A graph for NP-complete problems

Abstract

A weighted directed graph G_Q = (V, E) is defined for Q, the set of problems known to be NP-complete, with a vertex v_i in V being an NP-complete problem p_i in Q and the weight of an edge in E being the complexity of the transformation used to prove the NP-completeness of a problem. If the complexity of problem p₁ relative to problem p₂ is defined as the minimum complexity of a reduction from p₁ to p₂ (considering all paths from v₁ to v₂ in V) then the relative complexity graph of Q is the weighted graph G_QR = (V, E'), with the weight of edge e'_ij being the minimum complexity of p₁ relative to p₂. An O(n³) variant of the shortest path problem on directed graphs that is similar to the Floyd algorithm for all-pairs shortest paths is used to construct G_QR. G_QR can be updated with an O(n²) algorithm when a reduction with smaller complexity is found or a new edge is added to the graph, and with an O(n) algorithm if a new vertex (corresponding to the discovery of a new NP-complete problem) is added to the graph.