Inapproximability of Nondeterministic State and Transition Complexity Assuming P=!NP

Inapproximability results concerning minimization of nondeterministic finite automata relative to given deterministic finite automata were obtained only recently, modulo cryptographic assumptions [4]. Here we give upper and lower bounds on the approximability of this problem utilizing only the common assumption P ≠ NP, in the setup where the input is a finite language specified by a truth table. To this end, we derive an improved inapproximability result for the biclique edge cover problem. The obtained lower bounds on approximability can be sharpened in the case where the input is given as a deterministic finite automaton over a binary alphabet. This settles most of the open problems stated in [4]. Note that the biclique edge cover problem was recently studied by the authors as lower bound method for the nondeterministic state complexity of finite automata [5].

[1]  J. Orlin.  Contentment in graph theory: Covering graphs with cliques , 1977 .

[2]  李幼升,et al.  Ph , 1989 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Jean-Marc Champarnaud,et al.  A maxmin problem on finite automata , 1989, Discret. Appl. Math..

[5]  Georg Schnitger,et al.  Minimizing nfa's and regular expressions , 2007, J. Comput. Syst. Sci..

[6]  Hans Ulrich Simon,et al.  On Approximate Solutions for Combinatorial Optimization Problems , 1990, SIAM J. Discret. Math..

[7]  Markus Holzer,et al.  Finding Lower Bounds for Nondeterministic State Complexity Is Hard , 2006, Developments in Language Theory.

[8]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[9]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[10]  Viggo Kann,et al.  Polynomially Bounded Minimization Problems That Are Hard to Approximate , 1993, Nord. J. Comput..

[11]  Grzegorz Rozenberg,et al.  Developments in Language Theory II , 2002 .

[12]  Tao Jiang,et al.  Minimal NFA Problems are Hard , 1991, SIAM J. Comput..

[13]  Jérôme Amilhastre,et al.  FA Minimisation Heuristics for a Class of Finite Languages , 1999, WIA.

[14]  Markus Holzer,et al.  Computational Complexity of NFA Minimization for Finite and Unary Languages , 2007, LATA.

[15]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[16]  Markus Holzer,et al.  On the average state and transition complexity of finite languages , 2007, Theor. Comput. Sci..