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Title: On size-constrained minimum s–t cut problems and size-constrained 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 size-constrained minimum s–t cut problems and size-constrained dense subgraph problems. We introduce the minimum s–t cut with at-least-k vertices problem, the minimum s–t cut with at-most-k vertices problem, and the minimum s–t cut with exactly k vertices problem. We prove that they are NP-complete. Thus, they are not polynomially solvable unless P = NP. On the other hand, we also study the densest at-least-k-subgraph problem (DalkS) and the densest at-most-k-subgraph 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 n-1/k-1, where n is the number of vertices in the input graph. The second approximation algorithm for DamkS has an approximation ratio of O (nδ), for some δ < 1/3.
Authors:
 [1] ;  [2] ;  [3] ;  [2] ;  [4]
  1. Guangzhou Univ. (People's Republic of China); Fudan Univ. (People's Republic of China)
  2. North Carolina State Univ., Raleigh, NC (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  3. North Carolina State Univ., Raleigh, NC (United States)
  4. South China Univ. of Technology (People's Republic of China)
Publication Date:
OSTI Identifier:
1331085
Grant/Contract Number:
AC05-00OR22725
Type:
Accepted Manuscript
Journal Name:
Theoretical Computer Science
Additional Journal Information:
Journal Volume: 609; Journal Issue: P2; Journal ID: ISSN 0304-3975
Research Org:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
USDOE
Country of Publication:
United States
Language:
English
Subject:
At-least-k-subgraph problem; At-most-k-subgraph problem; approximation algorithm; The minimum s-t cut with at-least-k vertices problem; The minimum s-t cut with at-most-k vertices problem; The minimum s-t cut with exactly k vertices problem