Abstract
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 \(2^k\) 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.
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Acknowledgements
We thank the referees for helpful comments. The work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, 17K00029.
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Hirai, H., Namba, H. Shortest \((A+B)\)-Path Packing Via Hafnian. Algorithmica 80, 2478–2491 (2018). https://doi.org/10.1007/s00453-017-0334-0
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DOI: https://doi.org/10.1007/s00453-017-0334-0