On the complexity of the exact weighted independent set problem

Suppose we have a well-solved optimization problem, such as minimum spanning tree, maximum cut in planar graphs, minimum weight perfect matching, or maximum weight independent set in a bipartite graph. How hard is it to determine whether there exists a solution with a given weight ? Papadimitriou and Yannakakis showed in [PAPADIMITRIOU 82] that these so-called exact versions of the above optimization problems are NP-complete when the weights are encoded in binary. The question is then, what happens if the weights are « small », i.e., encoded in unary ? Contrary to the binary case, the answer to this question depends on the problem. – The exact spanning tree problem, and more generally, the exact arborescence problem are solvable in pseudo-polynomial time [BARAHONA 87]. – The exact cut problem is solvable in pseudo-polynomial time for planar and toroidal graphs [BARAHONA 87]. – The exact perfect matching problem is solvable in pseudo-polynomial time for planar graphs [BARAHONA 87], and more generally, for graphs that have a Pfaffian orientation1 (provided one is given). We recall that a matching of a graph G = (V,E) is a set E′ of pairwise non-adjacent edges of G. If |E′| = |V |/2, then E′ is said to be a perfect matching of G. Karzanov [KARZANOV 87] gives a polynomial-time