Shortest (A+B)-path packing via hafnian

Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect (A + B)-path packing problem is: given an undirected graph G and two disjoint node subsets A,B with even cardinalities, find shortest |A|/2 + |B|/2 disjoint paths whose ends are both in A or both in B. Besides its NP-hardness, we prove that this problem can be solved in randomized polynomial time if |A|+ |B| is fixed. Our algorithm basically follows the framework of Björklund and Husfeldt but uses a new technique: computation of hafnian modulo 2k combined with Gallai’s reduction from T -paths to matchings. We also generalize our technique for solving other path packing problems, and discuss its limitation.