Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)

(see paper for full abstract) Given a vertex-weighted directed graph $G=(V,E)$ and a set $T=\{t_1, t_2, \ldots t_k\}$ of $k$ terminals, the objective of the SCSS problem is to find a vertex set $H\subseteq V$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $i\neq j$. 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: - Our main algorithmic result is a $2^{O(k)}\cdot n^{O(\sqrt{k})}$ algorithm for planar SCSS, which is an improvement of a factor of $O(\sqrt{k})$ in the exponent over the algorithm of Feldman and Ruhl. - Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an $f(k)\cdot n^{o(\sqrt{k})}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. The following additional results put our upper and lower bounds in context: - In general graphs, we cannot hope for such a dramatic improvement over the $n^{O(k)}$ algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. - 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 $f(k)\cdot n^{o(k)}$ time algorithm for DSN on acyclic planar graphs.Comment: To appear in SICOMP. Extended abstract in SODA 2014. This version has a new author (Andreas Emil Feldmann), and the algorithm is faster and considerably simplified as compared to conference versio

A tight lower bound for steiner orientation

In the STEINER ORIENTATION problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs T. The question is whether we can orient the undirected edges in a way such that there is a directed s⇝t path for each terminal pair (s,t)∈T. Arkin and Hassin [DAM’02] showed that the STEINER ORIENTATION problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2 . From the viewpoint of exact algorithms, Cygan, Kortsarz and Nutov [ESA’12, SIDMA’13] designed an XP algorithm running in nO(k) time for all k≥1. Pilipczuk and Wahlström [SODA ’16] showed that the STEINER ORIENTATION problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS’01] the STEINER ORIENTATION problem does not admit an f(k)⋅no(k/logk) algorithm for any computable function f. That is, the nO(k) algorithm of Cygan et al. is almost optimal. In this paper, we give a short and easy proof that the nO(k) algorithm of Cygan et al. is asymptotically optimal, even if the input graph has genus 1. Formally, we show that the STEINER ORIENTATION problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)⋅no(k) time for any function f even if the underlying undirected graph has genus 1. We give a reduction from the GRID TILING problem which has turned out to be very useful in proving W[1]-hardness of several problems on planar graphs. As a result of our work, the main remaining open question is whether STEINER ORIENTATION admits the “square-root phenomenon” on planar graphs (graphs with genus 0): can one obtain an algorithm running in time f(k)⋅nO(k√) for PLANAR STEINER ORIENTATION, or does the lower bound of f(k)⋅no(k) also translate to planar graphs

Tight bounds for planar strongly connected steiner subgraph with fixed number of terminals (and extensions)

Given a vertex-weighted directed graph G = (V,E) and a set T = {t 1 ,t2,...,tk} 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 ti →; tj path for each i ≠ j. The problem is NP-hard, but Feldman and Ruhl (FOCS '99; SICOMP '06) gave a novel nO(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. Our main algorithmic result is a 2 O(k log k ) · n O(√k) algorithm for planar SCSS, which is an improvement of a factor of O(√k) in the exponent over the algorithm of Feldman and Ruhl. Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f(k) · no(√k) algorithm for any com-putable function /, unless the Exponential Time Hypothesis (ETH) fails. The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidth-based techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a grid-like fashion to tightly control the number of terminals in the created instance. The following additional results put our upper and lower bounds in context: Our 2O(k log k) · n o(√k) algorithm for planar directed graphs can be generalized to graphs excluding a fixed minor. In general graphs, we cannot hope for such a dramatic improvement over the nO(k) algorithm of Feldman and Ruhl: Assuming ETH, SCSS in general graphs does not have an f(k)· n o(k/ log k) algorithm for any computable function /. Feldman and Ruhl generalized their nO(k) algorithm to the more general Directed Steiner Forest (DSF) problem; here the task is to find a subgraph of minimum weight such that for every source si, there is a path to the corresponding terminal ti.We show that that, assuming ETH, there is no f(k) · no(k)time algorithm for DSF on acyclic planar graphs. Copyright © 2014 by the Society for Industrial and Applied Mathematics

Complexity of the Steiner Network Problem with Respect to the Number of Terminals

In the Directed Steiner Network problem we are given an arc-weighted digraph $G$, a set of terminals $T \subseteq V(G)$, and an (unweighted) directed request graph $R$ with $V(R)=T$. Our task is to output a subgraph $G' \subseteq G$ of the minimum cost such that there is a directed path from $s$ to $t$ in $G'$ for all $st \in A(R)$. It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time $|V(G)|^{o(|A(R)|)}$ even if $G$ is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time $|V(G)|^{o(|T|)}$ unless ETH fails, there is a significant gap in the complexity with respect to $|T|$ in the exponent. We show that Directed Steiner Network is solvable in time $f(R)\cdot |V(G)|^{O(c_g \cdot |T|)}$, where $c_g$ is a constant depending solely on the genus of $G$ and $f$ is a computable function. We complement this result by showing that there is no $f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)}$ algorithm for any function $f$ for the problem on general graphs, unless ETH fails

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$ demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known that this problem is notoriously hard as there is no $k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter $k$. For the bi-DSN$_\text{Planar}$ problem, the aim is to compute a planar optimum solution $N\subseteq G$ in a bidirected graph $G$, i.e., for every edge $uv$ of $G$ the reverse edge $vu$ exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for $k$. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN$_\text{Planar}$, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network $N\subseteq G$ needs to strongly connect a given set of $k$ terminals. It has been observed before that for SCSS a parameterized $2$-approximation exists when parameterized by $k$ [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for $k$ no parameterized $(2-\varepsilon)$-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for $k$