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SIAM J. on Discrete Mathematics

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2005

Volume 19, Issue 4, pp. 815-1073


Graph Minors and Reliable Single Message Transmission

Faith Ellen Fich, André Kündgen, Michael J. Pelsmajer, and Radhika Ramamurthi

SIAM J. Discrete Math. 19, pp. 815-847 (33 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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End-to-end communication considers the problem of sending messages from a sender $s$ to a receiver $r$ through an asynchronous, unreliable network, such as the Internet. We consider the problem of transmitting a single message from $s$ to $r$ through a network in which edges may fail and cannot recover. We assume that some $sr$-path survives, but we do not know which path it is. We are concerned with protocols that do not store information at intermediate nodes and that ensure that a message sent by $s$ will be recieved by $r$ (no matter which edges fail) without generating an infinite number of messages.
We explicitly characterize the family of networks for which there is such a protocol using headerless packets. This characterization is given in terms of forbidden rooted minors, which leads to a linear time recognition algorithm for this family of networks. We obtain a similar characterization for the family of networks in which a message can be broadcast from a single vertex $s$ to all other vertices. Finally, we show that there is a forbidden rooted minor characterization for the more general case when a header (containing routing information) of constant length is attached to the message, and we discuss the algorithmic consequences of this characterization.

Chain Decompositions of 4-Connected Graphs

Sean Curran, Orlando Lee, and Xingxing Yu

SIAM J. Discrete Math. 19, pp. 848-880 (33 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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In this paper we give a decomposition of a $4$-connected graph $G$ into nonseparating chains, which is similar to an ear decomposition of a $2$-connected graph. We also give an $O(|V(G)|^2|E(G)|)$ algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to $O(|V(G)|^3)$.This decomposition will be used to find four independent spanning trees in a $4$-connected graph.

Polylogarithmic Additive Inapproximability of the Radio Broadcast Problem

Michael Elkin and Guy Kortsarz

SIAM J. Discrete Math. 19, pp. 881-899 (19 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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The input for the radio broadcast problem is an undirected $n$-vertex graph $G$ and a source node $s$. The goal is to send a message from $s$ to the rest of the vertices in the minimum number of rounds. In a round, a vertex receives the message only if exactly one of its neighbors transmits. The radio broadcast problem admits an $O(\log^2 n)$ approximation [CW-87,KP-04]. [I. Chlamtac and O. Weinstein, in Proceedings of the IEEE INFOCOM, 1987, pp. 874-881; D. Kowalski and A. Pelc, in APPROX-RANDOM, Lecture Notes in Comput. Sci. 3122, Springer, Berlin, 2004, pp. 171-182].
In this paper we consider the additive approximation ratio of the problem. We prove that there exists a constant $c$ so that the problem cannot be approximated within an additive term of $c\log^2 n$, unless $NP\subseteq BTIME(n^{O(\log\log n)})$.

Computing Minimal Triangulations in Time O(nalpha log n) = o(n2.376)

Pinar Heggernes, Jan Arne Telle, and Yngve Villanger

SIAM J. Discrete Math. 19, pp. 900-913 (14 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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The problem of computing minimal triangulations of graphs, also called minimal fill, was introduced and solved in 1976 by Rose, Tarjan, and Lueker [SIAM J. Comput., 5 (1976), pp. 266-283] in time $O(nm)$ and thus $O(n^3)$ for dense graphs. Although the topic has received increasing attention since then and several new results on characterizing and computing minimal triangulations have been presented, this first time bound has remained the best. In this paper we introduce an $O(n^\alpha \log n)$ time algorithm for computing minimal triangulations, where $O(n^\alpha)$ is the time required to multiply two $n \times n$ matrices. The current best known $\alpha$ is less than $2.376$, and thus our result breaks the longstanding asymptotic time complexity bound for this problem. To achieve this result, we introduce and combine several techniques that are new to minimal triangulation algorithms, such as working on the complement of the input graph, graph search for a vertex set $A$ that bounds the size of the connected components when $A$ is removed, and matrix multiplication.

Short Answers to Exponentially Long Questions: Extremal Aspects of Homomorphism Duality

Jaroslav Nesetril and Claude Tardif

SIAM J. Discrete Math. 19, pp. 914-920 (7 pages) | Cited 7 times

Online Publication Date: August 01, 2006

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We prove that there exists a constant $k$ such that for every $n \geq 1$ there exists a directed core graph $H_n$ with at least $2^n$ vertices such that a directed graph $G$ is $H_n$-colorable if and only if every subgraph of $G$ with at most $kn\log(n)$ vertices is $H_n$-colorable. Our examples show that in general the "duals of relational structures" in the sense of [J. Nesetril and C. Tardif, J. Combin. Theory Ser. B, 80 (2000), pp. 80-97] can have superpolynomial size. The construction given in this paper gives a double exponential upper bound for such a construction. Here we improve this to an exponential upper bound.

An Application of Ramsey Theory to Coding for the Optical Channel

Navin Kashyap, Paul H. Siegel, and Alexander Vardy

SIAM J. Discrete Math. 19, pp. 921-937 (17 pages)

Online Publication Date: August 01, 2006

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In this paper, we analyze bi-infinite sequences over the alphabet $\{0,1,\ldots,q-1\}$, for an arbitrary $q \geq 2$, that satisfy the $q$-ary ghost pulse ($q$GP) constraint. A sequence $\x = {(x_k)}_{k \in \Z} \in \{0,1,\ldots,q-1\}^{\Z}$ satisfies the $q$GP constraint if for all $k,l,m \in \Z$ such that $x_k$, $x_l$ and $x_m$ are nonzero and equal, $x_{k+l-m}$ is also nonzero. This constraint arises in the context of coding for communication over a fiber optic medium. We show, using techniques from Ramsey theory, that if $\x$ satisfies the $q$GP constraint, then the set $\supp(\x) = \{l \in \Z:\ x_l \neq 0\}$ is the disjoint union of cosets of some subgroup, $k\Z$, of $\Z$, and a set of zero density. We provide much sharper results in the special cases of $q = 2$ and $q=3$. In the former case, we show that the corresponding binary ghost pulse constraint has zero capacity, and based on our results for the latter case, we conjecture that the capacity of the ternary ghost pulse constraint is also zero.

Dynamic TCP Acknowledgment: Penalizing Long Delays

Susanne Albers and Helge Bals

SIAM J. Discrete Math. 19, pp. 938-951 (14 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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We study the problem of acknowledging a sequence of data packets that are sent across a TCP connection. Previous work on the problem has focused mostly on the objective function that minimizes the sum of the number of acknowledgments sent and on the delays incurred for all of the packets. Dooly, Goldman, and Scott presented a deterministic $2$-competitive online algorithm and showed that this is the best competitiveness of a deterministic strategy. Recently Karlin, Kenyon, and Randall developed a randomized online algorithm that achieves an optimal competitive ratio of $e/(e-1) \approx 1.58$.
In this paper we investigate a new objective function that minimizes the sum of the number of acknowledgments sent and the maximum delay incurred for any of the packets. This function is especially interesting if a TCP connection is used for interactive data transfer between network nodes. The TCP acknowledgment problem with this new objective function is different in structure than the problem with the function considered previously. We develop a deterministic online algorithm that achieves a competitive ratio of $\pi^2/6 \approx 1.644$ and prove that no deterministic algorithm can have a smaller competitiveness. We also study a generalized objective function where delays are taken to the $p$th power for some positive integer $p$. Again we give tight upper and lower bounds on the best possible competitive ratio of deterministic online algorithms. The competitiveness is 1 plus an alternating sum of Riemann's zeta function and tends to 1.5 as $p\rightarrow \infty$. Finally, we consider randomized online algorithms and show that, for our first objective function, no randomized strategy can achieve a competitive ratio smaller than $3/(3 - 2/e)\approx 1.324$. For the generalized objective function we show a lower bound of $2/(2-1/e) \approx 1.225$.

Well-Covered Vector Spaces of Graphs

J. I. Brown and R. J. Nowakowski

SIAM J. Discrete Math. 19, pp. 952-965 (14 pages)

Online Publication Date: August 01, 2006

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For any field ${\bf F}$, the set of all functions $f : V(G) \rightarrow {\bf F}$ whose sum on each maximal independent set is constant forms a vector space over ${\bf F}$. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, such as nonempty circulant graphs of prime order, have bounded dimension.

On the Complexity of Some Enumeration Problems for Matroids

L. Khachiyan, E. Boros, K. Elbassioni, V. Gurvich, and K. Makino

SIAM J. Discrete Math. 19, pp. 966-984 (19 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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Let $M$ be a matroid defined by an independence oracle on ground set $S$, and let $A\subseteq S$. We present an incremental polynomial-time algorithm for enumerating all minimal (maximal) subsets of $S$ which span (do not span) $A$. Special cases of these problems include the generation of bases, circuits, hyperplanes, flats of given rank, circuits through a given element, generalized Steiner trees, and multiway cuts in graphs, as well as some other applications. We also consider some tractable and NP-hard generation problems related to systems of polymatroid inequalities and (generalized) packing and spanning in matroids.

Labelling Cayley Graphs on Abelian Groups

Sanming Zhou

SIAM J. Discrete Math. 19, pp. 985-1003 (19 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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For given integers $j \ge k \ge 1$, an $L(j,k)$-labelling of a graph $\Ga$ is an assignment of labels---nonnegative integers---to the vertices of $\Ga$ such that adjacent vertices receive labels that differ by at least $j$, and vertices distance two apart receive labels that differ by at least $k$. The span of such a labelling is the difference between the largest and the smallest labels used, and the minimum span over all $L(j,k)$-labellings of $\Ga$ is denoted by $\l_{j,k}(\Ga)$. The minimum number of labels needed in an $L(j,k)$-labelling of $\Ga$ is independent of $j$ and $k$, and is denoted by $\mu(\Ga)$. In this paper we introduce a general approach to $L(j,k)$-labelling Cayley graphs $\Ga$ over Abelian groups and deriving upper bounds for $\l_{j,k}(\Ga)$ and $\mu(\Ga)$. Using this approach we obtain upper bounds on $\l_{j,k}(\Ga)$ and $\mu(\Ga)$ for graphs $\Ga$ admitting a vertex-transitive Abelian group of automorphisms. Hypercubes $Q_d$ are examples of such graphs, and as consequences we obtain upper bounds for $\l_{j,k}(Q_d)$ and $\mu(Q_d)$. We also obtain the exact values of $\l_{j,k}(\Ga)$ ($2k \ge j \ge k$) and $\mu(\Ga)$ for some Hamming graphs $\Ga$. The result shows that, under certain arithmetic conditions, these two invariants rely only on $k$ and the orders of the two largest complete graph factors of the Hamming graph.

The Discrete Sine Transform and the Spectrum of the Finite $q$-ary Tree

Fabio Scarabotti

SIAM J. Discrete Math. 19, pp. 1004-1010 (7 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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Recently, He, Liu, and Strang [Stud. Appl. Math., 110 (2003), pp. 123-138] have computed the spectrum of the adjacency matrix of a class of finite trees. In this paper, we propose a different method and apply it to the slightly different class of finite $q$-ary trees.

Improved p-ary Codes and Sequence Families from Galois Rings of Characteristic p2

San Ling and Ferruh Özbudak

SIAM J. Discrete Math. 19, pp. 1011-1028 (18 pages)

Online Publication Date: August 01, 2006

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This paper explores the applications of a recent bound on some Weil-type exponential sums over Galois rings in the construction of codes and sequences. A family of codes over $\F_p$, mostly nonlinear, of length $p^{m+1}$ and size $p^2 \cdot p^{m ( D - \lfloor D/p^2 \rfloor )}$, where $1 \le D \le p^{m/2}$, is obtained. The bound on this type of exponential sums provides a lower bound for the minimum distance of these codes. Several families of pairwise cyclically distinct $p$-ary sequences of period $p(p^m-1)$ of low correlation are also constructed. They compare favorably with certain known $p$-ary sequences of period $p^m -1$. Even in the case $p=2$, one of these families is slightly larger than the family $Q(D)$ in section 8.8 in [T. Helleseth and P. V. Kumar, Handbook of Coding Theory, Vol. 2, North-Holland, 1998, pp. 1765-1853], while they share the same period and the same bound for the maximum nontrivial correlation.

Sparse Distance Preservers and Additive Spanners

Béla Bollobás, Don Coppersmith, and Michael Elkin

SIAM J. Discrete Math. 19, pp. 1029-1055 (27 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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For an unweighted graph $G = (V,E)$, $G' = (V,E')$ is a subgraph if $E' \subseteq E$, and $G' = (V',E',\omega)$ is a Steiner graph if $V \subseteq V'$, and for any pair of vertices $u,w \in V$, the distance between them in $G'$ (denoted $d_{G'}(u,w)$) is at least the distance between them in $G$ (denoted $d_G(u,w)$).
In this paper we introduce the notion of distance preserver. A subgraph (resp., Steiner graph) $G'$ of a graph $G$ is a subgraph (resp., Steiner) $D$-preserver of $G$ if for every pair of vertices $u,w \in V$ with $d_G(u,w) \ge D$, $d_{G'}(u,w) = d_G(u,w)$. We show that any graph (resp., digraph) has a subgraph $D$-preserver with at most $O(n^2/D)$ edges (resp., arcs), and there are graphs and digraphs for which any undirected Steiner $D$-preserver contains $\Omega(n^2/D)$ edges. However, we show that if one allows a directed Steiner (diSteiner) $D$-preserver, then these bounds can be improved. Specifically, we show that for any graph or digraph there exists a diSteiner $D$-preserver with $O({{n^2 \cdot \log D} \over {D \cdot \log n}})$ arcs, and that this result is tight up to a constant factor.
We also study $D$-preserving distance labeling schemes, that are labeling schemes that guarantee precise calculation of distances between pairs of vertices that are at a distance of at least $D$ one from another. We show that there exists a $D$-preserving labeling scheme with labels of size $O({{n} \over {D}} \log^2 n)$, and that labels of size $\Omega({{n} \over {D}} \log D)$ are required for any $D$-preserving labeling scheme.

The Two-Batch Liar Game over an Arbitrary Channel

Ioana Dumitriu and Joel Spencer

SIAM J. Discrete Math. 19, pp. 1056-1064 (9 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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We consider liar games in which player Paul must ask one full batch of questions, receive all answers, and then ask a second and final batch of questions. We show that the effect of this restriction is asymptotically negligible. The strategy for Paul is given explicitly.

Bisubmodular Function Minimization

Satoru Fujishige and Satoru Iwata

SIAM J. Discrete Math. 19, pp. 1065-1073 (9 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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This paper presents the first combinatorial polynomial algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. Since the rank functions of delta-matroids are bisubmodular, the scaling algorithm naturally leads to the first combinatorial polynomial algorithm for testing membership in delta-matroid polyhedra.
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