Adding Edges for Maximizing Weighted Reachability

• 出版年: Algorithms
• 主要な著者: Federico Corò, Gianlorenzo D'Angelo, Cristina M. Pinotti
• フォーマット: 論文
• 言語: 英語
• 出版事項: MDPI AG 2020-03-01
• 主題: graph augmentation; approximation algorithms; greedy algorithms; submodularity; dag; trees; dynamic programming
• オンライン･アクセス: https://www.mdpi.com/1999-4893/13/3/68

In this paper, we consider the problem of improving the reachability of a graph. We approach the problem from a graph augmentation perspective, in which a limited set size of edges is added to the graph to increase the overall number of reachable nodes. We call this new problem the <i>Maximum Connectivity Improvement</i> (MCI) problem. We first show that, for the purpose of solve solving MCI, we can focus on Directed Acyclic Graphs (DAG) only. We show that approximating the MCI problem on DAG to within any constant factor greater than <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>&#8722;</mo> <mn>1</mn> <mspace width="-1.111pt"></mspace> <mo>/</mo> <mspace width="-0.55542pt"></mspace> <mi>e</mi> </mrow> </semantics> </math> </inline-formula> is <inline-formula> <math display="inline"> <semantics> <mi mathvariant="italic">NP</mi> </semantics> </math> </inline-formula>-hard even if we restrict to graphs with a single source or a single sink, and the problem remains <inline-formula> <math display="inline"> <semantics> <mi mathvariant="italic">NP</mi> </semantics> </math> </inline-formula>-complete if we further restrict to unitary weights. Finally, this paper presents a dynamic programming algorithm for the MCI problem on trees with a single source that produces optimal solutions in polynomial time. Then, we propose two polynomial-time greedy algorithms that guarantee <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>&#8722;</mo> <mn>1</mn> <mspace width="-1.111pt"></mspace> <mo>/</mo> <mspace width="-0.55542pt"></mspace> <mi>e</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-approximation ratio on DAGs with a single source, a single sink or two sources.