A time- and space-optimal algorithm for the many-visits TSP

Kozma, L.; Mnich, Matthias; Vincze, Roland

doi:10.1137/1.9781611975482.106

Cited by 10 publications

(17 citation statements)

References 41 publications

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“…Lemma 2.2 (Berger et al [3], Lemma 2.4). A sequence of nonnegative integers {d v } v∈X is an outdegree sequence of an out-tree spanning X and rooted at r ∈ X if and only if…”

Section: Multisets Recall That a Multiset A Can Be Identified By Its ...mentioning

confidence: 99%

“…Yet another breakthrough came only recently, namely Berger, Kozma, Mnich and Vincze [2,3] improved the running time to O * (5 n ). Their main contribution is an idea that it is more convenient to use outbranchings (i.e.…”

Section: Related Workmentioning

confidence: 99%

“…As noted in [2,7], Fixed Degree Subgraph can be solved by a reduction to minimum cost flow. By applying Orlin's algorithm [23] we get the following.…”

Section: Reduction To Small Demandsmentioning

confidence: 99%

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Many visits TSP revisited

Kowalik¹,

Li²,

Nadara³

et al. 2020

Preprint

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We study the Many Visits TSP problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020 and runs in time and space O * (5 n ). They also show a polynomial space algorithm running in time O(16 n+o(n) ). In this work, we show three main results: * This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (S. Li, W. Nadara) and 677651 (Ł. Kowalik, M. Smulewicz).

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“…Lemma 2.2 (Berger et al [3], Lemma 2.4). A sequence of nonnegative integers {d v } v∈X is an outdegree sequence of an out-tree spanning X and rooted at r ∈ X if and only if…”

Section: Multisets Recall That a Multiset A Can Be Identified By Its ...mentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Many visits TSP revisited

Kowalik¹,

Li²,

Nadara³

et al. 2020

Preprint

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show abstract

“…The transportation problem. Observe that if we relax the connectivity and decomposability constraints from the unrestricted MV-mTSP variants (similarly to the single-salesman case [5,6,11]), the problem can be modelled as the Hitchcock transportation problem [20]. The transportation problem can be solved efficiently using a min-cost max-flow algorithm [13], even in strongly polynomial time [26,28].…”

Section: Approximation Algorithms For Many-visits Mtsp With Arbitrary...mentioning

confidence: 99%

Efficient Approximations for Many-Visits Multiple Traveling Salesman Problems

Bérczi¹,

Mnich²,

Vincze³

2022

Preprint

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A fundamental variant of the classical traveling salesman problem (TSP) is the so-called multiple TSP (mTSP), where a set of m salesmen jointly visit all cities from a set of n cities. The mTSP models many important real-life applications, in particular for vehicle routing problems. An extensive survey by Bektas (Omega 34(3), 2006) lists a variety of heuristic and exact solution procedures for the mTSP, which quickly solve particular problem instances.In this work we consider a further generalization of mTSP, the many-visits mTSP, where each city v has a request r(v) of how many times it should be visited by the salesmen. This problem opens up new real-life applications such as aircraft sequencing, while at the same time it poses several computational challenges. We provide multiple efficient approximation algorithms for important variants of the many-visits mTSP, which are guaranteed to quickly compute high-quality solutions for all problem instances.

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“…We cannot guess the topology of the spanning multigraph part of a solution in a manner similar to guessing a topology as in the case of digraphs, since trying every possibility already takes times 2 O( k r log k r ) . Instead, inspired by the work of Berger et al [8] (which guess a degree-sequence of a certain tree), we only guess a so called "occurrence sequence" of the spanning multigraph part of a solution. Let us first define a notion that we call an occurrence sequence.…”

Section: Guessing the Occurrence Sequence Of The Spanning Multigraph mentioning

confidence: 99%

On r-Simple k-Path and Related Problems Parameterized by k/r

Gutin

Wahlström

Zehavi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive integers r, k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r, k)-Monomial Detection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4 (k/r) log r · n O(1) for both problems. Gabizon et al. (2015) designed deterministic 2 O((k/r) log r) · n O(1) -time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the p-Set (r, q)-Packing problem, given a universe V , positive integers p, q, r, and a collection H of sets of size p whose elements belong to V , decide whether there exists a subcollection H of H of size q where each element occurs in at most r sets of H . Gabizon et al. obtained a deterministic 2 O((pq/r) log r) · n O(1) -time algorithm for p-Set (r, q)-Packing.The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2 o((k/r) log r) · (n + log k) O(1) -time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)-Packing.We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f (k/r)(n + log k) O(1) -time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2 O((k/r) 2 log(k/r)) · (n + log k) O(1) -time algorithm for r-Simple k-Path on digraphs and a 2 O(k/r) · (n + log k) O(1) -time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that p-Set (r, q)-Packing is FPT (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detection is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is non-canceling. For the special case of (r, k)-Monomial De...

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A time- and space-optimal algorithm for the many-visits TSP

Cited by 10 publications

References 41 publications

Many visits TSP revisited

Many visits TSP revisited

Efficient Approximations for Many-Visits Multiple Traveling Salesman Problems

On r-Simple k-Path and Related Problems Parameterized by k/r

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