Fast Hamiltonicity Checking Via Bases of Perfect Matchings

Cygan, Marek; Kratsch, Stefan; Nederlof, Jesper

doi:10.1145/3148227

Cited by 43 publications

(58 citation statements)

References 30 publications

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“…Our algorithm also naturally extends to higher dimensions, but one needs to track the connectivity requirements of the dynamic programming solutions more carefully, using representative sets and the rank-based approach [BCKN15;CKN18]. This gives a running time of 2 O(1/ε) d−1 n log O(1) n for any fixed dimension d 2.…”

Section: Our Contributionmentioning

confidence: 99%

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Kisfaludi-Bak¹,

Nederlof²,

Węgrzycki³

2020

Preprint

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We revisit the classic task of finding the shortest tour of n points in d-dimensional Euclidean space, for any fixed constant d 2. We determine the optimal dependence on ε in the running time of an algorithm that computes a (1 + ε)-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in 2 O(1/ε d−1 ) n log n time. This improves the previously smallest dependence on ε in the running time (1/ε) O(1/ε d−1 ) n log n of the algorithm by Rao and Smith (STOC 1998). We also show that a 2 o(1/ε d−1 ) poly(n) algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH).Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call sparsity-sensitive patching. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree.

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Section: Our Contributionmentioning

confidence: 99%

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Kisfaludi-Bak¹,

Nederlof²,

Węgrzycki³

2020

Preprint

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“…Applying the rank-based approach. Next we describe how we can use the rank-based approach [3,6] in our setting. A standard application of the rank-based approach works on a tree-decomposition of the underlying graph, where the bags represent vertex separators of the underlying graph.…”

Section: An Exact Algorithm For Tspmentioning

confidence: 99%

“…The number of matchings on B boundary points is |B| Θ(|B|) , which is again too much for our purposes. Fortunately, the rank-based approach [3,6] developed in recent years can be applied here. By applying this approach in a suitable manner, we then obtain our 2 O(n 1−1/d ) algorithm.…”

Section: Introductionmentioning

confidence: 99%

An ETH-Tight Exact Algorithm for Euclidean TSP

Berg

Bodlaender

Kisfaludi-Bak

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We study exact algorithms for EUCLIDEAN TSP in R d . In the early 1990s algorithms with n O( √ n)running time were presented for the planar case, and some years later an algorithm with n O(n 1−1/d ) running time was presented for any d 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on EUCLIDEAN TSP, except for a lower bound stating that the problem admits no 2 O(n 1−1/d−ε ) algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of EUCLIDEAN TSP by giving a 2 O(n 1−1/d ) algorithm and by showing that a 2 o(n 1−1/d ) algorithm does not exist unless ETH fails. affirmatively by Arora [1] who provided a PTAS with running time n(log n) O( √ d/ε) d−1 . Independently, Mitchell [18] designed a PTAS in R 2 . The running time was improved to 2 (1/ε) O(d) n + (1/ε) O(d) n log n by Rao and Smith [22]. Hence, the computational complexity of the approximation problem has essentially been settled. Results on exact algorithms for EUCLIDEAN TSP-these are the topic of our paper-are also quite different from those on the general problem. The best known algorithm for the general case runs, as already remarked, in exponential time, and there is no 2 o(n) algorithm under ETH due to classical reductions for HAMILTONIAN CYCLE [5, Theorem 14.6]. EUCLIDEAN TSP, on the other hand, is solvable in subexponential time. For the planar case this has been shown in the early 1990s by Kann [14] and independently by Hwang, Chang and Lee [12], who presented an algorithm with an n O( √ n) running time. Both algorithms use a divide-and-conquer approach that relies on finding a suitable separator. The approach taken by Hwang, Chang and Lee is based on considering a triangulation of the point set such that all segments of the tour appear in the triangulation, and then observing that the resulting planar graph has a separator of size O( √ n). Such a separator can be guessed in n O( √ n) ways, leading to a recursive algorithm with n O( √ n) running time. It seems hard to extend this approach to higher dimensions. Kann obtains his separator in a more geometric way, using the fact that in an optimal tour, there cannot be too many long edges that are relatively close together-see the Packing Property we formulate in Section 2. This makes it possible to compute a separator that is crossed by O( √ n) edges of an optimal tour, which can be guessed in n O( √ n)

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“…For the special case of Hamiltonian Cycle, a O(2 n ) algorithm is known [19,7] for general di-graphs from 1960s. It was recently improved to O(1.657 n ) for graphs [8], and to O(1.888 n ) for bipartite di-graphs [12]. For other results and more details we refer to Chapter 12 of [2].…”

Section: Inputmentioning

confidence: 99%

Parameterized Algorithms for Survivable Network Design with Uniform Demands

Bang‐Jensen¹,

Basavaraju²,

Klinkby³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph G and an integer r uv for every pair of vertices u, v ∈ V (G). The objective is to construct a subgraph H of minimum weight which contains r uv edge-disjoint (or node-disjoint) u-v paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. Consequently, there is a long line of research into exact-polynomial time algorithms as well as approximation algorithms for various restrictions of this problem.An important restriction of this problem is one where the connectivity demands are the same for every pair of vertices. In this paper, we first consider the edge-connectivity version of this problem which we call λ-Edge Connected Subgraph (λ-ECS). In this problem, the input is a λ-edge connected (di)graph G and an integer k and the objective is to check whether G contains a spanning subgraph H that is also λ-edge connected and H excludes at least k edges of G. In other words, we are asked to compute a maximum subset of edges, of cardinality at least k, which may be safely deleted from G without affecting its connectivity. If we replace λ-edge connectivity with λ-vertex connectivity we get the λ-Vertex Connected Subgraph (λ-VCS) problem.We show that λ-ECS is fixed-parameter tractable (FPT) for both graphs and digraphs even if the (di)graph has nonnegative real weights on the edges and the objective is to exclude from H, some edges of G whose total weight exceeds a prescribed value. In particular, we design an algorithm * This work is supported by PARAPPROX, ERC starting grant no. 306992. for the weighted variant of the problem with running time. We follow up on this result and obtain a polynomial compression for λ-ECS on unweighted graphs. As a direct consequence of our results, we obtain the first FPT algorithm for the parameterized version of the classical Minimum Equivalent Graph (MEG) problem. We also show that λ-VCS is FPT on digraphs; however the problem on undirected graphs remains open. Finally, we complement our algorithmic findings by showing that SNDP is W[1]-hard for both arc and vertex connectivity versions on digraphs. The core of our algorithms is composed of new combinatorial results on connectivity in digraphs and undirected graphs.

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Fast Hamiltonicity Checking Via Bases of Perfect Matchings

Abstract: For an even integer t ≥ 2, the Matching Connectivity matrix H t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices; an entry H t [ M 1 , M 2 ] is 1 if M 1 and … Show more

Cited by 43 publications

References 30 publications

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

An ETH-Tight Exact Algorithm for Euclidean TSP

Parameterized Algorithms for Survivable Network Design with Uniform Demands

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