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SIAM J. on Discrete Mathematics

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2006

Volume 20, Issue 4, pp. 811-1078


Uniform Formulae for Coefficients of Meromorphic Functions in Two Variables. Part I

Manuel Lladser

SIAM J. Discrete Math. 20, pp. 811-828 (18 pages) | Cited 2 times

Online Publication Date: December 05, 2006

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Uniform asymptotic formulae for arrays of complex numbers of the form $(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r$ and $s$ converge to infinity at a comparable rate. Our analysis is restricted to the case in which the generating function $F(z,w):=\sum f_{r,s} z^r w^s$ is meromorphic in a neighborhood of the origin. We provide uniform asymptotic formulae for the coefficients $f_{r,s}$ along directions in the $(r,s)$‐lattice determined by regular points of the singular variety of $F$. Our main result derives from the analysis of a one dimensional parameter‐varying integral describing the asymptotic behavior of $f_{r,s}$. We specifically consider the case in which the phase term of this integral has a unique stationary point; however, we allow the possibility that one or more stationary points of the amplitude term coalesce with this. Our results find direct application in certain problems associated to the Lagrange inversion formula as well as bivariate generating functions of the form $v(z)/(1-w\cdot u(z))$.

On Minimum Degree Implying That a Graph is H‐Linked

Ronald J. Gould, Alexandr Kostochka, and Gexin Yu

SIAM J. Discrete Math. 20, pp. 829-840 (12 pages) | Cited 8 times

Online Publication Date: December 05, 2006

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Given a fixed multigraph $H$, possibly containing loops, with $V(H) = \{h_1,\ldots,h_m\}$, we say that a graph $G$ is $H$‐linked if for every choice of $m$ vertices $v_1,\ldots,v_m$ in $G$, there exists a subdivision of $H$ in $G$ such that $v_i$ is the branch vertex representing $h_i$ (for all $i$). This generalizes the concept of $k$‐linked graphs (as well as a number of other well‐known path or cycle properties). In this paper we determine a sharp lower bound on $\delta(G)$ (which depends upon $H$) such that each graph $G$ on at least $10(|V(H)|+|E(H)|)$ vertices satisfying this bound is $H$‐linked.

Optimal Interleaving on Tori

Anxiao (Andrew) Jiang, Matthew Cook, and Jehoshua Bruck

SIAM J. Discrete Math. 20, pp. 841-879 (39 pages)

Online Publication Date: December 05, 2006

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This paper studies $t$‐interleaving on two‐dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A $t$‐interleaving of a graph is defined as a vertex coloring in which any connected subgraph of $t$ or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly $t$‐interleaved if its $t$‐interleaving number (the minimum number of colors needed for a $t$‐interleaving) meets the sphere‐packing lower bound, $\lceil t^2/2 \rceil$. We show that a torus is perfectly $t$‐interleavable if and only if its dimensions are both multiples of $\frac{t^2+1}{2}$ (if $t$ is odd) or $t$ (if $t$ is even). The next natural question is how much bigger the $t$‐interleaving number is for those tori that are not perfectly $t$‐interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its $t$‐interleaving number is at most just one more than the sphere‐packing lower bound. We also obtain bounds on $t$‐interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of $t$‐interleaving numbers for two‐dimensional tori. Each of our upper bounds is accompanied by an efficient $t$‐interleaving scheme that constructively achieves the bound.

On Budgeted Optimization Problems

Alpár Jüttner

SIAM J. Discrete Math. 20, pp. 880-892 (13 pages) | Cited 1 time

Online Publication Date: December 05, 2006

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In this paper we give a method for solving certain budgeted optimization problems in strongly polynomial time. The method can be applied to several known budgeted problems, and in addition we show two new applications. The first one extends Frederickson’s and Solis‐Oba’s result [G. N. Frederickson and R. Solis‐Oba, Combinatorica, 18 (1998), pp. 503–518] to (poly)matroid intersections from single matroids. The second one is the budgeted version of the minimum cost circulation problem.

0, 1/2‐Cuts and the Linear Ordering Problem: Surfaces That Define Facets

Samuel Fiorini

SIAM J. Discrete Math. 20, pp. 893-912 (20 pages) | Cited 3 times

Online Publication Date: December 05, 2006

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We find new facet‐defining inequalities for the linear ordering polytope generalizing the well‐known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as $\{0,\frac{1}{2}\}$‐cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have the same “shape” as the projective plane. Inspired by the classification of surfaces, a classic result in topology, we prove that a surface has facet‐defining $\{0,\frac{1}{2}\}$‐cuts of the same “shape” if and only if it is nonorientable.

Mod‐2 Cuts Generation Yields the Convex Hull of Bounded Integer Feasible Sets

C. Gentile, P. Ventura, and R. Weismantel

SIAM J. Discrete Math. 20, pp. 913-919 (7 pages) | Cited 1 time

Online Publication Date: December 05, 2006

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This paper focuses on the outer description of the convex hull of all integer solutions to a given system of linear inequalities. It is shown that if the given system contains lower and upper bounds for the variables, then the convex hull can be produced by iteratively generating so‐called mod‐2 cuts only. This fact is surprising and might even be counterintuitive, since many integer rounding cuts exist that are not mod‐2, i.e., representable as the $\{0,\frac{1}{2}\}$ combination of the given constraint system. The key, however, is that in general many more rounds of mod‐2 cut generation are necessary to produce the final description than in the traditional integer rounding procedure.

Factoring Finite Abelian Groups by Subsets with Maximal Span

Sándor Szabó

SIAM J. Discrete Math. 20, pp. 920-931 (12 pages)

Online Publication Date: December 05, 2006

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We say that a finite abelian group has the Rédei property if it does not admit factorization into two normalized subsets that both span the whole group. It will be shown that subgroups inherit the Rédei property from the group. Then four constructions are described to exhibit groups without the Rédei property. Using these we further narrow the list of $p$‐groups that might have the Rédei property.

Computing the Tutte Polynomial on Graphs of Bounded Clique‐Width

Omer Giménez, Petr Hliněný, and Marc Noy

SIAM J. Discrete Math. 20, pp. 932-946 (15 pages) | Cited 1 time

Online Publication Date: December 11, 2006

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The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree‐width. The notion of clique‐width extends the definition of cographs (graphs without induced $P_4$), and it is a more general notion than that of tree‐width. We show a subexponential algorithm (running in time $\exp{O(n^{1-\varepsilon})}\,$) for computing the Tutte polynomial on graphs of bounded clique‐width. In fact, our algorithm computes the more general $U$‐polynomial.

Multivariable Codes Over Finite Chain Rings: Serial Codes

E. Martínez‐Moro and I. F. Rúa

SIAM J. Discrete Math. 20, pp. 947-959 (13 pages) | Cited 2 times

Online Publication Date: December 11, 2006

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The structure of multivariate serial codes over a finite chain ring $R$ is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over $R$. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self‐dual codes over $R$ are studied.
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On Identities Concerning the Numbers of Crossings and Nestings of Two Edges in Matchings

Martin Klazar

SIAM J. Discrete Math. 20, pp. 960-976 (17 pages) | Cited 4 times

Online Publication Date: December 11, 2006

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Let $M,N$ be two matchings on $[2n]$ (possibly $M=N$). For an integer $l\ge 0$, let ${\cal T}(M,l)$ be the set of those matchings on $[2n+2l]$ which can be obtained from $M$ by successively adding $l$ times the first edge, and similarly for ${\cal T}(N,l)$. Let $s,t\in\{cr,ne\}$, where $cr$ is the statistic of the number of crossings in a matching and $ne$ is the statistic of the number of nestings (possibly $s=t$). We prove that if the statistics $s$ and $t$ coincide, respectively, on the sets of matchings ${\cal T}(M,l)$ and ${\cal T}(N,l)$ for $l=0,1$, then they coincide on these sets for every $l\ge 0$; similar identities hold for the joint statistic of $cr$ and $ne$. These results are instances of a general identity for crossings and nestings weighted by elements from an abelian group.
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On cost matrices with two and three distinct values of Hamiltonian paths and cycles

Santosh N. Kabadi and Abraham P. Punnen

SIAM J. Discrete Math. 20, pp. 977-998 (22 pages) | Cited 2 times

Online Publication Date: December 11, 2006

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A polynomial time testable characterization of cost matrices associated with a complete digraph on $n$ nodes such that all the Hamiltonian cycles (tours) have the same cost is well known. Tarasov [U.S.S.R. Comput. Maths. Math. Phys., 21 (1981), pp. 167–174.] obtained a characterization of cost matrices where tour costs take two distinct values. We provide a simple alternative characterization of such cost matrices, which can be tested in $O(n^2)$ time. We also provide analogous results where tours are replaced by Hamiltonian paths. When the cost matrix is skew‐symmetric, we provide polynomial time testable characterizations such that the tour costs take three distinct values. Corresponding results for the case of Hamiltonian paths are also given. Using these results, special instances of the asymmetric traveling salesman problem (ATSP) are identified that are solvable in polynomial time and that have improved constant factor approximation schemes. In particular, we observe that the 3/2 performance guarantee of the Christofides algorithm extends to all metric Hamiltonian symmetric matrices. Further, we identify special classes of ATSP for which polynomial $\epsilon$‐approximation algorithms are available for $\epsilon \in \{3/2, 4/3, 4\tau, \frac{3\tau^2}{2}, \frac{4+\delta}{3}\}$, where $\tau > 1/2$ and $\delta \geq 0$ are constants.

Toric Surface Codes and Minkowski Sums

John Little and Hal Schenck

SIAM J. Discrete Math. 20, pp. 999-1014 (16 pages) | Cited 6 times

Online Publication Date: December 11, 2006

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Toric codes are evaluation codes obtained from an integral convex polytope $P \subset {\mathbb R}^n$ and finite field ${\mathbb F}_q$. They are, in a sense, a natural extension of Reed–Solomon codes, and have been studied recently in [V. Diaz, C. Guevara, and M. Vath, Proceedings of Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289–300; Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, pp. 132–142], and [D. Joyner, Appl. Algebra Engrg. Comm. Comput., 15 (2004), pp. 63–79]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds in Hansen’s work and empirical results of Joyner; they also apply to previously unknown cases.

Online Bin Packing with Cardinality Constraints

Leah Epstein

SIAM J. Discrete Math. 20, pp. 1015-1030 (16 pages) | Cited 1 time

Online Publication Date: December 15, 2006

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We consider a one‐dimensional storage system where each container can store a bounded amount of capacity as well as a bounded number of items $k\geq 2$. This defines the (standard) bin packing problem with cardinality constraints, which is an important version of bin packing. Following previous work on the unbounded space online problem, we establish the exact best competitive ratio for bounded space online algorithms for every value of $k$. This competitive ratio is a strictly increasing function of $k$ which tends to $\Pi_\infty+1\approx 2.69103$ for large $k$. Lee and Lee showed in 1985 [J. ACM, 32 (1985), pp. 562–572] that the best possible competitive ratio for online bounded space algorithms for the classical bin packing problem is the sum of a series, and tends to $\Pi_\infty$ as the allowed space (number of open bins) tends to infinity. We further design optimal online bounded space algorithms for variable sized bin packing, where each allowed bin size may have a distinct cardinality constraint, and for the resource augmentation model. All algorithms achieve the exact best possible competitive ratio possible for the given problem and use constant numbers of open bins. Finally, we introduce unbounded space online algorithms with smaller competitive ratios than the previously known best algorithms for small values of $k$, for the standard cardinality constrained problem. These are the first algorithms with competitive ratio below 2 for $k=4,5,6$.

Set Systems with No Singleton Intersection

Peter Keevash, Dhruv Mubayi, and Richard M. Wilson

SIAM J. Discrete Math. 20, pp. 1031-1041 (11 pages)

Online Publication Date: December 15, 2006

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Let $\mathcal{F}$ be a $k$‐uniform set system defined on a ground set of size $n$ with no singleton intersection; i.e., no pair $A,B\in\mathcal{F}$ has $|A\cap B|=1$. Frankl showed that $|\mathcal{F}|\leq\binom{n-2}{k-2}$ for $k\geq4$ and $n$ sufficiently large, confirming a conjecture of Erdős and Sós. We determine the maximum size of $\mathcal{F}$ for $k=4$ and all $n$, and also establish a stability result for general $k$, showing that any $\mathcal{F}$ with size asymptotic to that of the best construction must be structurally similar to it.

Rota’s Basis Conjecture for Paving Matroids

Jim Geelen and Peter J. Humphries

SIAM J. Discrete Math. 20, pp. 1042-1045 (4 pages) | Cited 2 times

Online Publication Date: December 15, 2006

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Rota conjectured that, given $n$ disjoint bases of a rank‐n matroid M, there are n disjoint transversals of these bases that are all bases of M. We prove a stronger statement for the class of paving matroids.

Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes

Carles Padró and Ignacio Gracia

SIAM J. Discrete Math. 20, pp. 1046-1055 (10 pages)

Online Publication Date: December 15, 2006

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The matroid associated with a linear code is the representable matroid that is defined by the columns of any generator matrix. The matroid associated with a self‐dual code is identically self‐dual, but it is not known whether every identically self‐dual representable matroid can be represented by a self‐dual code. This open problem was proposed in [R. Cramer et al., Advances in Cryptology, Lecture Notes in Comput. Sci. 3621, Springer, New York, 2005, pp. 327–343], where it was proved to be equivalent to an open problem on the complexity of multiplicative linear secret sharing schemes. Some contributions to its solution are given in this paper. A new family of identically self‐dual matroids that can be represented by self‐dual codes is presented. Additionally, we prove that every identically self‐dual matroid on at most eight points is representable by a self‐dual code.

Higher‐Dimensional Packing with Order Constraints

Sándor P. Fekete, Ekkehard Köhler, and Jürgen Teich

SIAM J. Discrete Math. 20, pp. 1056-1078 (23 pages) | Cited 2 times

Online Publication Date: December 26, 2006

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We present a first exact study on higher‐dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higher‐dimensional generalizations of scheduling problems. Using graph‐theoretic structures to describe feasible solutions, we develop a novel exact branch‐and‐bound algorithm. This extends previous work by Fekete and Schepers; a key tool is a new order‐theoretic characterization of feasible extensions of a partial order to a given complementarity graph that is tailor‐made for use in a branch‐and‐bound environment. The usefulness of our approach is validated by computational results.
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