Inapproximability of the Max-cut Problem with Negative Weights

  • Hong, Sung-Pil (Department of Industrial Engineering, Seoul National University)
  • Published : 2008.05.31


We show that when a max-cut problem is allowed native-weight edges, to decide if the problem has a cut of a positive weight is NP-hard. This implies that there is no polynomial time algorithm which guarantees a cut whose objective value is no less than $1/p(<I>)$ times the optimum for any polynomially computable polynomial p, where denotes the encoding length of an instance I.



  1. Goemans, M. X. and D. P. Williamson, "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming," J. ACM 42 (1995) 1115-1145
  2. Matula, D. W. and F. Shahrokhi, Sparsest cuts and bottleneck in graphs, Discrete Appl. Math., 27 (1990) 113-123
  3. McCormick, S. T., M. R. Rao, and G. Rinaldi, Easy and difficult objective functions for max cut, Math. Program, Ser. B 94 (2003) 459-466
  4. Vazirani, V. V., Approximation algorithms, Springer, Berlin, 2001