The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

Burton, Benjamin A.; Cabello, Sergio; Kratsch, Stefan; Pettersson, William

doi:10.1137/18m1168704

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…We show the d = 2 case, and the higher dimensions follow again by taking the suspension, which changes the solution size linearly by doubling once for each suspension. It may be possible to prove this result with a modified argument based on the parameterized reduction from the Grid Tiling problem to the 2-Sphere Recognition problem in [17]. Here, we present a completely different reduction from the α×β-Clique problem defined below, as this gives us further hardness results for when the MBC d problem is parameterized with respect to both solution size and maximum coface degree (see Section 4.2.3).…”

Section: Solution Sizementioning

confidence: 99%

“…There might be a better parameterized algorithm than the one presented in this paper. In particular, we have the ETH-tight 2 O(k) n O( √ k) -time algorithm from Theorem 2 of B. Burton et al [17] that can recognize if a simplicial complex of size n contains a 2-sphere of size (at most) k as a sub-complex. This is interesting, because this problem is similar to the MBC 2 problem while also having a runtime close to what we are aiming for.…”

Section: Solution Size and Coface Degreementioning

confidence: 99%

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The Parameterized Complexity of Finding Minimum Bounded Chains

Blaser¹,

Brun²,

Salbu³

et al. 2021

Preprint

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Finding the smallest d-chain with a specific (d − 1)-boundary in a simplicial complex is known as the Minimum Bounded Chain (MBC d ) problem. The MBC d problem is NP-hard for all d ≥ 2. In this paper, we prove that it is also W[1]-hard for all d ≥ 2, if we parameterize the problem by solution size. We also give an algorithm solving the MBC 1 problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC d problem for all d. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

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Section: Solution Sizementioning

confidence: 99%

Section: Solution Size and Coface Degreementioning

confidence: 99%

The Parameterized Complexity of Finding Minimum Bounded Chains

Blaser¹,

Brun²,

Salbu³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Ivanov showed 2-Dim-Sphere is NP-hard [27]. Burton, Cabello, Kratsch, and Pettersson showed 2-Dim-Sphere is W [1]-Hard when parameterized by the number of 2-simplices of the sphere [8]. Burton, Cabello, Kratsch, and Pettersson also gave an algorithm for 2-Dim-Sphere that runs in 2 O(t) n O( √ t) time, where t is the number of triangles in the sphere.…”

Section: Related Workmentioning

confidence: 99%

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

Black¹,

Nayyeri²

2021

Preprint

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We consider the problem 2-Dim-Bounding-Surface. 2-Dim-Bounded-Surface asks whether or not there is a subcomplex S of a simplicial complex K homeomorphic to a given compact, connected surface bounded by a given subcomplex B ⊂ K. 2-Dim-Bounding-Surface is NP-hard. We show it is fixed-parameter tractable with respect to the treewidth of the 1-skeleton of the simplicial complex K.Using some of the techniques we developed for the 2-Dim-Bounded-Surface problem, we obtain fixed parameter tractable algorithms for other topological problems such as computing an optimal chain with a given boundary and computing an optimal chain in a given homology class.

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“…Most recently, it has been used to study the parameterized complexity of Steiner Tree in planar graphs [17,18]. The problem is also starting to be applied outside the strictly geometric setting: in computational topology [19] and for studying H -free graphs [20]. Our Theorem 2 is using gadgetry similar to other reductions in geometric intersection graphs, but it is ultimately a more standard reduction from the 3-SAT problem.…”

Section: Related Workmentioning

confidence: 99%

The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

2019

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Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let s ∈ P be a given source node. Each node p can transmit information to all other nodes within unit distance, provided p is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source s can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem-in the latter s must be able to reach every node within a specified number of hops-where we also consider how the complexity depends on the width w of the strip. We prove the following two lower bounds. First, we show that the regular version of the problem is W[1]-complete when parameterized by the solution size k. More precisely, we show that the problem does not admit an algorithm with running time f (k)n o(√ k) , unless ETH fails. The construction can also be used to show an f (w)n Ω(w) lower bound when we parameterize by the strip width w. Second, we prove that the hop-bounded version of the problem is NP-hard in strips of width 40. These results complement the algorithmic results in a companion paper (de Berg et al. in Algorithmica, submitted).

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The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

Cited by 6 publications

References 25 publications

The Parameterized Complexity of Finding Minimum Bounded Chains

The Parameterized Complexity of Finding Minimum Bounded Chains

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

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