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• W196127575 abstract "Given an edge-weighted directed graph $$G=(V,E)$$ on $$n$$ vertices and a set $$T={t_1, t_2, ldots t_p}$$ of $$p$$ terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set $$Hsubseteq E$$ of minimum weight such that $$G[H]$$ contains a $$t_{i}rightarrow t_j$$ path for each $$1le ine jle p$$ . The problem is NP-hard, but Feldman and Ruhl [FOCS ’99; SICOMP ’06] gave a novel $$n^{O(p)}$$ algorithm for the $$p$$ -SCSS problem. In this paper, we investigate the computational complexity of a variant of $$2$$ -SCSS where we have demands for the number of paths between each terminal pair. Formally, the $$2$$ -SCSS- $$(k_1, k_2)$$ problem is defined as follows: given an edge-weighted directed graph $$G=(V,E)$$ with weight function $$omega : Erightarrow mathbb {R}_{ge 0}$$ , two terminal vertices $$s, t$$ , and integers $$k_1, k_2$$ ; the objective is to find a set of $$k_1$$ paths $$F_1, F_2, ldots , F_{k_1}$$ from $$sleadsto t$$ and $$k_2$$ paths $$B_1, B_2, ldots , B_{k_2}$$ from $$tleadsto s$$ such that $$sum _{ein E} omega (e)cdot phi (e)$$ is minimized, where $$phi (e)= max Big {|{i : iin [k_1], ein F_i}| ; |{j : jin [k_2], ein B_j}|Big }$$ . For each $$kge 1$$ , we show the following: Our algorithm for $$2$$ -SCSS- $$(k,1)$$ relies on a structural result regarding the optimal solution followed by using the idea of a “token game similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the $$2$$ -SCSS- $$(k_1, k_2)$$ problem if $$min {k_1, k_2}ge 2$$ . Therefore $$2$$ -SCSS- $$(k,1)$$ is the most general problem one can attempt to solve with our techniques. To obtain the lower bound matching the algorithm, we reduce from a special variant of the Grid Tiling problem introduced by Marx [FOCS ’07; ICALP ’12]." @default.
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• W196127575 date "2014-01-01" @default.
• W196127575 modified "2023-09-24" @default.
• W196127575 title "A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands (Extended Abstract)" @default.
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• W196127575 doi "https://doi.org/10.1007/978-3-319-13524-3_14" @default.
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