Approximation algorithms for maximally balanced connected graph partition

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc $3$-BGP} admit a $5/4$-approximation and a $3/2$-approximation, respectively; but no approximability result exists for {\sc $k$-BGP} when $k \ge 4$, except a trivial $k$-approximation. In this paper, we present another $3/2$-approximation for our cardinality {\sc $3$-BGP} and then extend it to become a $k/2$-approximation for {\sc $k$-BGP}, for any constant $k \ge 3$. Furthermore, for {\sc $4$-BGP}, we propose an improved $24/13$-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.Comment: 23 pages, 7 figures, accepted for presentation at COCOA 2019 (Xiamen, China

Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs (With erratum)

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP$_2$) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algorithm in time $O(N\log N)$ for finding a minimum weight non-separating path between two given nodes in a grid graph of $N$ nodes with positive weight. This result leads to a 5/4-approximation algorithm for the BCP$_2$ problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP$_2$ problem on grid graphs. Based on the exact algorithm and a rounding technique, we show an approximation scheme, which is a fully polynomial time approximation scheme for fixed number of rows.Comment: With erratu

Exploring neighborhoods in large metagenome assembly graphs reveals hidden sequence diversity

Genomes computationally inferred from large metagenomic data sets are often incomplete and may be missing functionally important content and strain variation. We introduce an information retrieval system for large metagenomic data sets that exploits the sparsity of DNA assembly graphs to efficiently extract subgraphs surround- ing an inferred genome. We apply this system to recover missing content from genome bins and show that substantial genomic se- quence variation is present in a real metagenome. Our software implementation is available at https://github.com/spacegraphcats/ spacegraphcats under the 3-Clause BSD License

Graph Minors and Parameterized Algorithm Design

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique

Approximation hardness of dominating set problems in bounded degree graphs

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree

Approximation hardness of edge dominating set problems

We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than 7/6. The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs