Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH2k(NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH2(NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner’s conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP. 1

An Improved Exact Algorithm for the Domatic Number Problem

The 3-domatic number problem asks whether a given graph can be partitioned intothree dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695^n (up to polynomial factors). This result improves the previous bound of 2.8805^n, which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Delta(G) >= 5. Our new randomized algorithm employs Schoening's approach to constraint satisfaction problems.Comment: 9 pages, a two-page abstract of this paper is to appear in the Proceedings of the Second IEEE International Conference on Information & Communication Technologies: From Theory to Applications, April 200

Distance domatic numbers for grid graphs

We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.Comment: 27 pages, 12 figure

Subgraph Domatic Problem and Writing Capacity of Memory Devises with Restricted State Transitions

A code design problem for memory devises with restricted state transitions is formulated as a combinatorial optimization problem that is called a subgraph domatic partition (subDP) problem. If any neighbor set of a given state transition graph contains all the colors, then the coloring is said to be valid. The goal of a subDP problem is to find a valid coloring with the largest number of colors for a subgraph of a given directed graph. The number of colors in an optimal valid coloring gives the writing capacity of a given state transition graph. The subDP problems are computationally hard; it is proved to be NP-complete in this paper. One of our main contributions in this paper is to show the asymptotic behavior of the writing capacity $C(G)$ for sequences of dense bidirectional graphs, that is given by C(G)=Omega(n/ln n) where n is the number of nodes. A probabilistic method called Lovasz local lemma (LLL) plays an essential role to derive the asymptotic expression.Comment: 7 page

Bounds on Asymptotic Rate of Capacitive Crosstalk Avoidance Codes for On-chip Buses

In order to prevent the capacitive crosstalk in on-chip buses, several types of capacitive crosstalk avoidance codes have been devised. These codes are designed to prohibit transition patterns prone to the capacity crosstalk from any consecutive two words transmitted to on-chip buses. This paper provides a rigorous analysis on the asymptotic rate of (p,q)-transition free word sequences under the assumption that coding is based on a pair of a stateful encoder and a stateless decoder. The symbols p and q represent k-bit transition patterns that should not be appeared in any consecutive two words at the same adjacent k-bit positions. It is proved that the maximum rate of the sequences equals to the subgraph domatic number of (p,q)-transition free graph. Based on the theoretical results on the subgraph domatic partition problem, a pair of lower and upper bounds on the asymptotic rate is derived. We also present that the asymptotic rate 0.8325 is achievable for the (10,01)-transition free word sequences.Comment: 10 pages, 2 figures, submitted to ISIT 201

Optimally Approximating the Coverage Lifetime of Wireless Sensor Networks

We consider the problem of maximizing the lifetime of coverage (MLCP) of targets in a wireless sensor network with battery-limited sensors. We first show that the MLCP cannot be approximated within a factor less than $\ln n$ by any polynomial time algorithm, where $n$ is the number of targets. This provides closure to the long-standing open problem of showing optimality of previously known $\ln n$ approximation algorithms. We also derive a new $\ln n$ approximation to the MLCP by showing a $\ln n$ approximation to the maximum disjoint set cover problem (DSCP), which has many advantages over previous MLCP algorithms, including an easy extension to the $k$-coverage problem. We then present an improvement (in certain cases) to the $\ln n$ algorithm in terms of a newly defined quantity "expansiveness" of the network. For the special one-dimensional case, where each sensor can monitor a contiguous region of possibly different lengths, we show that the MLCP solution is equal to the DSCP solution, and can be found in polynomial time. Finally, for the special two-dimensional case, where each sensor can monitor a circular area with a given radius around itself, we combine existing results to derive a $1+\epsilon$ approximation algorithm for solving MLCP for any $\epsilon >0$.Comment: submitted to IEEE/ACM Transactions on Networking, 17 page

k-Tuple Restrained Domination in Graphs

For $k \ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \setminus S$ is adjacent to at least $k$ vertices in $S$; that is, $|N_G[v] \cap S| \ge k$ for every vertex $v$ of $G$ where $N_G[v]$ denotes the closed neighborhood of $v$ which consists of $v$ and all neighbors of $v$. A $k$-tuple restrained dominating set of $G$ is a $k$-tuple dominating set $S$ of $G$ with the additional property that every vertex outside $S$ has at least $k$ neighbors outside $S$. The minimum cardinality of a $k$-tuple restrained dominating set of $G$ is the $k$-tuple restrained domination number of $G$. When $k=1$, the $k$-tuple restrained domination number is the well-studied restrained domination number. In this paper, we determine the $k$-tuple restrained domination number of several classes of graphs. Tight upper bounds on the $k$-tuple restrained domination number of a general graph are established. We present basic properties of the $k$-tuple restrained domatic number of a graph which is the maximum number of the classes of a partition of $V(G)$ into $k$-tuple restrained dominating sets of $G$

Contracting Graphs to Split Graphs and Threshold Graphs

We study the parameterized complexity of Split Contraction and Threshold Contraction. In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges. We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input split graph to a threshold graph, is FPT when parameterized by the number of contractions. To give a complete picture, we show that these two problems admit no polynomial kernels unless NP\subseteq coNP/poly.Comment: 14 pages, 4 figure

Cubic Graphs with Total Domatic Number at Least Two

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of $G$. In this paper we would like to characterize the cubic graphs with total domatic number at least two.Comment: 6 pages, 5 figure

On cobweb posets and their combinatorially admissible sequences

The main purpose of this article is to pose three problems which are easy to be formulated in an elementary way. These problems which are specifically important also for the new class of partially ordered sets seem to be not yet solved.Comment: 16 pages, 9 figures, affiliated to The Internet Gian Carlo Rota Polish Seminar: 16 pages, 9 figures, affiliated to The Internet Gian Carlo Rota Polish Seminar http://ii.uwb.edu.pl/akk/sem/sem_rota.ht