Abstract
A weighted directed graph GQ = (V, E) is defined for Q, the set of problems known to be NP-complete, with a vertex vi in V being an NP-complete problem pi 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 GQR = (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 GQR. GQR 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.
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