Showing papers in "SIAM Journal on Discrete Mathematics in 2017"
TL;DR: General bounds for the maximum possible size of a hypergraph with no Berge-$G$ as a subhypergraph are proved and an analogue of the Kovari--Sos--Turan theorem is proved.
Abstract: Let $E(G)$ and $V(G)$ denote the edge set and vertex set of a (hyper)graph $G$. Let $G$ be a graph and $\mathcal{H}$ be a hypergraph. We say that a hypergraph $\mathcal{H}$ is a Berge-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subset f(e)$. This generalizes the established definitions of “Berge path” and “Berge cycle” to general graphs. For a fixed graph $G$ we examine the maximum possible size of a hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove general bounds for this maximum when $G$ is an arbitrary graph. We also consider the specific case when $G$ is a complete bipartite graph and prove an analogue of the Kovari--Sos--Turan theorem. In case $G$ is $C_4$, we improve the bounds given by Gyori and Lemons [Discrete Math., 312, (2012), pp. 1518--1520].
97 citations
TL;DR: A suite of implementations of these algorithms with a ready-to-use, platform-agnostic interface based on Docker containers and the AlgoRun framework are provided, so that interested computational scientists can easily perform similar tests with inputs from their own research areas on their own computers or through a convenient Web interface.
Abstract: Finding inclusion-minimal hitting sets (MHSs) for a given family of sets is a fundamental combinatorial problem with applications in domains as diverse as Boolean algebra, computational biology, and data mining. Although many algorithms are available in the literature to generate these MHSs, application papers typically consider only a few before selecting one (or introducing a novel algorithm), suggesting the need for a comprehensive survey and performance comparison. We introduce several of these applications, discussing how MHS generation is applied in each domain and which algorithms have been used, providing a unified view of these applications for researchers from diverse areas. We survey twenty-one algorithms for MHS generation from across a variety of domains, considering their history, classification, and useful features. We provide the results of a comprehensive suite of benchmarks of public software implementations of seventeen of these algorithms, including six we implemented ourselves in C++,...
65 citations
TL;DR: In this paper, it was shown that for any fixed dense graph and bounded-degree tree on the same number of vertices, a modest random perturbation of the graph will typically contain a copy of the tree.
Abstract: We show that for any fixed dense graph $G$ and bounded-degree tree $T$ on the same number of vertices, a modest random perturbation of $G$ will typically contain a copy of $T$. This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c=c(\alpha,\Delta)$ such that if $G$ is an $n$-vertex graph with minimum degree at least $\alpha n$, and $T$ is an $n$-vertex tree with maximum degree at most $\Delta$, then if we add $cn$ uniformly random edges to $G$, the resulting graph will contain $T$ asymptotically almost surely (as $n\to\infty$). Our proof uses a lemma concerning the decomposition of a dense graph into superregular pairs of comparable sizes, which may be of independent interest.
61 citations
TL;DR: In this paper, the treewidth bound was improved to O(sqrt{(g+1)(k+1)n}) by showing that the number of crossings per edge is at most polylogarithmic.
Abstract: We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$ and that these bounds are tight up to a constant factor. In the special case of $g=0$, so-called $k$-planar graphs, the treewidth bound is $O(\sqrt{(k+1)n})$, which is tight and improves upon a known $O((k+1)^{3/4}n^{1/2})$ bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g
58 citations
TL;DR: In this paper, it was shown that the size-Ramsey number of a monochromatic path of length 2/3-α n can be computed in polynomial time.
Abstract: The size-Ramsey number ${R}{F}$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any coloring of the edges of $G$ with two colors yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le {R}(){P_n} \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, ${R}{P_n} \ge (1+\sqrt{2})n-O(1)$, due to Bollobas, and the upper bound, ${R}(){P_n} \le 91n$, due to Letzter. Next we study long monochromatic paths in an edge-colored random graph $\mathcal{G}(n,p)$ with $pn \to \infty$. Let $\alpha > 0$ be an arbitrarily small constant. Recently, Letzter showed that asymptotically almost surely (a.a.s.) any $2$-edge coloring of $\mathcal{G}(n,p)$ yields a monochromatic path of length $(2/3-\alpha)n$, which is optimal. Extending this result, we show that ...
56 citations
TL;DR: In this article, the maximum number of fixed points in a monotone Boolean network with interaction graph is studied and upper and lower bounds on the number of vertices in the largest sublattice of the network.
Abstract: Given a digraph $G$, much attention has focused on the maximum number $\phi(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound $\phi(G)\leq 2^{\tau}$, where $\tau$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $\phi_m(G)$ of fixed points in a monotone Boolean network with interaction graph $G$. We establish new upper and lower bounds on $\phi_m(G)$ that depend on the cycle structure of $G$. In addition to $\tau$, the involved parameters are the maximum number $
u$ of vertex-disjoint cycles, and the maximum number $
u^*$ of vertex-disjoint cycles verifying some additional technical conditions. We improve the feedback bound $2^\tau$ by proving that $\phi_m(G)$ is at most the largest sublattice of $\{0,1\}^\tau$ without chain of size $
u+2$, and without another forbidden pattern described by two disjoin...
42 citations
TL;DR: This sequel to a paper entitled Variations on the sum-product problem is quantitatively improved as well as generalize a method from it to give a near-optimal bound for a new expander.
Abstract: This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: $\exists a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| \gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$.
40 citations
TL;DR: In this article, the minimum distance of narrow-sense primitive BCH codes with special Bose distance was shown to be Ω(m-2 1/2 ) where m-2 2/2 is the size of the BCH code.
Abstract: Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well studied subclass of BCH codes, there are very few theoretical results on the minimum distance. In this paper, we present new results on the minimum distance of narrow-sense primitive BCH codes with special Bose distance. We prove that for a prime power $q$, the $q$-ary narrow-sense primitive BCH code with length $q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \le m-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$. This is achieved by employing the beautiful theory of sets of quadratic forms, symmetric bilinear forms, and alternating bilinear forms over finite fields, which can be best described using the framework of association schemes.
36 citations
TL;DR: It is proved that Walksat, a popular randomized satisfiability algorithm, fails on random $k$-SAT formulas not very far above clause/variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters.
Abstract: Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random $k$-SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, beyond intuition, there has been scant rigorous evidence that “practical” algorithms are affected by these phase transitions. In this paper we prove that Walksat, a popular randomized satisfiability algorithm, fails on random $k$-SAT formulas not very far above clause/variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters. Specifically, we prove that Walksat is ineffective with high probability (w.h.p.) if $m/n>c2^k\ln^2k/k$, where $m$ is the number of clauses, $n$ is the number of variables, and $c>0$ is an absolute constant. By comparison, Walksat is known to find satisfying assignments in linear time w.h.p. if $m/n 0$ [A. Coja-Oghlan and A. Frieze, SIAM J. Comput., 43 (2...
35 citations
TL;DR: A relaxed version of Neumann-Lara's conjecture that every planar digraph with digirth at least three is 2-colourable is proved.
Abstract: Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colorable.
35 citations
TL;DR: In this article, the authors study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics.
Abstract: We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${CAT}(0)$ metric of Billera--Holmes--Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen--Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.
TL;DR: Two asymptotic upper bounds on CAN$(t,k,v)$ are established that are tighter than the known bounds, and a two-stage bound is derived that employs the Lov‐asz local lemma and the conditional Lov\'aszLocal lemma distribution.
Abstract: Covering arrays find important application in software and hardware interaction testing. For practical applications it is useful to determine or bound the minimum number of rows, $\mathsf{CAN}(t,k,v)$, in a covering array for given values of the parameters $t,k$, and $v$. Asymptotic upper bounds for $\mathsf{CAN}(t,k,v)$ have been established using the Stein--Lovasz--Johnson strategy and the Lovasz local lemma. A series of improvements on these bounds is developed in this paper. First an estimate for the discrete Stein--Lovasz--Johnson bound is derived. Then using alteration, the Stein--Lovasz--Johnson bound is improved upon, leading to a two-stage construction algorithm. Bounds from the Lovasz local lemma are improved upon in a different manner, by examining group actions on the set of symbols. Two asymptotic upper bounds on $\mathsf{CAN}(t,k,v)$ are established that are tighter than the known bounds. A two-stage bound is derived that employs the Lovasz local lemma and the conditional Lovasz local lemma ...
TL;DR: It is shown that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties.
Abstract: In a series of four papers we prove the following relaxation of the Loebl--Komlos--Sos conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac{1}{2}+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemeredi regularity lemma: We decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follo...
TL;DR: The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook lengths as mentioned in this paper.
Abstract: The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In [A. H. Morales, I. Pak, and G. Panova, Hook Formulas for Skew Shapes I. $q$-Analogues and Bijections] we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.
TL;DR: It is proved that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cd...
Abstract: For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a graph with at least two edges, $H$-free Edge Completion is NP-complete if and only if $H$ is a graph with at least two nonedges, and $H$-free Edge Editing is NP-complete if and only if $H$ is a graph with at least three vertices. Our result on $H$-free Edge Editing resolves a conjecture by Alon and Stav [Theoret. Comput. Sci., 2009, pp. 4920--4927]. Additionally, we prove that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cd...
TL;DR: In this article, the authors study properties and invariants of matrix codes endowed with the rank metric and relate them to the covering radius, and give upper bounds on the cover radius of a code by applying different combinatorial methods.
Abstract: In this paper we study properties and invariants of matrix codes endowed with the rank metric and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. The various bounds are then applied to the classes of maximal-rank-distance and quasi-maximal-rank-distance codes.
TL;DR: The notion of resolving sets in a graph was introduced by Slater [Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Util. Math., Winnipeg, 1975, pp. 549--559] and Harary and Melter [Ars Combin, 2 (1976), pp. 191--195] as a way of uniquely identifying every vertex in the graph.
Abstract: The notion of resolving sets in a graph was introduced by Slater [Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Util. Math., Winnipeg, 1975, pp. 549--559] and Harary and Melter [Ars Combin., 2 (1976), pp. 191--195] as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices $x$ and $y$ there is a vertex in the set which has distinct distances to $x$ and $y$. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud [Algorithmica, 2016, pp. 1--31] showed that this problem is NP-complete even on interval graphs. They complemented this ...
TL;DR: This is the third of a series of four papers in which the following relaxation of the Loebl--Komlos--Sos conjecture is proved: for every $\alpha>0$ there exists a number $k_0$ such that for every $k>k-vertex graph $G$ every vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph.
Abstract: This is the third of a series of four papers in which we prove the following relaxation of the Loebl--Komlos--Sos conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the fourth paper, the refined structure will be used for embedding the tree $T$.
TL;DR: In this paper, the authors studied the parameterized complexity of the directed Steiner Tree problem on directed sparse graphs and showed that both the directed and the undirected versions are W2-hard on general graphs and hence unlikely to be fixed parameter tractable.
Abstract: We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W2-hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs.
TL;DR: The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph G, is defined as follows: independent exponential clocks of rate 1 are associated with the vertices of G, one to each vertex.
Abstract: The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph G, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of G, one to each vertex. Initially, one vertex of G knows the rumour. Whenever the clock of a vertex x rings, it calls a random neighbour y: if x knows the rumour and y does not, then x tells y the rumour (a push operation), and if x does not know the rumour and y knows it, y tells x the rumour (a pull operation). The average spread time of G is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of G is the smallest time t such that with probability at least 1 − 1∕n, after time t all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times 1, 2, …, has been studied extensively.
TL;DR: This work discusses several parametrizations of the space of circular planar electrical networks, and shows how to test if a network with n nodes is well-connected by checking that $\binom{n}{2}$ minors of the $n\times n$ response matrix are positive.
Abstract: We discuss several parametrizations of the space of circular planar electrical networks. With any circular planar network we associate a canonical minimal network with the same response matrix, called a “standard” network. The conductances of edges in a standard network can be computed as a biratio of Pfaffians constructed from the response matrix. The conductances serve as coordinates that are compatible with the cell structure of circular planar networks in the sense that one conductance degenerates to $0$ or $\infty$ when moving from a cell to a boundary cell. We also show how to test if a network with $n$ nodes is well-connected by checking that $\binom{n}{2}$ minors of the $n\times n$ response matrix are positive; Colin de Verdiere had previously shown that it was sufficient to check the positivity of exponentially many minors. For standard networks with $m$ edges, positivity of the conductances can be tested by checking the positivity of $m+1$ Pfaffians.
TL;DR: The main result is an $O(n^2)$ algorithm (where n = |A \cup B|$) for the popular matching problem in this model, where each vertex has a preference list ranking its neighbors.
Abstract: We are given a bipartite graph $G = (A \cup B, E)$ where each vertex has a preference list ranking its neighbors: In particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference list of any $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number of vertices that prefer $M$ to $M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is $\mathsf{NP}$-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b$'s list and $b$ desires to be matched to any of them. Our main result is an $O(n^2)$ algorithm (where $n = |A \cup B|$) for the popular matching problem in this model. Note th...
TL;DR: This is the last of a series of four papers in which it is proved that any graph satisfying the conditions of the above approximate version of the Loebl--Komlos--Sos conjecture contains one of ten specific configurations.
Abstract: This is the last of a series of four papers in which we prove the following relaxation of the Loebl--Komlos--Sos conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$, every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$ and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure and proved that any graph satisfying the conditions of the above approximate version of the Loebl--Komlos--Sos conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.
TL;DR: A parameterized approximation algorithm for MMVC and its weighted variant and the approximation ratio of this algorithm cannot be achieved by polynomial-time algorithms unless P = NP, and its running time can be limited by tight conditional lower bounds.
Abstract: The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest studying the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximum Minimal Vertex Cover (MMVC) problem, which has applications to wireless ad hoc networks and whose minimization version, Vertex Cover, is one of the most studied problems in the field of parameterized complexity. We first present tight conditional lower bounds for the running time of any algorithm for MMVC or its weighted variant. Next, we develop a parameterized approximation algorithm for MMVC and its weighted variant. The approximation ratio of this algorithm cannot be achieved by polynomial-time algorithms unless P = NP, and its running...
TL;DR: This work describes the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets, and determines the geometry of Minkowski- and Cayley sums of anti-blocking poly topes.
Abstract: To every poset $P$, Stanley [Discrete Comput. Geom., 1 (1986), pp. 9--23] associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of $P$. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer [J. Combin. Theory Ser. A, 118 (2011), pp. 1322--1333] introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labeled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we c...
TL;DR: A primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities is given, which finds a schedule of cost at most four times the constructed dual solution.
Abstract: We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal and Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the linear program is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a polynomial time $(4+\epsilon )$-app...
TL;DR: In this paper, it was shown that the set of simple cycles on a graph can be viewed as a set of words whose letters, the edges of the graph, obey a specific commutation rule.
Abstract: Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including an immanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function from an abelianization procedure.
TL;DR: It is proved that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general $k-Hypergraph Vertex Cover and $k$-Set Packing, so it is NP-hard to approximate them within a factor of $\Omega (k)$ and $\widetilde \Omega(k)$, respectively.
Abstract: Given an undirected graph $G = (V_G, E_G)$ and a fixed “pattern” graph $H = (V_H, E_H)$ with $k$ vertices, we consider the $H$-Transversal and $H$-Packing problems. The former asks to find the smallest $S \subseteq V_G$ such that the subgraph induced by $V_G \setminus S$ does not have $H$ as a subgraph, and the latter asks to find the maximum number of pairwise disjoint $k$-subsets $S_1, \ldots, S_m \subseteq V_G$ such that the subgraph induced by each $S_i$ has $H$ as a subgraph. We prove that if $H$ is 2-connected, $H$-Transversal and $H$-Packing are almost as hard to approximate as general $k$-Hypergraph Vertex Cover and $k$-Set Packing, so it is NP-hard to approximate them within a factor of $\Omega (k)$ and $\widetilde \Omega (k)$, respectively. We also show that there is a 1-connected $H$ where $H$-Transversal admits an $O(\log k)$-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of $H$-Transversal where $H$ is a (family of) cycles, we m...
TL;DR: In this paper, it was shown that the density of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities of the SNF over the matrix.
Abstract: We show that the density $\mu$ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities $\mu_{p^s}$ of SNF over $\Bbb{Z}/p^s\Bbb{Z}$ with $p$ a prime and $...