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2012

Volume 26, Issue 2 (partial)


On Optimal Weighted Balanced Clusterings: Gravity Bodies and Power Diagrams

Andreas Brieden and Peter Gritzmann

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.

Packing Tight Hamilton Cycles in Uniform Hypergraphs

Deepak Bal and Alan Frieze

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 $\ell<k/2$.

A Fractional Analogue of Brooks' Theorem

Andrew D. King, Linyuan Lu, and Xing Peng

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 $\chi_f(G)\leq \Delta(G)- \frac{2}{67}$.

Trait-Dependent Extinction Leads to Greater Expected Biodiversity Loss

Beáta Faller and Mike Steel

SIAM J. Discrete Math. 26, pp. 472-481 (10 pages)

Online Publication Date: April 10, 2012

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We use a classical combinatorial inequality to establish a Markov inequality for multivariate binary Markov processes on trees. We then apply this result, alongside the Fortuin–Kasteleyn–Ginibre (FKG) inequality, to compare the expected loss of biodiversity under two models of species extinction. One of these models is the generalized version of an earlier model in which extinction is influenced by some trait that can be classified into two states and which evolves on a tree according to a Markov process. Since more than one trait can affect the rates of species extinction, it is reasonable to allow, in the generalized model, $k$ binary states that influence extinction rates. We compare this model to one that has matching marginal extinction probabilities for each species but for which the species extinction events are stochastically independent.

All Alternating Groups $A_n$ with $n\geq12$ Have Polytopes of Rank $\lfloor\frac{n-1}{2}\rfloor$

Maria Elisa Fernandes, Dimitri Leemans, and Mark Mixer

SIAM J. Discrete Math. 26, pp. 482-498 (17 pages)

Online Publication Date: April 19, 2012

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Using the correspondence between abstract regular polytopes and string C-groups, in a recent paper [M. E. Fernandes, D. Leemans, and M. Mixer, J. Combin. Theory Ser. A, 119 (2012), pp. 42–56], we constructed an abstract regular polytope of rank $r$, for each $r \geq 3$, with automorphism group isomorphic to $A_{2r+3}$ when $r$ is odd, and $A_{2r+1}$ when $r$ is even. In this paper, the remaining cases are completed. It is shown that every group $A_n$, with $n$ sufficiently large, acts on at least one abstract regular polytope of rank $\lfloor\frac{n-1}{2}\rfloor$. We conjecture that this is the highest possible rank for $n\geq 12$.

Linear Time Split Decomposition Revisited

Pierre Charbit, Fabien de Montgolfier, and Mathieu Raffinot

SIAM J. Discrete Math. 26, pp. 499-514 (16 pages)

Online Publication Date: April 26, 2012

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Given a family $\mathcal{F}$ of subsets of a ground set $V$, its orthogonal is defined to be the family of subsets that do not overlap any element of $\mathcal{F}$. Using this tool we revisit the problem of designing a simple linear time algorithm for undirected graph split (also known as 1-join) decomposition.

On the Size of Lattice Simplices with a Single Interior Lattice Point

Gennadiy Averkov

SIAM J. Discrete Math. 26, pp. 515-526 (12 pages)

Online Publication Date: April 26, 2012

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Let $\mathcal{T}^d$ be the set of all $d$-dimensional simplices $T$ in $\mathbb{R}^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d$ is finite up to affine transformations that preserve $\mathbb{Z}^d$. It is known that when $d$ grows, the maximum volume of the simplices $T \in \mathcal{T}^d$ becomes extremely large. We improve and refine bounds on the size of $T \in \mathcal{T}^d$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T \in \mathcal{T}^d$ can be decomposed into an ascending chain of faces $G_1 \subseteq \cdots \subseteq G_d=T$ such that for every $i \in \{1,\ldots,d\}$, $G_i$ is $i$-dimensional and the size of $G_i$ is bounded from above in terms of $i$ and $d$. The bound on the size of $G_i$ is double exponential in $i$. The presented upper bounds are asymptotically tight on the log-log scale.

A $q$-Analogue of the Addressing Problem of Graphs by Graham and Pollak

Saori Watanabe, Kota Ishii, and Masanori Sawa

SIAM J. Discrete Math. 26, pp. 527-536 (10 pages)

Online Publication Date: April 26, 2012

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In this paper we consider a $q$-ary extension of the classical binary addressing problem of graphs which was originally posed by Graham and Pollak [Bell System Tech. J., 50 (1971), pp. 2495–2519]. A lower bound for the minimum length of addressings is presented in terms of eigenvalues of distance matrices. The bound is sharp for complete graphs and $r$-hypertrees but not for the Petersen graph. The determinant of the distance matrices of $r$-hypertrees is explicitly calculated, as a generalization of beautiful theorems by Graham and Pollak and Sivasubramanian [Linear Algegra Appl., 431 (2009), pp. 1234–1248] on trees and $3$-hypertrees. The $q$-ary addressings are applied to the decomposition of the complete graphs. We give an alternative proof of the Liu–Schwenk theorem [Congr. Numer., 81 (1991), pp. 129–142] on the decomposition of the complete graphs into edge-disjoint complete multipartite graphs.

Hitting Times for Random Walks with Restarts

Svante Janson and Yuval Peres

SIAM J. Discrete Math. 26, pp. 537-547 (11 pages)

Online Publication Date: May 03, 2012

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The time it takes a random walker in a lattice to reach the origin from another vertex $x$ has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $\gamma(x,0)$ (minimized over restarting strategies) is finite; it was called the “grade” of $x$ by Dumitriu, Tetali, and Winkler. They showed that in a more general setting, the grade (a variant of the “Gittins index”) plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu, Tetali, and Winkler on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $\gamma(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.

Multidimensional Kruskal–Katona Theorem

Boris Bukh

SIAM J. Discrete Math. 26, pp. 548-554 (7 pages)

Online Publication Date: May 03, 2012

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We present a generalization of a version of the Kruskal–Katona theorem due to Lovász. A shadow of a $d$-tuple $(S_1,\dots,S_d)\in\binom{X}{r}^d$ consists of $d$-tuples $(S_1',\dots,S_d')\in\binom{X}{r-1}^d$ obtained by removing one element from each of the $S_i$. We show that if a family $\mathcal{F}\subset\binom{X}{r}^d$ has size $|\mathcal{F}|=\binom{x}{r}^d$ for a real number $x\geq r$, then the shadow of $\mathcal{F}$ has size at least $\binom{x}{r-1}^d$.

The Incidence Hopf Algebra of Graphs

Brandon Humpert and Jeremy L. Martin

SIAM J. Discrete Math. 26, pp. 555-570 (16 pages)

Online Publication Date: May 03, 2012

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The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs, and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.

Cops and Robber with Constraints

Fedor V. Fomin, Petr A. Golovach, and Paweł Prałat

SIAM J. Discrete Math. 26, pp. 571-590 (20 pages)

Online Publication Date: May 03, 2012

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Cops and robber is a classical pursuit-evasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem's complexity and combinatorial properties with constraining the following natural game parameters: fuel, the number of steps each cop can make; cost, the total sum of steps along edges all cops can make; and time, the number of rounds of the game.

Algorithms for Finding a Maximum Non-k-Linked Graph

Yusuke Kobayashi and Yuichi Yoshida

SIAM J. Discrete Math. 26, pp. 591-604 (14 pages)

Online Publication Date: May 08, 2012

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A graph with at least $2k$ vertices is said to be $k$-linked if for any two ordered $k$-tuples $(s_1, \dots , s_k)$ and $(t_1, \dots , t_k)$ of $2k$ distinct vertices, there exist pairwise vertex-disjoint paths $P_1, \dots , P_k$ such that $P_i$ connects $s_i$ and $t_i$ for $i=1, \dots , k$. For a given graph $G$, we consider the problem of finding a maximum induced subgraph of $G$ that is not $k$-linked. This problem is a common generalization of computing vertex-connectivity and testing $k$-linkedness of $G$, and it is closely related to the concept of $H$-linkedness. In this paper, we give the first polynomial-time algorithm for the case of $k=2$, whereas a similar problem to find a maximum induced subgraph without $2$-vertex-disjoint paths connecting fixed terminal pairs is NP-hard. For the case of general $k$, we give an $(8k-2)$-additive approximation algorithm. We also investigate the computational complexity of the edge-disjoint case and the directed case.

De Bruijn Sequences for Fixed-Weight Binary Strings

Frank Ruskey, Joe Sawada, and Aaron Williams

SIAM J. Discrete Math. 26, pp. 605-617 (13 pages)

Online Publication Date: May 08, 2012

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De Bruijn sequences are circular strings of length $2^n$ whose length $n$ substrings are the binary strings of length $n$. Our focus is on creating circular strings of length $\binom{n}{w}$ for the binary strings of length $n$ with weight (number of $1$s) equal to $w$. In this case, each fixed-weight string can be encoded by its first $n{-}1$ bits since the final bit is redundant. For this reason, we construct circular strings of length $\binom{n-1}{w}+\binom{n-1}{w-1}$ whose length $n{-}1$ substrings are the binary strings of length $n{-}1$ with weight $w$ or $w{-}1$. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of $n$ bits are generated in constant amortized time while using $O(n \log n)$-space. This article's results were also used to create de Bruijn sequences for binary strings of length $n$ with a specified maximum weight.

Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Arnab Bhattacharyya, Elena Grigorescu, Madhav Jha, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff

SIAM J. Discrete Math. 26, pp. 618-646 (29 pages)

Online Publication Date: May 15, 2012

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Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. Transitive-closure spanners are used in access control, property testing and data structures. We show a connection between $2$-TC-spanners and local monotonicity filters. A local monotonicity filter, introduced by Saks and Seshadhri [SIAM J. Comput., pp. 2897–2926], is a randomized algorithm that, given access to an oracle for an almost monotone function $f : \{1,2,\dots,m\}^d \to \mathbb{R}$, can quickly evaluate a related function $g : \{1,2,\dots,m\}^d \to \mathbb{R}$ which is guaranteed to be monotone. Furthermore, the filter can be implemented in a distributed manner. We show that an efficient local monotonicity filter implies a sparse 2-TC-spanner of the directed hypergrid, providing a new technique for proving lower bounds for local monotonicity filters. Our connection is, in fact, more general: an efficient local monotonicity filter for functions on any partially ordered set (poset) implies a sparse 2-TC-spanner of the directed acyclic graph corresponding to the poset. We present nearly tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid. These bounds imply stronger lower bounds for local monotonicity filters that nearly match the upper bounds of Saks and Seshadhri.

Extending Fractional Precolorings

Daniel Král', Matjaž Krnc, Martin Kupec, Borut Lužar, and Jan Volec

SIAM J. Discrete Math. 26, pp. 647-660 (14 pages)

Online Publication Date: May 15, 2012

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For every $d\ge 3$ and $k\in\{2\}\cup[3,\infty)$, we determine the smallest $\varepsilon$ such that every fractional $(k+\varepsilon)$-precoloring of vertices at mutual distance at least $d$ of a graph $G$ with fractional chromatic number equal to $k$ can be extended to a proper fractional $(k+\varepsilon)$-coloring of $G$. Our work complements analogous results of Albertson for ordinary colorings and those of Albertson and West for circular colorings.
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