Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Abstract

Given a vertex-weighted directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . .t k } of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆ V of minimum weight such that G[H] contains a t i → t j path for each i = j. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel n O(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs.• Feldman and Ruhl generalized their n O(k) algorithm to the more general DIRECTED STEINER NETWORK (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source s i there is a path to the corresponding terminal t i . We show that, assuming ETH, there is no f (k) · n o(k) time algorithm for DSN on acyclic planar graphs.All our lower bounds hold for the edge-unweighted version, while the algorithm works for the more general vertex-(un)weighted version. IntroductionThe STEINER TREE (ST) problem is one of the earliest and most fundamental problems in combinatorial optimization: given an undirected graph G = (V, E) and a set T ⊆ V of terminals, the objective is to find a tree of minimum size which connects all the terminals. The ST problem is believed to have been first formally defined by Gauss in a letter in 1836, and the first combinatorial formulation is attributed independently to Hakimi [43] and Levin [52] in 1971. The ST problem is known to be NP-complete, and was in fact part of Karp's original list [49] of 21 NP-complete problems. In the directed version of the ST problem, called DIRECTED STEINER TREE (DST), we are also given a root vertex r and the objective is to find a minimum size arborescence which connects the root r to each terminal from T . An easy reduction from SET COVER shows that the DST problem is also NP-complete.Steiner-type problems arise in the design of networks. Since many networks are symmetric, the directed versions of Steiner-type problems were mostly of theoretical interest. However, in recent years, it has been observed [65, 67] that the connection cost in various networks such as satellite or radio networks are not symmetric. Therefore, directed graphs form the most suitable model for such networks. In addition, Ramanathan [65] also used the DST problem to find low-cost multicast trees, which have applications in point-to-multipoint communication in high bandwidth networks. We refer the interested reader to Winter [69] for a survey on applications of Steiner problems in networks.In this paper we consider two well-studied Steiner-type problems in directed graphs, namely the STRONGLY CONNECTED STEINER SUBGRAPH and the DIRECTED STEINER NETWORK problems. In the (vertex-unweighted) STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem, given a directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . . ,t k } of k terminals, the objective is to find a set S ⊆ V of minimum size such that G[S] contains a t i → t j path...

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

“…This result is an example of the so-called "square-root phenomenon": planarity often allows runtimes that improve the exponent by a square root factor in terms of the parameter when compared to the general case [28,50,40,47,41,52,55,54,51]. Interestingly though, Chitnis et al [14] show that under ETH, no f (k) • n o(k) time algorithm can compute the optimum solution to DSN planar . Thus assuming a bidirected input graph in Theorem 4 is necessary (under ETH) to obtain a factor of O( √ k) in the exponent of n.…”

“…It stands out that to compute optimum solutions, this theorem rules out runtimes for which the dependence of the exponent of n is o( √ k), while for the general DSN problem, as mentioned above, the both necessary and sufficient dependence of the exponent is linear in k [24,14]. Could it be that bi-DSN Planar is just as hard as DSN when computing optimum solutions?…”

“…It is possible to obtain approximation factors O(n 2/3+ε ) and O(k 1/2+ε ) though [3,9,25]. For settings where the number k of demands is fairly small, one may aim for algorithms that only have a mild exponential runtime blow-up in k, i.e., a runtime of the form f (k) • n O (1) , where f (k) is some function independent of n. If an algorithm computing the optimum solution with such a runtime exists for a computable function f (k), then the problem is called fixed-parameter tractable (FPT) for parameter k. However it is unlikely that DSN is FPT for this well-studied parameter, as it is known to be W [1]-hard [31] for k. In fact one can show [14,22] that under the Exponential Time Hypothesis (ETH) there is no algorithm computing the optimum in time f (k) • n o(k) for any function f (k) independent of n. ETH assumes that there is no 2 o(n) time algorithm to solve 3SAT [33,34]. The best we can hope for is therefore a so-called XP-algorithm computing the optimum in time n O(k) , and this was also shown to exist by Feldman and Ruhl [24].…”

On Subexponential Parameterized Algorithms for Steiner Tree and Directed Subset TSP on Planar Graphs

“…The algorithmic results of this form are thus very problem-specific, exploiting nontrivial observations on the structure of the solution or invoking other tools tailored to the problem's nature. Recent results include algorithms for Subset TSP [29], Multiway Cut [28,33], unweighted Steiner Tree parameterized by the number of edges of the solution [38,39], Strongly Connected Steiner Subgraph, [9], Subgraph Isomorphism [21], facility location problems [35], Odd Cycle Transversal [32], and 3-Coloring parameterized by the number of vertices with degree ≥ 4 [1].…”

“…. , (s k , t k ), find a subgraph of minimum weight that contains an s i → t i path for every i) can be solved in time n O(k) on general graphs, and there is no f (k)n o(k) time algorithm even on planar graphs [9]. However, this problem is not genuinely planar, as the pairs (s i , t i ) can encode arbitrary connections that do not respect planarity.…”