Relating Partial and Complete Solutions and the Complexity of Computing Smallest Solutions

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism ? between two isomorphic graphs is as hard as computing ? itself.Th is result optimally improves upon a result of Gal et al. We establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.W e also show that computing the lexicographically first four-coloring for planar graphs is ?2p-hard.Th is result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P ? NP. We discuss this application to non-self-reducibility and provide a general related result.

[1]  Richard J. Lipton,et al.  Computing from partial solutions , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[2]  Ker-I Ko.  On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[3]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[4]  Alan L. Selman.  Natural Self-Reducible Sets , 1988, SIAM J. Comput..

[5]  José L. Balcázar.  Self-Reducibility , 1990, J. Comput. Syst. Sci..

[6]  Christos H. Papadimitriou,et al.  On the complexity of unique solutions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[7]  Larry Stockmeyer,et al.  Planar 3-colorability is polynomial complete , 1973, SIGA.

[8]  C. P. Schnorr.  On self-transformable combinatorial problems , 1981 .

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

[10]  Claus-Peter Schnorr,et al.  Optimal Algorithms for Self-Reducible Problems , 1976, ICALP.

[11]  Alan J. Demers,et al.  Some Comments on Functional Self-Reducibility and the NP Hierarchy , 1976 .

[12]  Harald Hempel,et al.  The Operators min and max on the Polynomial Hierarchy , 2000, Int. J. Found. Comput. Sci..

[13]  Mark W. Krentel.  The Complexity of Optimization Problems , 1988, J. Comput. Syst. Sci..

[14]  Heribert Vollmer.  On Different Reducibility Notions for Function Classes , 1994, STACS.

[15]  Edith Hemaspaandra,et al.  P-Selektive Sets and Reducing Search to Decision vs Self-Reducibility , 1996, J. Comput. Syst. Sci..

[16]  Paul Young,et al.  Self-Reducibility: Effects of Internal Structure on Computational Complexity , 1990 .

[17]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[18]  K. Appel,et al.  Every planar map is four colorable. Part II: Reducibility , 1977 .

[19]  Samir Khuller,et al.  Planar Graph Coloring is not Self-Reducible, Assuming P != NP , 1991, Theor. Comput. Sci..

[20]  Klaus W. Wagner.  More Complicated Questions About Maxima and Minima, and Some Closures of NP , 1987, Theor. Comput. Sci..