Finding Detours is Fixed-Parameter Tractable

Bezáková, Ivona; Curticapean, Radu; Dell, Holger; Fomin, Fedor V.

doi:10.1137/17m1148566

Cited by 21 publications

(30 citation statements)

References 38 publications

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“…Fomin et al [7] showed that an algorithm for k-(s, t)-Path can be used to solve the Long (s, t)-Path and Long Cycle problems. Bezáková et al [2] showed that an algorithm for k-(s, t)-Path can be used to solve the Exact Detour problem. Using our algorithm for k-(s, t)-Path instead the algorithm of [15] gives faster algorithms for these problems.…”

Section: Introductionmentioning

confidence: 99%

Faster deterministic parameterized algorithm for k-Path

Tsur

2019

Theoretical Computer Science

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In the k-Path problem, the input is a directed graph G and an integer k ≥ 1, and the goal is to decide whether there is a simple directed path in G with exactly k vertices. We give a deterministic algorithm for k-Path with time complexity O * (2.554 k ). This improves the previously best deterministic algorithm for this problem of Zehavi [ESA 2015] whose time complexity is O * (2.597 k ). The technique used by our algorithm can also be used to obtain faster deterministic algorithms for k-Tree, r-Dimensional k-Matching, Graph Motif, and Partial Cover.

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Section: Introductionmentioning

confidence: 99%

Faster deterministic parameterized algorithm for k-Path

Tsur

2019

Theoretical Computer Science

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“…Similar ideas have successfully been applied to several problems (see e.g. [2,5,10,19,20,32]). Our results also fall in the framework of "above/below" parameterization, with the remark that our parameter of interest is not the value to be optimized but a structural property of the input, which we parameterize near its "critical value".…”

Section: Related Workmentioning

confidence: 96%

“…For each vertex v ∈ S \ R S , we claim that N G (v) ∩ R C = ∅. This follows from the fact that N G (v) = N H (v 1 ) = N H (v 2 ) and Claim 3.5 (5), using that v / ∈ R S implies v 1 , v 2 / ∈ R S . Hence the (up to two) neighbors that v ∈ S \ R S has in C on the Hamiltonian cycle F do not belong to R C , while Claim 3.5 (5) ensures that all vertices of N G (v) are saturated by H and hence belong to C .…”

Section: Claim 34 If There Is a Path Cover Of S In G Having C -Endpmentioning

confidence: 96%

“…) and Claim 3.5(5), using that v / ∈ R S implies v 1 , v 2 / ∈ R S . Hence the (up to two) neighbors that v ∈ S \ R S has in C on the Hamiltonian cycle F do not belong to R C , while Claim 3.5 (5) ensures that all vertices of N G (v) are saturated by H and hence belong to C . For each vertex v ∈ S \ R S , for each edge from v to C ∩ C incident on v in F , we insert the corresponding edge into F 2 .…”

Section: :9mentioning

confidence: 99%

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Hamiltonicity Below Dirac’s Condition

Jansen

Kozma

Nederlof

2019

Graph-Theoretic Concepts in Computer Science

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Dirac's theorem (1952) is a classical result of graph theory, stating that an n-vertex graph (n ≥ 3) is Hamiltonian if every vertex has degree at least n/2. Both the value n/2 and the requirement for every vertex to have high degree are necessary for the theorem to hold.In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time c k · n O(1) , for some fixed constant c, if at least n − k vertices have degree at least n/2, or if all vertices have degree at least n/2 − k. The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH).The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with O(k) vertices can be found in polynomial time. ACM Subject ClassificationTheory of computation → Graph algorithms analysis, Theory of computation → Parameterized complexity and exact algorithms 23:3 states this algorithmic result as a corollary of structural theorems. He does not describe the details of the algorithm or its analysis, but these are not hard to reconstruct.) Here we improve the running time of Häggkvist's algorithm to the stated (asymptotically optimal) c k · n O(1) by more efficiently solving the arising path-cover subproblem. Statement of resultsOur first result shows that if a graph has a "relaxed" Dirac property, it can be compressed while preserving its Hamiltonicity. Theorem 1.1. Let G be an n-vertex graph such that at least n − k vertices of G have degree at least n/2. There is a deterministic algorithm that, given G, constructs in time O(n 3 ) a 3k-vertex graph G , such that G is Hamiltonian if and only if G is Hamiltonian.Equivalently stated in the language of parameterized complexity, the Hamiltonian cycle problem parameterized by k has a kernel with a linear number of vertices. To determine the Hamiltonicity of a graph G, we simply apply the algorithm of Theorem 1.1 to compress G, and use an exponential-time algorithm (for instance, the Held-Karp algorithm) to solve Hamiltonian Cycle directly on the compressed graph. We thus obtain the following result.Corollary 1.2. If at least n − k vertices of an n-vertex graph G have degree at least n/2, then Hamiltonian Cycle with input G can be solved in deterministic time O(8 k · k 2 + n 3 ).As an alternative, we may also use an approach based on inclusion-exclusion [26] to solve the reduced Hamiltonian cycle instance, achieving the overall running time O(8 k · k 3 + n 3 ), with polynomial space.Our result for the second relaxation of Dirac's theorem is as follows.

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“…From a more general perspective, our work belongs to a popular subfield of Parameterized Complexity concerning parameterization above/below specified guarantees. In addition to [10,7], the parameterized complexity of paths and cycles above some guarantees was studied in [2,16], and [8].…”

Section: Longest Cycle Above Madmentioning

confidence: 99%

Longest Cycle above Erdős-Gallai Bound

Fomin¹,

Golovach²,

Sagunov³

2022

Preprint

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In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2 O(k) • n O(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G) + k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

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Finding Detours is Fixed-Parameter Tractable

Cited by 21 publications

References 38 publications

Faster deterministic parameterized algorithm for k-Path

Faster deterministic parameterized algorithm for k-Path

Hamiltonicity Below Dirac’s Condition

Longest Cycle above Erdős-Gallai Bound

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