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2007

Volume 21, Issue 3, pp. 551-821


The Integer Knapsack Cover Polyhedron

Hande Yaman

SIAM J. Discrete Math. 21, pp. 551-572 (22 pages)

Online Publication Date: July 11, 2007

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We study the integer knapsack cover polyhedron which is the convex hull of the set of vectors $x\in \mathbb{Z}_{+}^{n}$ that satisfy $C^{T}x\geq b$, with $C\in \mathbb{Z}_{++}^{n}$ and $b\in \mathbb{Z}_{++}$. We present some general results about the nontrivial facet-defining inequalities. Then we derive specific families of valid inequalities, namely, rounding, residual capacity, and lifted rounding inequalities, and identify cases where they define facets. We also study some known families of valid inequalities called 2-partition inequalities and improve them using sequence-independent lifting.

Recognizing Chordal Probe Graphs and Cycle-Bicolorable Graphs

Anne Berry, Martin Charles Golumbic, and Marina Lipshteyn

SIAM J. Discrete Math. 21, pp. 573-591 (19 pages) | Cited 3 times

Online Publication Date: July 18, 2007

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A graph $G=(V,E)$ is a chordal probe graph if its vertices can be partitioned into two sets, $P$ (probes) and $N$ (non-probes), where $N$ is a stable set and such that $G$ can be extended to a chordal graph by adding edges between non-probes. We give several characterizations of chordal probe graphs, first, in the case of a fixed given partition of the vertices into probes and non-probes, and second, in the more general case where no partition is given. In both of these cases, our results are obtained by introducing new classes, namely, $N$-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. $N$-triangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it $N$-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is $O(|P||E|)$, thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is $O(|E|^2)$.

Prefix Reversals on Binary and Ternary Strings

Cor Hurkens, Leo van Iersel, Judith Keijsper, Steven Kelk, Leen Stougie, and John Tromp

SIAM J. Discrete Math. 21, pp. 592-611 (20 pages) | Cited 1 time

Online Publication Date: July 25, 2007

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Given a permutation $\pi$, the application of prefix reversal $f^{(i)}$ to $\pi$ reverses the order of the first $i$ elements of $\pi$. The problem of sorting by prefix reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Discrete Math., 27 (1979), pp. 47–57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings; and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.

Approximation Algorithms for Network Design with Metric Costs

Joseph Cheriyan and Adrian Vetta

SIAM J. Discrete Math. 21, pp. 612-636 (25 pages) | Cited 3 times

Online Publication Date: July 25, 2007

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We study undirected networks with edge costs that satisfy the triangle inequality. Let $n$ denote the number of nodes. We present an $O(1)$-approximation algorithm for a generalization of the metric-cost subset $k$-node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each $k=1,2,\dots,n-1$, there exists a $k$-node connected graph whose cost is within a factor of ${ 22\/}$ of the cost of any simple $k$-edge connected graph. Based on our $O(1)$-approximation algorithm, we present an $O(\log r_{\max})$-approximation algorithm for the metric-cost node-connectivity survivable network design problem, where $r_{\max}$ denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704–720] recently proved, assuming NP$\nsubseteq\;$DTIME($n^{polylog(n)}$), a hardness-of-approximation lower bound of $2^{\log^{1-\epsilon}n}$ for the subset $k$-node-connectivity problem, where $\epsilon$ denotes a small positive number.

Locating Servers for Reliability and Affine Embeddings

Kenneth A. Berman

SIAM J. Discrete Math. 21, pp. 637-646 (10 pages)

Online Publication Date: July 25, 2007

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Consider the problem of locating servers in a network for the purpose of storing data, performing an application, etc., so that at least one server will be available to clients even if up to $k$ component failures occur throughout the network. Letting $G = (V,E)$ be the graph with vertex set $V$ and edge set $E$ representing the topology of the network, and letting $L \subseteq V$ be a set of potential locations for the servers, a fundamental problem is to determine a minimum-size set $S \subseteq L$ such that the network remains connected to $S$ even if up to $k$ component failures occur throughout the network. We say that such a set $S$ is $k$-fault-tolerant. In this paper we present an algebraic characterization of $k$-fault-tolerant sets in terms of affine embeddings of $G$ in $k$-dimensional Euclidean space. Employing this characterization, we present a polynomial-time Monte Carlo algorithm for computing a minimum-size $k$-fault-tolerant subset $S$ of $L$. In fact, we solve the following more general problem for directed networks: given a digraph $G = (V,E)$ (an undirected graph is equivalent to a symmetric digraph) and a subset $L \subseteq V$, we find a $k$-fault-tolerant subset $S$ of $L$ having minimum cost, where a unary integer cost $c(v)$ is associated with locating a server at vertex $v \in V$.

Discrete Lines and Wandering Paths

A. Vince

SIAM J. Discrete Math. 21, pp. 647-661 (15 pages)

Online Publication Date: August 01, 2007

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The problem of finding an approximation to a geometric line by a discrete line using pixels is ubiquitous in computer graphics applications. We show that this discrete line problem in ${\mathbb R}^{n+1}$, for grids of any shape, is equivalent to a geometry problem in ${\mathbb R}^n$ concerning the minimization of the distance that a certain type of closed polygonal path wanders from the origin. This geometry problem is solved completely in dimension 1 (corresponding to 2-dimensional grids), and two simple and efficient algorithms provide near optimum solutions in higher dimensions.

The Maximum Induced Bipartite Subgraph Problem with Edge Weights

Denis Cornaz and A. Ridha Mahjoub

SIAM J. Discrete Math. 21, pp. 662-675 (14 pages) | Cited 1 time

Online Publication Date: August 29, 2007

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Given a graph $G=(V,E)$ with nonnegative weights on the edges, the maximum induced bipartite subgraph problem (MIBSP) is to find a maximum weight bipartite subgraph $(W,E[W])$ of $G$. Here $E[W]$ is the edge set induced by $W$. An edge subset $F\subseteq E$ is called independent if there is an induced bipartite subgraph of $G$ whose edge set contains $F$. Otherwise, it is called dependent. In this paper we characterize the minimal dependent sets, that is, the dependent sets that are not contained in any other dependent set. Using this, we give an integer linear programming formulation for MIBSP in the natural variable space, based on an associated class of valid inequalities called dependent set inequalities. Moreover, we show that the minimum dependent set problem with nonnegative weights can be reduced to the minimum circuit problem in a directed graph, and can then be solved in polynomial time. This yields a polynomial-time separation algorithm for the dependent set inequalities as well as a polynomial-time cutting plane algorithm for solving the linear relaxation of the problem. We also discuss some polyhedral consequences.

On The Chromatic Number of Geometric Hypergraphs

Shakhar Smorodinsky

SIAM J. Discrete Math. 21, pp. 676-687 (12 pages) | Cited 5 times

Online Publication Date: September 05, 2007

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A finite family $\mathcal{R}$ of simple Jordan regions in the plane defines a hypergraph $H=H(\mathcal{R})$ where the vertex set of $H$ is $\mathcal{R}$ and the hyperedges are all subsets $S \subset \R$ for which there is a point $p$ such that $S = \{r \in \mathcal{R} | p \in r\}$. The chromatic number of $H(\mathcal{R})$ is the minimum number of colors needed to color the members of $\mathcal{R}$ such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs and obtain the following results: (i) Any hypergraph that is induced by a family of $n$ simple Jordan regions such that the maximum union complexity of any $k$ of them (for $1\leq k \leq m$) is bounded by $\mathcal{U}(m)$ and $\frac{\mathcal{U}(m)}{m}$ is a nondecreasing function is $O(\frac{\mathcal{U}(n)}{n})$-colorable. Thus, for example, we prove that any finite family of pseudo-discs can be colored with a constant number of colors. (ii) Any hypergraph induced by a finite family of planar discs is four colorable. This bound is tight. In fact, we prove that this statement is equivalent to the four-color theorem. (iii) Any hypergraph induced by $n$ axis-parallel rectangles is $O(\log n)$-colorable. This bound is asymptotically tight. Our proofs are constructive. Namely, we provide deterministic polynomial-time algorithms for coloring such hypergraphs with only “few” colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs). As an application of (i) and (ii) we obtain simple constructive proofs for the following: (iv) Any set of $n$ Jordan regions with near linear union complexity admits a conflict-free (CF) coloring with polylogarithmic number of colors. (v) Any set of $n$ axis-parallel rectangles admits a CF-coloring with $O(\log^2(n))$ colors.

Labelings of Graphs with Fixed and Variable Edge-Weights

Robert Babilon, Vít Jelínek, Daniel Král', and Pavel Valtr

SIAM J. Discrete Math. 21, pp. 688-706 (19 pages) | Cited 3 times

Online Publication Date: September 12, 2007

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Motivated by $L(p,q)$-labelings of graphs, we introduce a notion of $\lambda$-graphs: a $\lambda$-graph $G$ is a graph with two types of edges: 1-edges and $x$-edges. For a parameter $x\in[0,1]$, a proper labeling of $G$ is a labeling of vertices of $G$ by nonnegative reals such that the labels of the endvertices of a 1-edge differ by at least 1 and the labels of the endvertices of an $x$-edge differ by at least $x$; $\lambda_G(x)$ is the smallest real such that $G$ has a proper labeling by labels from the interval $[0,\lambda_G(x)]$. We study properties of the function $\lambda_G(x)$ for finite and infinite $\lambda$-graphs and establish the following results: if the function $\lambda_G(x)$ is well defined, then it is a piecewise linear function of $x$ with finitely many linear parts. Surprisingly, the set $\Lambda(\alpha,\beta)$ of all functions $\lambda_G$ with $\lambda_G(0)=\alpha$ and $\lambda_G(1)=\beta$ is finite for any $\alpha\le\beta$. We also prove a tight upper bound on the number of segments for finite $\lambda$-graphs $G$ with convex functions $\lambda_G(x)$.

Large Complete Bipartite Subgraphs In Incidence Graphs Of Points And Hyperplanes

Roel Apfelbaum and Micha Sharir

SIAM J. Discrete Math. 21, pp. 707-725 (19 pages)

Online Publication Date: September 12, 2007

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We show that if the number $I$ of incidences between $m$ points and $n$ planes in $\mathbb{R}^3$ is sufficiently large, then the incidence graph (which connects points to their incident planes) contains a large complete bipartite subgraph involving $r$ points and $s$ planes, so that $rs \ge \frac{I^2}{mn} - a(m+n)$, for some constant $a>0$. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only $\frac{I^2}{mn}-\frac{m+n}{16}$ incidences. We also take some steps towards generalizing this result to higher dimensions.

Constructing Finite Field Extensions with Large Order Elements

Qi Cheng

SIAM J. Discrete Math. 21, pp. 726-730 (5 pages) | Cited 1 time

Online Publication Date: September 12, 2007

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In this paper, we present an algorithm that, given a fixed prime power $q$ and a positive integer $N$, finds an integer $n \in [N, 2qN]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order greater than $ 5.8^{n / \log_q n}$, in time polynomial in $N$. We present another algorithm that finds an integer $n \in [N, N+O(N^{0.77})]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order at least $ 5.8^{\sqrt{n}}$, in time polynomial in $N$. Our result is inspired by the recent AKS primality testing algorithm [M. Agrawal, N. Kayal, and N. Saxena, Ann. of Math. (2), 160 (2004), pp. 781–793] and the subsequent improvements [P. Berrizbeitia, Math. Comp., 74 (2005), pp. 2043–2059, Q. Cheng, in Proceedings of the 23rd Annual International Cryptology Conference (CRYPTO 2003), D. Boneh, ed., Lecture Notes in Comput. Sci. 2729, Springer-Verlag, Berlin, 2003, pp. 338–348, D. J. Bernstein, Math. Comp., 76 (2007), pp. 389–403].

A Two-Set Problem on Coloring the Integers

Jeffrey A. Ryan

SIAM J. Discrete Math. 21, pp. 731-736 (6 pages)

Online Publication Date: September 19, 2007

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For positive integers $m,r$ and a system of inequalities $\Re$, define $f(m,r,\Re)$ to be the minimum integer $n$ such that for every coloring of $\{1,2,\ldots,n\}$ with $r$ colors, there exist two monochromatic subsets $X,Y \subseteq [1,n]$ (but not necessarily of the same color) which satisfy: (i) $\Re$, (ii) the largest number in $X$ is less than the smallest number in $Y$, (iii) $|X|=|Y|=m$. Let $L_{X}=-2x_{1}+x_{m-1}+x_{m}$ for $x_{1},x_{m-1},x_{m}\in X$, $L_{Y}=-2y_{1}+y_{m-1}+y_{m}$ for $y_{1},y_{m-1},y_{m}\in Y$, and let $\Re :=L_{X} \leq L_{Y}$. In this paper we prove that $f(m,r,\Re)=5m-3$ and consider the corresponding question for zero-sum sets and generalize our result in the sense of the Erdős–Ginzburg–Ziv theorem.

Crossing Stars in Topological Graphs

Gábor Tardos and Géza Tóth

SIAM J. Discrete Math. 21, pp. 737-749 (13 pages) | Cited 1 time

Online Publication Date: September 26, 2007

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Let $G$ be a graph without loops or multiple edges drawn in the plane. It is shown that, for any $k$, if $G$ has at least $C_k n$ edges and $n$ vertices, then it contains three sets of $k$ edges, such that every edge in any of the sets crosses all edges in the other two sets. Furthermore, two of the three sets can be chosen such that all $k$ edges in the set have a common vertex.

Optimal Lee-Type Local Structures in Cartesian Products of Cycles and Paths

Simon Špacapan

SIAM J. Discrete Math. 21, pp. 750-762 (13 pages) | Cited 1 time

Online Publication Date: September 26, 2007

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We define the neighborhood of an $r$-ball $B(u,r)$ centered on $u\in G$ as the set of all vertices $x$, such that $d(u,x)=r+1$, and denote it by $N(u,r)$. We call a set $X$ of pairwise disjoint $r$-balls an optimal local structure for $B(u,r)$ if $N(u,r)\subset \bigcup X$ and no $r$-ball from $X$ intersects $B(u,r)$. We prove the nonexistence of an optimal local structure in $G=C_{q_1} \square C_{q_2} \square \cdots \square C_{q_n}$ for any $B(u,r)\subset G$, where $n\geq 3,$ $r\geq n$, and $q_i\geq 2r+1$ for $i=1,\ldots,n$. In particular, this confirms the nonexistence of perfect Lee codes with parameters $n\geq 3,$ $e \geq n$, and $q\geq 2e+1$. We also prove that if $q_i$ is even for $i=1,\ldots,n$, $\sum_{i=1}^n q_i/2$ is odd, and $2r+1=\sum_{i=1}^n q_i/2$, then for every $r$-ball in $G$ there is an optimal local structure.

Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

Domingo Gómez, Jaime Gutierrez, and Álvar Ibeas

SIAM J. Discrete Math. 21, pp. 763-784 (22 pages)

Online Publication Date: September 26, 2007

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In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph's node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph's diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gröbner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter $d$, all of them associated to irreducible monomial ideals.

Constructions of Optical Orthogonal Codes from Finite Geometry

T. L. Alderson and Keith E. Mellinger

SIAM J. Discrete Math. 21, pp. 785-793 (9 pages)

Online Publication Date: September 28, 2007

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The link between finite geometry and various classes of error-correcting codes is well known. Arcs in projective spaces, for instance, have a close tie to linear MDS codes as well as the high-performing low-density parity-check codes. In this article, we demonstrate a connection between arcs and optical orthogonal codes (OOCs), a class of nonlinear binary codes used for many modern communication applications. Using arcs and Baer subspaces of finite projective spaces, we construct some infinite classes of OOCs with auto-correlation and cross-correlation both larger than 1.

Avoiding Monochromatic Sequences With Special Gaps

Bruce M. Landman and Aaron Robertson

SIAM J. Discrete Math. 21, pp. 794-801 (8 pages)

Online Publication Date: September 28, 2007

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For $S \subseteq \mathbb{Z}^+$ and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,\dots,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},\dots,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$.

On Rota's Basis Conjecture

Jim Geelen and Kerri Webb

SIAM J. Discrete Math. 21, pp. 802-804 (3 pages) | Cited 3 times

Online Publication Date: October 31, 2007

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Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-$n$ matroid $M$, then there are $n$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of $M$. We prove the weaker result that there are $O(\sqrt n)$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases. We also prove that if $(B_1,\ldots,B_k)$ are disjoint bases of a rank-$n$ matroid with $n> \binom{k+1}{2}$, then there are $n$ disjoint independent transversals of $(B_1,\ldots,B_k)$.

On Extremal $k$-Graphs Without Repeated Copies of 2-Intersecting Edges

Yeow Meng Chee and Alan C. H. Ling

SIAM J. Discrete Math. 21, pp. 805-821 (17 pages)

Online Publication Date: October 31, 2007

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The problem of determining extremal hypergraphs containing at most $r$ isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turán numbers. In this paper, we determine the size of extremal $k$-uniform hypergraphs containing at most one pair of 2-intersecting edges for $k\in\{3,4\}$. We give a complete solution when $k=3$ and an almost complete solution (with eleven exceptions) when $k=4$.
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