Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Dvořák, Pavel; Feldmann, Andreas; Knop, Dušan; Masařík, Tomáš; Toufar, Tomáš; Veselý, Pavel

doi:10.1137/18m1209489

Cited by 13 publications

(9 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the parameterization by the number of Steiner vertices in the optimum solution, a folklore result says that Steiner Tree is W [2]-hard (cf. [22,28]). However a parameterized approximation scheme exists for this parameter [28].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Parameterized Complexity of the Survivable Network Design Problem

Feldmann¹,

Mukherjee²,

Leeuwen³

2021

Preprint

View full text Add to dashboard Cite

For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph G with edge costs, a set R of terminal vertices, and an integer demand ds,t for every terminal pair s, t ∈ R. The task is to compute a subgraph H of G of minimum cost, such that there are at least ds,t disjoint paths between s and t in H. Depending on the type of disjointness we obtain several variants of SNDP that have been widely studied in the literature: if the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals, i.e., they may only share terminals.In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size , the sum of demands D, the number of terminals k, and the maximum demand dmax. Using simple, elegant arguments, we prove the following results.• We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter : both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard (even in the uniform single-source case with k = 3). • We identify some special cases of VC-SNDP that are FPT:when dmax ≤ 3 for parameter , on locally bounded treewidth graphs (e.g., planar graphs) for parameter , and on graphs of treewidth tw for parameter tw + D, which is in contrast to a result by Bateni et al. [JACM 2011] who show NP-hardness for tw = 3 (even if dmax = 1, i.e., the Steiner Forest problem). • The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with dmax = 1 on directed graphs, and is FPT parameterized by k [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where dmax = 2, is W[1]-hard, even when parameterized by (which is always larger than k). * Supported by the Czech Science Foundation GA ČR (grant #19-27871X), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).

show abstract

Section: Introductionmentioning

confidence: 99%

“…[22,28]). However a parameterized approximation scheme exists for this parameter [28]. Similar results have been found for special cases of DSN [19].…”

Section: Introductionmentioning

confidence: 99%

The Parameterized Complexity of the Survivable Network Design Problem

Feldmann¹,

Mukherjee²,

Leeuwen³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Dvorák et.al. [10] gave an efficient parameterized approximation scheme, parameterized by the number of terminals, for constructing a Steiner tree on an undirected weighted graph, and claim that such a scheme is unlikely to exist for directed graphs for the same parameter.…”

Section: Introductionmentioning

confidence: 99%

Parameterized and Approximation Algorithms for the Steiner Arborescence Problem on a Hypercube

Mahapatra¹,

Narayanan²,

Narayanaswamy³

et al. 2021

Preprint

View full text Add to dashboard Cite

We address the problem of computing a Steiner Arborescence on a directed hypercube. The directed hypercube enjoys a special connectivity structure among its node set {0, 1} m , but its exponential in m size renders traditional Steiner tree algorithms inefficient. Even though the problem was known to be NP-complete, parameterized complexity of the problem was unknown. With applications in evolutionary/phylogenetic tree reconstruction algorithms and incremental algorithms for computing a property on multiple input graphs, any algorithm for this problem would open up new ways to study these applications. In this paper, we present the first algorithms, to the best our knowledge, that prove the problem to be fixed parameter tractable (FPT) wrt two natural parameters -number of input terminals and penalty of the arborescence. These parameters along with the special structure of the hypercube offer different trade-offs in terms of running time tractability vs. approximation guarantees that are interestingly additive in nature.Given any directed m-dimensional hypercube Q m , rooted at the zero node 0 m , and a set of input terminals R ⊆ {0, 1} m that needs to be spanned by the Steiner arborescence, we prove that the problem is FPT wrt the penalty parameter q (the extra cost over m incurred by an arborescence), by providing a randomized algorithm that computes an optimal arborescence T in O q 4 4 q(q+1) + q |R| m 2 with probability at least 4 −q . If we trade-off exact solution for an additive approximation one, then we can exploit a connection between the minimum vertex cover of an associated conflict graph and penalty q opt of optimal Steiner arborescence to design a parameterized approximation algorithm with better running time -this

show abstract

“…In their pioneering work Lokshtanov et al showed, inter alia, that various parameterized problems which do not 1 admit "classical" polynomial kernels have α-approximate kernels (also called "lossy kernels") of polynomial size. This line of work has attracted much attention, and finding lossy kernels of polynomial size, especially for problems which don't admit classical kernels of polynomial size, is an important-and challenging-area of current research in parameterized algorithms [13,8,7,9].…”

Section: Introductionmentioning

confidence: 99%

$α$-approximate Reductions: a Novel Source of Heuristics for Better Approximation Algorithms

Manne,

Philip,

Saurabh

et al. 2021

Preprint

View full text Add to dashboard Cite

Lokshtanov et al. [STOC 2017] introduced lossy kernelization as a mathematical framework for quantifying the effectiveness of preprocessing algorithms in preserving approximation ratios. αapproximate reduction rules are a central notion of this framework. We propose that carefully crafted α-approximate reduction rules can yield improved approximation ratios in practice, while being easy to implement as well. This is distinctly different from the (theoretical) purpose for which Lokshtanov et al. designed α-approximate Reduction Rules. As evidence in support of this proposal we present a new 2-approximate reduction rule for the Dominating Set problem. This rule, when combined with an approximation algorithm for Dominating Set, yields significantly better approximation ratios on a variety of benchmark instances as compared to the latter algorithm alone.The central thesis of this work is that α-approximate reduction rules can be used as a tool for designing approximation algorithms which perform better in practice. To the best of our knowledge, ours is the first exploration of the use of α-approximate reduction rules as a design technique for practical approximation algorithms. We believe that this technique could be useful in coming up with improved approximation algorithms for other optimization problems as well.

show abstract

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Cited by 13 publications

References 33 publications

The Parameterized Complexity of the Survivable Network Design Problem

The Parameterized Complexity of the Survivable Network Design Problem

Parameterized and Approximation Algorithms for the Steiner Arborescence Problem on a Hypercube

$α$-approximate Reductions: a Novel Source of Heuristics for Better Approximation Algorithms

Contact Info

Product

Resources

About