TY - JOUR
T1 - Complexity and approximability of the happy set problem
AU - Asahiro, Yuichi
AU - Eto, Hiroshi
AU - Hanaka, Tesshu
AU - Lin, Guohui
AU - Miyano, Eiji
AU - Terabaru, Ippei
N1 - Funding Information:
This work was partially supported by the Natural Sciences and Engineering Research Council of Canada , the Grants-in-Aid for Scientific Research of Japan ( KAKENHI ) Grant Numbers JP17K00016 and JP17K00024 , JP19K21537 , and JST CREST JPMJR1402 .
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/4/18
Y1 - 2021/4/18
N2 - In this paper we study the approximability of the MAXIMUM HAPPY SET problem (MaxHS) and the computational complexity of MaxHS on graph classes: For an undirected graph G=(V,E) and a subset S⊆V of vertices, a vertex v is happy if v and all its neighbors are in S; otherwise unhappy. Given an undirected graph G=(V,E) and an integer k, the goal of MaxHS is to find a subset S⊆V of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a (2Δ+1)-approximation algorithm for MaxHS on graphs with maximum degree Δ. Next, we show that the approximation ratio can be improved to Δ if the maximum degree Δ of the input graph is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to block graphs, or interval graphs. We prove nevertheless that MaxHS on bipartite graphs or on cubic graphs remains NP-hard.
AB - In this paper we study the approximability of the MAXIMUM HAPPY SET problem (MaxHS) and the computational complexity of MaxHS on graph classes: For an undirected graph G=(V,E) and a subset S⊆V of vertices, a vertex v is happy if v and all its neighbors are in S; otherwise unhappy. Given an undirected graph G=(V,E) and an integer k, the goal of MaxHS is to find a subset S⊆V of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a (2Δ+1)-approximation algorithm for MaxHS on graphs with maximum degree Δ. Next, we show that the approximation ratio can be improved to Δ if the maximum degree Δ of the input graph is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to block graphs, or interval graphs. We prove nevertheless that MaxHS on bipartite graphs or on cubic graphs remains NP-hard.
UR - http://www.scopus.com/inward/record.url?scp=85103285503&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85103285503&partnerID=8YFLogxK
U2 - 10.1016/j.tcs.2021.03.023
DO - 10.1016/j.tcs.2021.03.023
M3 - Article
AN - SCOPUS:85103285503
SN - 0304-3975
VL - 866
SP - 123
EP - 144
JO - Theoretical Computer Science
JF - Theoretical Computer Science
ER -