eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-11-28
46:1
46:20
10.4230/LIPIcs.ISAAC.2019.46
article
A 21/16-Approximation for the Minimum 3-Path Partition Problem
Chen, Yong
1
Goebel, Randy
2
Su, Bing
3
Tong, Weitian
4
https://orcid.org/0000-0002-9815-2330
Xu, Yao
5
Zhang, An
1
Department of Mathematics, Hangzhou Dianzi University, Hangzhou, Zhejiang, China
Department of Computing Science, University of Alberta, Edmonton, Alberta T6G 2E8, Canada
School of Economics and Management, Xi'an Technological University, Xi'an, Shaanxi, China
Department of Computer Science, Eastern Michigan University, Ypsilanti, Michigan 48197, USA
Department of Computer Science, Kettering University, Flint, Michigan 48504, USA
The minimum k-path partition (Min-k-PP for short) problem targets to partition an input graph into the smallest number of paths, each of which has order at most k. We focus on the special case when k=3. Existing literature mainly concentrates on the exact algorithms for special graphs, such as trees. Because of the challenge of NP-hardness on general graphs, the approximability of the Min-3-PP problem attracts researchers' attention. The first approximation algorithm dates back about 10 years and achieves an approximation ratio of 3/2, which was recently improved to 13/9 and further to 4/3. We investigate the 3/2-approximation algorithm for the Min-3-PP problem and discover several interesting structural properties. Instead of studying the unweighted Min-3-PP problem directly, we design a novel weight schema for l-paths, l in {1, 2, 3}, and investigate the weighted version. A greedy local search algorithm is proposed to generate a heavy path partition. We show the achieved path partition has the least 1-paths, which is also the key ingredient for the algorithms with ratios 13/9 and 4/3. When switching back to the unweighted objective function, we prove the approximation ratio 21/16 via amortized analysis.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol149-isaac2019/LIPIcs.ISAAC.2019.46/LIPIcs.ISAAC.2019.46.pdf
3-path partition
exact set cover
approximation algorithm
local search
amortized analysis