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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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A Fractional Analogue of Brooks' Theorem SIAM J. Discrete Math. 26, pp. 452-471 (20 pages) Online Publication Date: April 10, 2012
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Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $\chi_f(G)$ is at least $\Delta(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $\omega(G)$ equals the maximum degree $\Delta(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$, while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result: If a connected graph $G$ with $\Delta(G)\geq 4$ is not one of the graphs listed above, then we have $\chi_f(G)\leq \Delta(G)- \frac{2}{67}$. |
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On Optimal Weighted Balanced Clusterings: Gravity Bodies and Power Diagrams SIAM J. Discrete Math. 26, pp. 415-434 (20 pages) Online Publication Date: April 04, 2012
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We study weighted clustering problems in Minkowski spaces under balancing constraints with a view towards separation properties. First, we introduce the gravity polytopes and more general gravity bodies that encode all feasible clusterings and indicate how they can be utilized to develop efficient approximation algorithms for quite general, hard to compute objective functions. Then we show that their extreme points correspond to strongly feasible power diagrams, certain specific cell complexes, whose defining polyhedra contain the clusters, respectively. Further, we characterize strongly feasible centroidal power diagrams in terms of the local optima of some ellipsoidal function over the gravity polytope. The global optima can also be characterized in terms of the separation properties of the corresponding clusterings. |
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A Deterministic Algorithm for the Frieze–Kannan Regularity Lemma SIAM J. Discrete Math. 26, pp. 15-29 (15 pages) Online Publication Date: January 03, 2012
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The Frieze–Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. R. Williams recently asked if one can construct a partition satisfying the conditions of the Frieze–Kannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an $\tilde O(n^{\omega})$ time algorithm for constructing such a partition, where $\omega < 2.376$ is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze–Kannan regularity lemma. |
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Forbidden Induced Subgraphs of Double-split Graphs SIAM J. Discrete Math. 26, pp. 1-14 (14 pages) Online Publication Date: January 03, 2012
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In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under the operation of taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs. |
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Griggs and Yeh's Conjecture and $L(p,1)$-labelings SIAM J. Discrete Math. 26, pp. 145-168 (24 pages) Online Publication Date: February 16, 2012
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An $L(p,1)$-labeling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geqslant p$ if dist$(x,y)=1$ and $|f(x)-f(y)|\geqslant 1$ if dist$(x,y)=2$, where dist$(x,y)$ is the distance between the two vertices $x$ and $y$ in the graph. The span of an $L(p,1)$-labeling $f$ is the difference between the largest and the smallest labels used by $f$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geqslant 2$ has an $L(2,1)$-labeling with span at most $\Delta^2$. We settle this conjecture for $\Delta$ sufficiently large. More generally, we show that for any positive integer $p$ there exists a constant $\Delta_p$ such that every graph with maximum degree $\Delta\geqslant \Delta_p$ has an $L(p,1)$-labeling with span at most $\Delta^2$. This yields that for each positive integer $p$, there is an integer $C_p$ such that every graph with maximum degree $\Delta$ has an $L(p,1)$-labeling with span at most $\Delta^2+C_p$. |
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Connected Domination and Spanning Trees with Many Leaves SIAM J. Discrete Math. 13, pp. 202-211 (10 pages) Online Publication Date: August 01, 2006
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Let G=(V,E) be a connected graph. A connected dominating set $S \subset V$ is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted $\gamma_c(G)$, is the minimum cardinality of a connected dominating set. Alternatively, $|V|-\gamma_c(G)$ is the maximum number of leaves in a spanning tree of $G$. Let $\delta$ denote the minimum degree of G. We prove that $\gamma_c(G) \leq |V| \frac{\ln(\delta+1)}{\delta+1}(1+o_\delta(1))$. Two algorithms that construct a set this good are presented. One is a sequential polynomial time algorithm, while the other is a randomized parallel algorithm in RNC. |
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SIAM J. Discrete Math. 26, pp. 384-398 (15 pages) Online Publication Date: March 15, 2012
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We consider a version of the secretary problem where each candidate has an identical twin. The aim, as in the classical problem, is to choose with the largest possible probability a top candidate, i.e., one of the best twins. We find an optimal stopping time for such a choice, the probability of success the optimal stopping time yields, and their asymptotic behavior. |
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SIAM J. Discrete Math. 26, pp. 353-374 (22 pages) Online Publication Date: March 13, 2012
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We prove that asymptotically (as $n\to\infty$) almost all graphs with $n$ vertices and $C_dn^{2-\frac{1}{2d}} \log^{\frac{1}{d}} n$ edges are universal with respect to the family of all graphs with maximum degree bounded by $d$. Moreover, we provide an efficient deterministic embedding algorithm for finding copies of bounded degree graphs in graphs satisfying certain pseudorandom properties. We also prove a counterpart result for random bipartite graphs, where the threshold number of edges is even smaller but the embedding is randomized. |
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SIAM J. Discrete Math. 26, pp. 30-38 (9 pages) Online Publication Date: January 17, 2012
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A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint translates of $V$. In other words, $V$ is a tile if and only if there is a subset $A \subseteq \mathbb{F}_2^n$ such that $V+A = \mathbb{F}_2^n$ uniquely (i.e., $v + a = v' + a'$ implies that $v=v'$ and $a=a'$, where $v,v' \in V$ and $a,a' \in A$). In some problems in coding theory and hashing we are given a putative tile $V$ and wish to know whether or not it is a tile. In this paper we give two computational criteria for certifying that $V$ is not a tile. The first involves the impossibility of a bin-packing problem, and the second involves the infeasibility of a linear program. We apply both criteria to a list of putative tiles given by Gordon, Miller, and Ostapenko [IEEE Trans. Inform. Theory, 56 (2010), pp. 984–991] in the context of hashing to find close matches, to show that none of them are, in fact, tiles. |
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On a Dispersion Problem in Grid Labeling SIAM J. Discrete Math. 26, pp. 39-51 (13 pages) Online Publication Date: January 24, 2012
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Given $k$ labelings of a finite $d$-dimensional cubical grid, define the combined distance between two labels to be the sum of the $\ell_1$-distance between the two labels in each labeling. We want to construct $k$ labelings which maximize the minimum combined distance between any two labels. When $d=1$, this can be interpreted as placing $n$ nonattacking rooks in a $k$-dimensional chessboard of size $n$ in such a way to maximize the minimum $\ell_1$-distance between any two rooks. Rook placements are also known as Latin hypercube designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a $k$-dimensional chessboard of size $n$ as certain lattice-like structures for certain well-chosen values of $n$. Then, we extend these constructions to any values of $n$ using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the two-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of $k$ labelings of $d$-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least $3$ is at least as hard as graph isomorphism. |
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Labeling Planar Graphs without 4,5-Cycles with a Condition on Distance Two SIAM J. Discrete Math. 26, pp. 52-64 (13 pages) Online Publication Date: January 26, 2012
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Wegner conjectured that for each planar graph $G$ with maximum degree $\Delta$ at least 4, $\chi(G^2)\leq\Delta+5$ if $4\leq\Delta\leq7$, and $\chi(G^2)\leq\lfloor \frac{3\Delta}{2}\rfloor +1$ if $\Delta\geq8$. Let $G$ be a planar graph without 4- and 5-cycles. In this paper, we discuss the $L(p,q)$-labeling of $G$ and show that $\lambda_{p,q}(G)\leq(2q-1)\Delta+6p+6q-6$ and $\lambda_{p,q}(G)\leq\max\{(2q-1)\Delta+6p+2q-4,9(2q-1)+8p-4,6(2q-1)+10p-5\},$ where $p$ and $q$ are positive integers with $p\geq q$. As a corollary, $\chi(G^2)\leq\Delta+7$ if $\Delta\leq7$, $\chi(G^2)\leq14$ if $\Delta=8$, and $\chi(G^2)\leq\Delta+5$ if $\Delta\geq9$. |
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Formulae for the Alon–Tarsi Conjecture SIAM J. Discrete Math. 26, pp. 65-70 (6 pages) Online Publication Date: January 26, 2012
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The sign of a Latin square is $-1$ if it has an odd number of rows and columns that are odd permutations; otherwise it is $+1$. Let $L^{\text{\scshape{even}}}_n$ and $L^{\text{\scshape{odd}}}_n$ be, respectively, the number of Latin squares of order $n$ with sign $+1$ and $-1$. The Alon–Tarsi conjecture asserts that $L^{\text{\scshape{even}}}_n \neq L^{\text{\scshape{odd}}}_n$ when $n$ is even. We prove that $L_n^{\text{\scshape{even}}}-L_n^{\text{\scshape{odd}}}=(-1)^{n(n-1)/2} \sum_{A \in B_n} (-1)^{\sigma_0(A)} \det(A)^n$, where $B_n$ is the set of $n \times n$ $\,(0,1)$-matrices and $\sigma_0(A)$ is the number of $0$ elements in $A$. We use this formula to give another proof of the Alon–Tarsi conjecture for $n=p-1$ for odd prime $p$. |
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SIAM J. Discrete Math. 26, pp. 77-88 (12 pages) Online Publication Date: February 09, 2012
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A queue layout of a graph consists of a linear ordering $\sigma$ of its vertices and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to $\sigma$. We show that the $n$-dimensional hypercube $Q_n$ has a layout into $n-\lfloor \log_2 n \rfloor$ queues for all $n\ge 1$. On the other hand, for every $\varepsilon>0$, every queue layout of $Q_n$ has more than $(\frac{1}{2}-\varepsilon) n-O(1/\varepsilon)$ queues and, in particular, more than $(n-2)/3$ queues. This improves previously known upper and lower bounds on the minimal number of queues in a queue layout of $Q_n$. For the lower bound we employ a new technique of out-in representations and contractions which may be of independent interest. |
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Parameterized Complexity of Arc-Weighted Directed Steiner Problems SIAM J. Discrete Math. 25, pp. 583-599 (17 pages) Online Publication Date: June 27, 2011
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We start a systematic parameterized computational complexity study of three NP-hard network design problems on arc-weighted directed graphs: directed Steiner tree, strongly connected Steiner subgraph, and directed Steiner network. We investigate their parameterized complexities with respect to the three parameterizations: “number of terminals,” “an upper bound on the size of the connecting network,” and the combination of these two. We achieve several parameterized hardness results as well as some fixed-parameter tractability results, in this way extending previous results of Feldman and Ruhl [SIAM J. Comput., 36 (2006), pp. 543–561]. |
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SIAM J. Discrete Math. 26, pp. 404-414 (11 pages) Online Publication Date: March 29, 2012
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Monotone monadic strict NP (MMSNP) is a class of computational problems that is closely related to the class of constraint satisfaction problems for constraint languages over finite domains. It is known that one of those classes has a complexity dichotomy if and only if the other class has. Whereas the dichotomy conjecture has been verified for several subclasses of constraint satisfaction problems, little is known about the the computational complexity for subclasses of MMSNP. In this paper we completely classify the complexity of MMSNP for the case where the obstructions are monochromatic and where loops in the input are forbidden. That is, we determine the computational complexity of natural partition problems of the following type. For fixed sets of finite structures ${\cal S}_1, \dots, {\cal S}_k$, decide whether a given loopless structure can be vertex-partitioned into $k$ parts such that for each $i \leq k$ none of the structures in ${\cal S}_i$ is homomorphic to the $i$th part. |
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Packing Tight Hamilton Cycles in Uniform Hypergraphs SIAM J. Discrete Math. 26, pp. 435-451 (17 pages) Online Publication Date: April 04, 2012
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We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\leq\ell\leq k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices, and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}\setminus E_i|=\ell$. We define a class of $(\epsilon,p)$-regular hypergraphs, that includes random hypergraphs, for which we can prove the existence of a decomposition of almost all edges into type $\ell$ Hamilton cycles, where $\ell<k/2$. |
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SIAM J. Discrete Math. 25, pp. 1562-1588 (27 pages) Online Publication Date: November 22, 2011
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Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the types of answers that one can hope to achieve in this setting. |
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On Security of Statistical Databases SIAM J. Discrete Math. 25, pp. 1778-1791 (14 pages) Online Publication Date: December 15, 2011
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A statistical database (SDB) is a database that is used to return statistical information derived from the records to user queries for statistical data analysis. Sometimes, by correlating enough statistics, confidential data (stored in an SDB) about an individual can be inferred. Examples of confidential information stored in an SDB might be salaries or data concerning the medical history of individuals. An important problem is to provide security to SDBs against the disclosure of confidential information. An SDB is said to be secure if no protected data can be inferred from the available queries. One of the security-control methods suggested in the literature consists of query restriction: the security problem is to limit the use of the SDB, introducing a control mechanism, such that no protected data can be obtained from the available queries. Chin and Ozsoyoglu [IEEE Trans. Software Engrg., 8 (1982), pp. 574–582] introduced a control mechanism, called AUDIT EXPERT, where only SUM queries, that is, only certain sums of individual records, are available for the users. This SUM query model leads to several challenging optimization problems. Assume there are $n$ numeric records $\{z_1,\ldots,z_n\}$ stored in database. A natural problem is to maximize the number of answerable SUM queries, that is, the number of subset sums of $\{z_1,\ldots,z_n\}$ (possibly with some additional constraints), that can be returned, such that none of numbers $z_i$ (or sums of subsets of size not exceeding a specified number) can be inferred from these queries. In this paper we give tight bounds for this number under constraints on size and dimension of query subsets. |
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SIAM J. Discrete Math. 26, pp. 239-249 (11 pages) Online Publication Date: February 23, 2012
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We show that for all integers $m \geqslant 4$ there exists a $2m\times 2m\times m$ latin cuboid that cannot be completed to a $2m\times 2m\times 2m$ latin cube. We also show that for all even $m>2$ there exists a $(2m{-}1)\times(2m{-}1)\times(m{-}1)$ latin cuboid that cannot be extended to any $(2m{-}1)\times(2m{-}1)\times m$ latin cuboid. |
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Large $B_d$-Free and Union-free Subfamilies SIAM J. Discrete Math. 26, pp. 71-76 (6 pages) Online Publication Date: February 09, 2012
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For a property $\Gamma$ and a family of sets ${\mathcal F}$, let $f({\mathcal F},\Gamma)$ be the size of the largest subfamily of ${\mathcal F}$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f({\mathcal F},\Gamma)$ over all families of size $m$. A family ${\mathcal F}$ is said to be $B_d$-free if it has no subfamily ${\mathcal F}'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family ${\mathcal F}$ is $a$-union-free if $F_1\cup \dots \cup F_a \neq F_{a+1}$ whenever $F_1,\dots,F_{a+1}$ are distinct sets in ${\mathcal F}$. We verify a conjecture of Erdős and Shelah that $f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$. |
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