Tight lower bounds for certain parameterized NP-hard problems

  • Authors:

  • Jianer Chen;Benny Chor;Mike Fellows;Xiuzhen Huang;David Juedes;Iyad A. Kanj;Ge Xia

  • Affiliations:

  • Department of Computer Science, Texas A&M University, College Station, TX;School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel;School of Electrical Engineering and Computer Science, University of Newcastle, University Drive, Callaghan, Australia;Department of Computer Science, Texas A&M University, College Station, TX;School of EE&CS, Ohio University, Athens, OH;School of CTI, DePaul University, Chicago, IL;Department of Computer Science, Lafayette College, Easton, PA and Department of Computer Science, Texas A&M University, College Station, TX

  • Venue:
  • Information and Computation
  • Year:
  • 2005

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Abstract

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time nO(k)mO(1), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t - 1)- st level W[t-1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, hitting set, set cover, and feature set, cannot be solved in time no(k)mO(1), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes WEIGHTED q-SAT (for any fixed q ≥ 2), CLIQUE, INDEPENDENT SET, and DOMINATING SET, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either nkmO(1) or O(nk).