m\times n$, and let $p$ be the density of ${\bf A}$ (i.e., the number of ones divided by $m\cdot n$). We show that ${\bf A}$ can be approximated in the cut norm within $\varepsilon\cdot mnp$ by a sum of cut matrices (of rank 1), where the number of summands is independent of the size $m\cdot n$ of ${\bf A}$, provided that ${\bf A}$ satisfies a certain boundedness condition. This decomposition can be computed in polynomial time. This result extends the work of Frieze and Kannan [Combinatorica, 19 (1999), pp. 175–220] to sparse matrices. As an application, we obtain efficient $1-\varepsilon$ approximation algorithms for “bounded” instances of MAX CSP problems.

SC-Hamiltonicity and Its Linkages with Strong Hamiltonicity of a Graph

Daniel K. Benvenuti and Abraham P. Punnen

SIAM J. Discrete Math. 23, pp. 2035-2041 (7 pages)

Online Publication Date: January 06, 2010

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In this paper, we provide a complete characterization of undirected SC-Hamiltonian graphs that are not strongly Hamiltonian. This conclusively settles a conjecture by Kryński [Discrete Appl. Math., 55 (1994), pp. 87–89], which was later disproved by Kabadi and Punnen [Discrete Math., 271 (2003), pp. 129–139] with a counterexample. We show that the Kabadi–Punnen counterexample is the only class of graphs where Kryński's conjecture is false, thereby proving the conjecture for all other graphs.

Direct Product Factorization of Bipartite Graphs with Bipartition-reversing Involutions

Ghidewon Abay-Asmerom, Richard H Hammack, Craig E. Larson, and Dewey T. Taylor

SIAM J. Discrete Math. 23, pp. 2042-2052 (11 pages)

Online Publication Date: January 15, 2010

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Given a connected bipartite graph $G$, we describe a procedure which enumerates and computes all graphs $H$ (if any) for which there is a direct product factorization $G\cong H\times K_2$. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Brešar, Imrich, Klavžar, Rall, and Zmazek [Finite and infinite hypercubes as direct products, Australas. J. Combin., 36 (2006), pp. 83–90, and Hypercubes as direct products, SIAM J. Discrete Math., 18 (2005), pp. 778–786].

Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko

SIAM J. Discrete Math. 23, pp. 2053-2078 (26 pages)

Online Publication Date: January 15, 2010

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Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant $k$, we present a $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $(\frac{1}{5}-\epsilon)$-approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k-1}+\epsilon}$ for $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$-approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$.

Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees

Hsueh-I Lu

SIAM J. Discrete Math. 23, pp. 2079-2092 (14 pages)

Online Publication Date: January 15, 2010

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We address the problem of designing compact routing tables for an unlabeled connected $n$-node planar network $G$. For each node $r$ of $G$, the designer is given a routing spanning tree $T_r$ of $G$ rooted at $r$, which specifies the routes for sending packets from $r$ to the rest of $G$. Each node $r$ of $G$ is equipped with ports $1,2,\ldots,\mathit{deg}_r$, where $\mathit{deg}_r$ is the degree of $r$ in $T_r$. Each port of $r$ is supposed to be assigned to a neighbor of $r$ in $T_r$ in a one-to-one manner. For each node $v$ of $G$ with $v\neq r$, let $\mathit{port}_r(v)$ be the port to which $r$ should forward packets with destination $v$. Under the assumption that the designer has the freedom to determine the label and the port assignment of each node in $G$, the routing table design problem is to design a compact routing table $R_r$ for each node $r$ such that $\mathit{port}_r(v)$ can be determined merely from $R_r$ and the label of $v$. Compact routing tables for various network topologies have been extensively studied in the literature. Planar networks are particularly important for routing with geometric metrics. Based upon four-page decompositions of $G$, Gavoille and Hanusse gave the best previously known polynomial-time computable result for this problem with linear-space routing tables, where the time complexity is measured under the conventional unit-cost RAM model of computation: Each $\mathit{port}_r(v)$ is computable from $R_r$ and the label of $v$ in $O(\log^{2+\epsilon}n)$ time for any positive constant $\epsilon$. The number of bits required to encode each $R_r$ is at most $8n+o(n)$. The time required to compute each $R_r$ is $O(n)$. Based on orderly spanning trees of $G$, our design achieves the following improved bounds without increasing the time complexity for computing each $R_r$: Each $\mathit{port}_r(v)$ is computable from $R_r$ and the label of $v$ in $O(\log^{1+\epsilon}n)$ time for any positive constant $\epsilon$. The number of bits required to encode each $R_r$ is at most $7.181n+o(n)$. The overall code length of all $n$ routing tables is at most $7n^2+o(n^2)$ bits.

On the Distribution of Orbits of $\mathrm{PGL_2(q)}$ in ${\mathbbF}_{q^n}$ and the Klapper Conjecture

Igor E. Shparlinski

SIAM J. Discrete Math. 23, pp. 2093-2099 (7 pages)

Online Publication Date: January 15, 2010

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Motivated by a conjecture of Klapper [Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993], we study the distribution of elements $\xi$ of a finite field $\mathbb{F}_{q^n}$ of $q^n$ elements under the action of the transformations $\xi\to(a\xi+b)/(c\xi+d)$ for matrices $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\mathrm{PGL_2(q)}$. We slightly improve a result of Niederreiter and Winterhof [Finite Fields Appl., 9 (2003), pp. 458–471] towards this conjecture. On the other hand, we also show that the original conjecture is false as stated.

Pancyclicity of Restricted Hypercube-Like Networks under the Conditional Fault Model

Sun-Yuan Hsieh and Chia-Wei Lee

SIAM J. Discrete Math. 23, pp. 2100-2119 (20 pages)

Online Publication Date: January 15, 2010

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A graph $G$ is said to be conditional $k$-edge-fault pancyclic if after removing $k$ faulty edges from $G$, under the assumption that each node is incident to at least two fault-free edges, the resulting graph contains a cycle of every length from its girth to $|V(G)|$. In this paper, we consider the common properties of a wide class of interconnection networks, called restricted hypercube-like networks, from which their conditional edge-fault pancyclicity can be determined. We then apply our technical theorems to show that several multiprocessor systems, including $n$-dimensional locally twisted cubes, $n$-dimensional generalized twisted cubes, recursive circulants $G(2^{n},4)$ for odd $n$, $n$-dimensional crossed cubes, and $n$-dimensional twisted cubes for odd $n$, are all conditional $(2n-5)$-edge-fault pancyclic.

Repetition Error Correcting Sets: Explicit Constructions and Prefixing Methods

Lara Dolecek and Venkat Anantharam

SIAM J. Discrete Math. 23, pp. 2120-2146 (27 pages)

Online Publication Date: January 15, 2010

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In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number $r$ of repetitions, in the sense that no two strings in the code can give rise to the same string after $r$ repetitions. We propose explicit number theoretic constructions of such subsets. In the case of $r=1$ repetition, the proposed construction is asymptotically optimal. For $r\geq1$, the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to $r$ repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number $r$ of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any $r$ repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible.

Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, and Birgit Vogtenhuber

SIAM J. Discrete Math. 23, pp. 2147-2155 (9 pages)

Online Publication Date: January 15, 2010

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We consider a variation of a problem stated by Erdős and Szekeres in 1935 about the existence of a number $f^{\mathrm{ES}}(k)$ such that any set $S$ of at least $f^{\mathrm{ES}}(k)$ points in general position in the plane has a subset of $k$ points that are the vertices of a convex $k$-gon. In our setting the points of $S$ are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of $S$ in its interior. We show that any sufficiently large bichromatic set of points in $\mathbb{R}^2$ in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).

On the Complexity of Finding a Sun in a Graph

Chính T. Hoàng

SIAM J. Discrete Math. 23, pp. 2156-2162 (7 pages)

Online Publication Date: January 22, 2010

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The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs (graphs containing no house, hole, or domino), have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun is NP-complete.

On the Hull Number of Triangle-Free Graphs

Mitre C. Dourado, Fábio Protti, Dieter Rautenbach, and Jayme L. Szwarcfiter

SIAM J. Discrete Math. 23, pp. 2163-2172 (10 pages)

Online Publication Date: January 22, 2010

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A set of vertices $C$ in a graph is convex if it contains all vertices which lie on shortest paths between vertices in $C$. The convex hull of a set of vertices $S$ is the smallest convex set containing $S$. The hull number $h(G)$ of a graph $G$ is the smallest cardinality of a set of vertices whose convex hull is the vertex set of $G$. For a connected triangle-free graph $G$ of order $n$ and diameter $d$ at least 4, we prove that $h(G)\leq(n-d+3)/3$ if $G$ has minimum degree at least 3 and that $h(G)\leq2(n-d+5)/7$, if $G$ is cubic. Furthermore for a connected graph $G$ of order $n$, girth $g$ at least 5, minimum degree at least 2, and diameter $d$, we prove $h(G)\leq2+(n-d-1)/\left\lceil\frac{g-1}{2}\right\rceil$. All bounds are best possible.

On the Classification of Type II Codes of Length 24

Noam D. Elkies and Scott Duke Kominers

SIAM J. Discrete Math. 23, pp. 2173-2177 (5 pages)

Online Publication Date: February 03, 2010

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We give a new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkov's approach to the classification of the root systems of Type II lattices in $\mathbb{R}^{24}$ and gives a new instance of the analogy between lattices and codes.

Asymptotic Enumeration of $k$-Edge-Colored $k$-Regular Graphs

Jeanette C. McLeod

SIAM J. Discrete Math. 23, pp. 2178-2197 (20 pages)

Online Publication Date: February 03, 2010

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Let $P(k,n)$ be the collection of all sets of $k$ disjoint perfect matchings in a complete graph with $2n$ vertices. We prove that if $k=o(n^{5/6})$, then $|P(k,n)|\sim\frac{1}{k!}(\frac{(2n)!}{2^{n}n!})^{k}(\frac{(2n)!}{(2n)^k(2n-k)!})^{n}\cdot(1-\frac{k}{2n})^{n/2}e^{k/4}$ for $n\rightarrow\infty$. This improves upon an existing result of Bollobás [Combinatorics, London Math. Soc. Lecture Note Ser. 52, Cambridge University Press, Cambridge, New York, 1981, pp. 80–102] who solved this problem for constant $k$, and a more recent result of Lieby et al. [Combin. Probab. Comput., 18 (2009), pp. 533–549] where an estimate is obtained for $k=o(n^{1/3})$.

A Short Note on an Advance in Estimating the Worst-Case Performance Ratio of the MPS Algorithm

Giuseppe Paletta and Francesca Vocaturo

SIAM J. Discrete Math. 23, pp. 2198-2203 (6 pages)

Online Publication Date: February 03, 2010

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Recently, a new approximation algorithm for the nonpreemptive scheduling of independent jobs on $m$ identical parallel processors has appeared in the literature. The algorithm, named $MPS$ (MultiProcessor Scheduling), combines partial solutions which satisfy suitable properties. Its performance ratio is bounded by $\frac{z+1}{z}-\frac{1}{mz}$, where $z$ represents the number of initial partial solutions provided by the algorithm. This note presents an advanced estimate of $z$ and, consequently, an improved worst-case performance ratio of the $MPS$ algorithm.

A New Proof of the $H$-Coloring Dichotomy

Mark H. Siggers

SIAM J. Discrete Math. 23, pp. 2204-2210 (7 pages)

Online Publication Date: February 03, 2010

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In this paper, we present a new proof of the $H$-coloring dichotomy, which was first proved by Hell and Nešetřil in 1990, and then was reproved by Bulatov in 2005. Our proof is much shorter than the original proof and avoids the algebraic machinery of Bulatov's proof.

The Joints Problem in $\mathbb{R}^n$

René Quilodrán

SIAM J. Discrete Math. 23, pp. 2211-2213 (3 pages)

Online Publication Date: February 03, 2010

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We show that given a collection of $A$ lines in $\mathbb{R}^n$, $n\geq2$, the maximum number of their joints (points incident to at least $n$ lines whose directions form a linearly independent set) is $O(A^{n/(n-1)})$. An analogous result for smooth algebraic curves is also proven.
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