Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

Improved Approximation Algorithm for Minimum-Weight (1,m)(1,m)--Connected Dominating Set

The classical minimum connected dominating set (MinCDS) problem aims to find a minimum-size subset of connected nodes in a network such that every other node has at least one neighbor in the subset. This problem is drawing considerable attention in the field of wireless sensor networks because connected dominating sets can serve as virtual backbones of such networks. Considering fault-tolerance, researchers developed the minimum $k$-connected $m$-fold CDS (Min$(k,m)$CDS) problem. Many studies have been conducted on MinCDSs, especially those in unit disk graphs. However, for the minimum-weight CDS (MinWCDS) problem in general graphs, algorithms with guaranteed approximation ratios are rare. Guha and Khuller designed a $(1.35+\varepsilon)\ln n$-approximation algorithm for MinWCDS, where $n$ is the number of nodes. In this paper, we improved the approximation ratio to $2H(\delta_{\max}+m-1)$ for MinW$(1,m)$CDS, where $\delta_{\max}$ is the maximum degree of the graph

Sizes of Minimum Connected Dominating Sets of a Class of Wireless Sensor Networks

We consider an important performance measure of wireless sensor networks, namely, the least number of nodes, N, required to facilitate routing between any pair of nodes, allowing other nodes to remain in sleep mode in order to conserve energy. We derive the expected value and the distribution of N for single dimensional dense networks

Theory and Techniques for Synthesizing a Family of Graph Algorithms

Although Breadth-First Search (BFS) has several advantages over Depth-First Search (DFS) its prohibitive space requirements have meant that algorithm designers often pass it over in favor of DFS. To address this shortcoming, we introduce a theory of Efficient BFS (EBFS) along with a simple recursive program schema for carrying out the search. The theory is based on dominance relations, a long standing technique from the field of search algorithms. We show how the theory can be used to systematically derive solutions to two graph algorithms, namely the Single Source Shortest Path problem and the Minimum Spanning Tree problem. The solutions are found by making small systematic changes to the derivation, revealing the connections between the two problems which are often obscured in textbook presentations of them.Comment: In Proceedings SYNT 2012, arXiv:1207.055

Maximal Clique Enumeration and Related Tools for Microarray Data Analysis

The purpose of this study was to investigate the utility of exact maximal clique enumeration in DNA microarray analysis, to analyze and improve upon existing exact maximal clique enumeration algorithms, and to develop new clique-based algorithms to assist in the analysis as indicated during the course of the study. As a first test, microarray data sets comprised of pre-classified human lung tissue samples were obtained through the Critical Assessment of Microarray Data Analysis (CAMDA) conference. A combination of exact maximal clique enumeration and approximate dominating set was used to attempt to classify the samples. In another test, maximal clique enumeration was used for a priori clustering of microarray data from Mus musculus (mouse). Cliques from this graph, though smaller than the anticipated groups of co-regulated genes, exhibited a high degree of overlap. Many genes within the overlap are either known or suspected to be involved in one or more gene regulatory networks. Experimental tests of four exact maximal clique enumeration algorithms on graphs derived from Mus musculus data normalized by either RMA or MAS 5.0 software were performed. A branch and bound Bron and Kerbosch algorithm was shown to perform the best on the widest range of inputs. A base Bron and Kerbosch algorithm was faster on very sparse graphs, but slowed considerably as edge density increased. Both the Kose and greedy algorithms were significantly slower than both Bron and Kerbosch algorithms on all inputs. Means to improve further the branch and bound Bron and Kerbosch algorithm were then considered. Two preprocessing rules and more exacting bounds were added to the algorithm both together and separately. The low degree preprocessing rule was found to improve performance most consistently, though significant improvement was only observed with the sparsest graphs, where improvement is least necessary. Finally, a first attempt at developing an algorithm that would integrate genes that were likely excluded from a clique as a result of noise into the appropriate group was made. Initial testing of the resulting paraclique algorithm revealed that the algorithm maintains the desired high level of inter-group edge density while expanding the core clique to a more acceptable size. Research in this area is ongoing