On the approximability of the maximum induced matching problem

In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3

Minimizing Flow Time in the Wireless Gathering Problem

We address the problem of efficient data gathering in a wireless network through multi-hop communication. We focus on the objective of minimizing the maximum flow time of a data packet. We prove that no polynomial time algorithm for this problem can have approximation ratio less than \Omega(m^{1/3) when $m$ packets have to be transmitted, unless $P = NP$. We then use resource augmentation to assess the performance of a FIFO-like strategy. We prove that this strategy is 5-speed optimal, i.e., its cost remains within the optimal cost if we allow the algorithm to transmit data at a speed 5 times higher than that of the optimal solution we compare to

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph $G=(U,V,E)$ is convex if the vertices in $V$ can be linearly ordered such that for each vertex $u\in U$, the neighbors of $u$ are consecutive in the ordering of $V$. An induced matching $H$ of $G$ is a matching such that no edge of $E$ connects endpoints of two different edges of $H$. We show that in a convex bipartite graph with $n$ vertices and $m$ weighted edges, an induced matching of maximum total weight can be computed in $O(n+m)$ time. An unweighted convex bipartite graph has a representation of size $O(n)$ that records for each vertex $u\in U$ the first and last neighbor in the ordering of $V$. Given such a compact representation, we compute an induced matching of maximum cardinality in $O(n)$ time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in $O(n)$ time. If no compact representation is given, the cover can be computed in $O(n+m)$ time. All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of $O(n^2)$

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complet

High-Throughput SNP Genotyping by SBE/SBH

Despite much progress over the past decade, current Single Nucleotide Polymorphism (SNP) genotyping technologies still offer an insufficient degree of multiplexing when required to handle user-selected sets of SNPs. In this paper we propose a new genotyping assay architecture combining multiplexed solution-phase single-base extension (SBE) reactions with sequencing by hybridization (SBH) using universal DNA arrays such as all $k$-mer arrays. In addition to PCR amplification of genomic DNA, SNP genotyping using SBE/SBH assays involves the following steps: (1) Synthesizing primers complementing the genomic sequence immediately preceding SNPs of interest; (2) Hybridizing these primers with the genomic DNA; (3) Extending each primer by a single base using polymerase enzyme and dideoxynucleotides labeled with 4 different fluorescent dyes; and finally (4) Hybridizing extended primers to a universal DNA array and determining the identity of the bases that extend each primer by hybridization pattern analysis. Our contributions include a study of multiplexing algorithms for SBE/SBH genotyping assays and preliminary experimental results showing the achievable tradeoffs between the number of array probes and primer length on one hand and the number of SNPs that can be assayed simultaneously on the other. Simulation results on datasets both randomly generated and extracted from the NCBI dbSNP database suggest that the SBE/SBH architecture provides a flexible and cost-effective alternative to genotyping assays currently used in the industry, enabling genotyping of up to hundreds of thousands of user-specified SNPs per assay.Comment: 19 page