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

Marx, Dániel; Pilipczuk, Marcin; Pilipczuk, Michał

doi:10.1109/focs.2018.00052

“…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 .…”

Section: Our Resultsmentioning

confidence: 98%

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

Chitnis

¹

,

Feldmann

²

,

Manurangsi

³

2021

ACM Trans. Algorithms

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The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ 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 -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution 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) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-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 .

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“…On the side of upper bounds, the improvement often stems from the fact that planar graphs have (recursive) planar separators of size O( √ n), and the theory of bidimensionality provides an elegant framework for a similar speedup in the parameterized setting for some problems [12]. However, in many cases these algorithms rely on highly problem-specific arguments [6,28,21,30,2,23,14]. The lower bounds are conditional to the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi, and Zane [19] and follow from careful reductions from problems displaying this phenomenon, e.g., Planar 3-Coloring, k-Clique, or Grid Tiling.…”

Section: Introductionmentioning

confidence: 99%

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

Cohen-Addad

¹

,

Verdière

²

,

Marx

³

et al. 2021

J. ACM

3

1

0

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We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

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“…Given the amount of attention the planar version of Steiner-type problems received from the viewpoint of approximation (see, e.g., [2,3,11,26,32]) and the availability of techniques for parameterized algorithms on planar graphs (see, e.g., [6,27,40,50,59]), it is natural to explore SCSS and DSN restricted to planar graphs 1 . In general, one can have the expectation that the problems restricted to planar graphs become easier, but sophisticated techniques might be needed to exploit planarity.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

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

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

Self Cite

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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...

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

Cited by 27 publications

References 39 publications

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

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

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