On sizeconstrained minimum s–t cut problems and sizeconstrained dense subgraph problems
In some application cases, the solutions of combinatorial optimization problems on graphs should satisfy an additional vertex size constraint. In this paper, we consider sizeconstrained minimum s–t cut problems and sizeconstrained dense subgraph problems. We introduce the minimum s–t cut with atleastk vertices problem, the minimum s–t cut with atmostk vertices problem, and the minimum s–t cut with exactly k vertices problem. We prove that they are NPcomplete. Thus, they are not polynomially solvable unless P = NP. On the other hand, we also study the densest atleastksubgraph problem (DalkS) and the densest atmostksubgraph problem (DamkS) introduced by Andersen and Chellapilla [1]. We present a polynomial time algorithm for DalkS when k is bounded by some constant c. We also present two approximation algorithms for DamkS. In conclusion, the first approximation algorithm for DamkS has an approximation ratio of n1/k1, where n is the number of vertices in the input graph. The second approximation algorithm for DamkS has an approximation ratio of O (n^{δ}), for some δ < 1/3.
 Authors:

^{[1]};
^{[2]};
^{[3]};
^{[2]};
^{[4]}
 Guangzhou Univ. (People's Republic of China); Fudan Univ. (People's Republic of China)
 North Carolina State Univ., Raleigh, NC (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 North Carolina State Univ., Raleigh, NC (United States)
 South China Univ. of Technology (People's Republic of China)
 Publication Date:
 OSTI Identifier:
 1331085
 Grant/Contract Number:
 AC0500OR22725
 Type:
 Accepted Manuscript
 Journal Name:
 Theoretical Computer Science
 Additional Journal Information:
 Journal Volume: 609; Journal Issue: P2; Journal ID: ISSN 03043975
 Research Org:
 Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 Atleastksubgraph problem; Atmostksubgraph problem; approximation algorithm; The minimum st cut with atleastk vertices problem; The minimum st cut with atmostk vertices problem; The minimum st cut with exactly k vertices problem