Shortest two disjoint paths in conservative graphs (2307.12602v3)
Abstract: We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown recently by Schlotter and Seb\H{o} (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, there are no cycles in $G$ with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in $G$.
- Irrelevant vertices for the Planar Disjoint Paths problem. Journal of Combinatorial Theory, Series B, 122:815–843, 2017. doi:10.1016/j.jctb.2016.10.001.
- Negative-weight single-source shortest paths in near-linear time. In FOCS 2022: Proceedings of the 63rd Annual IEEE Symposium on Foundations of Computer Science, pages 600–611. IEEE Computer Society, 2022. doi:10.1109/FOCS54457.2022.00063.
- Shortest two disjoint paths in polynomial time. SIAM Journal on Computing, 48(6):1698–1710, 2019. doi:10.1137/18M1223034.
- A procedure for determining a family of miminum-cost network flow patterns. Technical Report ORO-TP-15, Operations Research Office, Johns Hopkins University, 1960.
- The planar directed k𝑘kitalic_k-vertex-disjoint paths problem is fixed-parameter tractable. In FOCS 2013: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pages 197–206. IEEE Computer Society, 2013. doi:10.1109/FOCS.2013.29.
- Shortest vertex-disjoint two-face paths in planar graphs. ACM Trans. Algorithms, 7(2):19:1–19:12, 2011. doi:10.1145/1921659.1921665.
- Disjoint paths in a planar graph – a general theorem. SIAM Journal on Discrete Mathematics, 5(1):112–116, 1992. doi:10.1137/0405009.
- Masao Iri. A new method for solving transportation-network problems. Journal of the Operations Research Society of Japan, 3:27–87, 1960.
- William S. Jewell. Optimal flow through networks. Operations Research, 10(4):476–499, 1962.
- Richard M. Karp. On the computational complexity of combinatorial problems. Networks, 5(1):45–68, 1975.
- On shortest disjoint paths in planar graphs. Discrete Optimization, 7(4):234–245, 2010. doi:10.1016/j.disopt.2010.05.002.
- An exponential time parameterized algorithm for planar disjoint paths. In STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, page 1307–1316. Association for Computing Machinery, 2020. doi:10.1145/3357713.3384250.
- James F. Lynch. The equivalence of theorem proving and the interconnection problem. ACM SIGDA Newsletter, 5:31–36, 1975. doi:10.1145/1061425.1061430.
- Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65–110, 1995. doi:10.1006/jctb.1995.1006.
- Odd paths, cycles and T𝑇Titalic_T-joins: Connections and algorithms, 2022. arXiv:2211.12862.
- Alexander Schrijver. Finding k𝑘kitalic_k disjoint paths in a directed planar graph. SIAM Journal on Computing, 23(4):780–788, 1994. doi:10.1137/S0097539792224061.
- Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, 2003.
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