The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

Bevern, René van; Fluschnik, Till; Mertzios, George B.; Molter, Hendrik; Sorge, Manuel; Suchý, Ondřej

doi:10.1016/j.disopt.2018.05.002

Cited by 9 publications

(17 citation statements)

References 39 publications

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“…Proposition 1. Let G = (R ⊎ B, E) be a graph with two distinct vertices s, t ∈ B, k ≥ 2, l ≥ 0 be two integers, and 1) , we can decide whether there is a k-short ℓ-secluded or ℓ-unsecluded st-path only containing vertices in B.…”

Section: Upper Boundsmentioning

confidence: 99%

“…The open neighborhood of a path consists of all vertices that are not contained in the path but adjacent to at least one vertex in the path. Formally, we study the following 2 × 2 problems: Overview of our results: NP-c., W [1]/W [2]-h., p-NP-h., noPK abbreviate NPcomplete, W [1]/W [2]-hard, para-NP-hard, no polynomial kernel, respectively. a (even on planar graphs, Thm.…”

Section: Introductionmentioning

confidence: 99%

“…Fomin et al [9], building upon the work of Chechik et al [4], studied the parameterized complexity of Secluded Path (in its weighted version). They prove Secluded Path to be W [1]-hard when parameterized by the length of the path (which refers to the value k − 1 in SSP). Moreover, they prove Secluded Path to be in FPT when parameterized by the size of the closed neighborhood of the path (which refers to the value k + ℓ in SSP), but admits no polynomial kernel when parameterized by the combined parameter size of the closed neighborhood of the path, treewidth and maximum degree of the underlying graph, unless coNP ⊆ NP / poly.…”

Section: Introductionmentioning

confidence: 99%

“…Van Bevern et al [1] studied the problems of finding st-separators with small closed neighborhood ("secluded") and of finding small st-separators with small open neighborhood ("small secluded"). They motivated to distinguish between the size of the subgraph in question and the size of the open neighborhood.…”

Section: Introductionmentioning

confidence: 99%

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On the computational complexity of length- and neighborhood-constrained path problems

Luckow

Fluschnik

2020

Information Processing Letters

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Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident with the path. Herein, we consider short and long paths on the one side, and small and large neighborhoods on the other side-yielding four decision problems. We show that all four problems are NP-complete, even in planar graphs with small maximum degree. Moreover, we study all four variants when parameterized by a bound k on the length of the path, by a bound ℓ on the size of neighborhood, and by k + ℓ.

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Section: Upper Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the computational complexity of length- and neighborhood-constrained path problems

Luckow

Fluschnik

2020

Information Processing Letters

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“…Van Bevern et al [4] studied several classical graph optimization problems in both the "secluded" (small closed neighborhood) and the "small secluded" (small set with small open neighborhood) variants. Amongst others, they prove that while finding a secluded s-t separator with small closed neighborhood remains solvable in polynomial time, finding a small secluded s-t separator is NP-complete.…”

Section: Related Workmentioning

confidence: 99%

Parameterized algorithms and data reduction for the short secluded s‐t‐path problem

2019

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Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, , the SHORT SECLUDED PATH problem is to find a simple s-t-path with at most k vertices and neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and . We also obtain a 2 O(tw) ⋅ 2 ⋅ n-time algorithm for n-vertex graphs of treewidth tw, which yields subexponential-time algorithms in several graph classes. ]. This version contains full proof details, new kernelization results with respect to the feedback vertex number as parameter, and the algorithm for graphs of bounded treewidth has been generalized to a more general problem variant and accelerated. Networks. 2020;75:34-63. wileyonlinelibrary.com/journal/net © 2019 Wiley Periodicals, Inc. 34 VAN BEVERN ET AL. 35 TABLE 1 Overview of our results par. Positive results Negative results vc Size vc O(r) -kernel in K r,r -subgraph-free graphs (Theorem 3.8) No polynomial kernel and WK[1]-hard w.r.t. vc (Theorem 3.1) fes Size poly(fes)-kernel (Theorem 5.15) fvs O(fvs ⋅ (k + ) 2 )-vertex kernel (Theorem 5.4) No kernel with size poly(fvs + ) (Theorem 5.20) tw 2 O(tw) ⋅ 2 ⋅ n-time algorithm (Theorem 4.2) No kernel with size poly(tw + k + ) even in planar graphs with const. Δ (Theorem 4.14)Herein, n, tw, vc, fes, fvs, and Δ denote the number of vertices, treewidth, vertex cover number, feedback edge number, feedback vertex number, and maximum degree of the input graph, respectively. FIGURE 1Overview on the existence of polynomial kernelization. Gray: no polynomial-size kernel unless coNP ⊆ NP/poly. White: polynomial-size kernel exists. An arrow from parameter p to p ′ means that the value of p can be upper-bounded by a polynomial in p ′ [25]. Thus, hardness results for p ′ also hold for p and polynomial-size kernels for p also hold for p ′ call a problem fixed-parameter tractable if it can be solved in f (k) ⋅ n O(1) time on inputs of length n and some function f depending only on some parameter k. In contrast to an algorithm that merely runs in polynomial time for fixed k, say, in O(n k ) time, which is intractable even for small values of k, fixed-parameter algorithms can solve NP-hard problems quickly if k is small.Provably effective polynomial-time data reduction. Parameterized complexity theory also provides a framework for data reduction with performance guarantees-problem kernelization [16,21,27,47].Kernelization allows for provably effective polynomial-time data reduction. Note that a result of the form "our polynomial-time data reduction algorithm reduces the input size by at least one bit, preserving optimality of solutions" is impossible for NP-hard problems unless P = NP. In contrast, a kernelization algorithm reduces a problem instance into an equivalent one (the problem kernel) whose size depends o...

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Enumerating Isolated Cliques in Temporal Networks

Molter

Niedermeier

Renken

2019

Complex Networks and Their Applications VIII

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Isolation is a concept from the world of clique enumeration that is mostly used to model communities that do not have much contact to the outside world. Herein, a clique is considered isolated if it has few edges connecting it to the rest of the graph. Motivated by recent work on enumerating cliques in temporal networks, we lift the isolation concept to this setting. We discover that the addition of the time dimension leads to six distinct natural isolation concepts. Our main contribution is the development of fixed-parameter enumeration algorithms for five of these six clique types employing the parameter "degree of isolation". On the empirical side, we implement and test these algorithms on (temporal) social network data, obtaining encouraging preliminary results.

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The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

Cited by 9 publications

References 39 publications

On the computational complexity of length- and neighborhood-constrained path problems

On the computational complexity of length- and neighborhood-constrained path problems

Parameterized algorithms and data reduction for the short secluded s‐t‐path problem

Enumerating Isolated Cliques in Temporal Networks

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