Some simplified NP-complete problems

It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of SIMPLE MAX CUT (MAX CUT with edge weights restricted to value 1), and, as a corollary, the completeness of the OPTIMAL LINEAR ARRANGEMENT problem. We then show that even if the domains of the NODE COVER and DIRECTED HAMILTONIAN PATH problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For GRAPH 3-COLORABILITY, NODE COVER, and UNDIRECTED HAMILTONIAN CIRCUIT, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.

[1]  Brian W. Kernighan,et al.  An efficient heuristic procedure for partitioning graphs , 1970, Bell Syst. Tech. J..

[2]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[3]  Ronald L. Graham,et al.  Bounds on multiprocessing anomalies and related packing algorithms , 1972, AFIPS '72 (Spring).

[4]  Sartaj Sahni.  Some Related Problems from Network Flows, Game Theory and Integer Programming , 1972, SWAT.

[5]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[6]  David S. Johnson,et al.  Fast Allocation Algorithms , 1972, SWAT.

[7]  Amir Pnueli,et al.  Permutation Graphs and Transitive Graphs , 1972, JACM.

[8]  Jeffrey D. Ullman,et al.  Worst-case analysis of memory allocation algorithms , 1972, STOC.

[9]  Fanica Gavril,et al.  Algorithms for a maximum clique and a maximum independent set of a circle graph , 1973, Networks.

[10]  Sartaj Kumar Sahni,et al.  On The Knapsack And Other Computationally Related Problems , 1973 .

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

[12]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[13]  P. Herrmann.  ON REDUCIBILITY AMONG COMBINATORIAL PROBLEMS , 1973 .

[14]  Jeffrey D. Ullman,et al.  Polynomial complete scheduling problems , 1973, SOSP '73.

[15]  Ravi Sethi,et al.  Complete register allocation problems , 1973, SIAM J. Comput..

[16]  D. Adolphson.  Optimal linear-ordering. , 1973 .

[17]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.