New inapproximability bounds for TSP

Karpiński, Marek; Lampis, Michael; Schmied, Richard

doi:10.1016/j.jcss.2015.06.003

Cited by 86 publications

(79 citation statements)

References 18 publications

(39 reference statements)

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“…There are even strict bounds on how close approximation algorithms 4 can get to the optimal path of E-TSP (Karpinski, Lampis, & Schmied, 2013), and many approximation algorithms require more than the linear time observed in human performance studies.…”

Section: Previous Workmentioning

confidence: 99%

The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

Carruthers¹,

Stege²,

Masson³

2018

The Journal of Problem Solving

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The role that the mental, or internal, representation plays when people are solving hard computational problems has largely been overlooked to date, despite the reality that this internal representation drives problem solving. In this work we investigate how performance on versions of two hard computational problems differs based on what internal representations can be generated. Our findings suggest that problem solving performance depends not only on the objective difficulty of the problem, and of course the particular problem instance at hand, but also on how feasible it is to encode the goal of the given problem. A further implication of these findings is that previous human performance studies using NP-hard problems may have, surprisingly, underestimated human performance on instances of problems of this class. We suggest ways to meaningfully frame human performance results on instances of computationally hard problems in terms of these problems' computational complexity, and present a novel framework for interpreting results on problems of this type. The framework takes into account the limitations of the human cognitive system, in particular as it applies to the generation of internal representations of problems of this class.

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Section: Previous Workmentioning

confidence: 99%

The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

Carruthers¹,

Stege²,

Masson³

2018

The Journal of Problem Solving

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“…We remark that the strongest known negative result for the general ATSP with the triangle inequality is due to Karpinski et al [8]. They prove that unless = , the ATSP with triangle inequality cannot have a polynomial time approximation algorithm with performance guarantee better than 75/74.…”

Section: Known Resultsmentioning

confidence: 89%

“…Given an -approximate solution to the flowshop instance, we can obtain a TSP path ALGĜ forĜ with cost Using the result of the Karpinski et al [8] for the ATSP, we derive the following theorem. Proof.…”

Section: A More Efficient Embeddingmentioning

confidence: 99%

No-Wait Flowshop Scheduling Is as Hard as Asymmetric Traveling Salesman Problem

Mucha

Sviridenko

2016

Mathematics of OR

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In this paper, we study the classical no-wait flowshop scheduling problem with makespan objective (F no-wait C max in the standard three-field notation). This problem is well known to be a special case of the asymmetric traveling salesman problem (ATSP) and as such has an approximation algorithm with logarithmic performance guarantee. In this work, we show a reverse connection, we show that any polynomial time -approximation algorithm for the no-wait flowshop scheduling problem with makespan objective implies the existence of a polynomial time 1 + -approximation algorithm for the ATSP for any > 0. This, in turn, implies that all nonapproximability results for the ATSP (current or future) will carry over to its special case. In particular, it follows that the no-wait flowshop problem is APX-hard, which is the first nonapproximability result for this problem.

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“…RPP is strongly NP-hard [27,41], its special case with R = E is the polynomial-time solvable Chinese Postman problem [17,18]. Containing the metric Traveling Salesman Problem as a special case, RPP is APX-hard [37]. There is a folklore polynomial-time 3/2-approximation based on the Christofides-Serdyukov algorithm for the metric Traveling Salesman Problem [11,46] (we refer to arc routing surveys [7,22] for a detailed algorithmic description).…”

Section: Related Workmentioning

confidence: 99%

“…We aim for (1 + ε)-approximations for all ε > 0. Unfortunately, containing the metric Traveling Salesman Problem as a special case, RPP is APX-hard [37]. Thus, finding such approximations typically requires exponential time, we present data reduction rules for this task.…”

mentioning

confidence: 99%

On $$(1+\varepsilon )$$ -approximate Data Reduction for the Rural Postman Problem

Bevern

Fluschnik

Tsidulko

2019

Mathematical Optimization Theory and Operations Research

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Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and cost d of edges traversed additionally to the required ones. Thus, in particular, RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I with 2b + O(c/ε) vertices in O(n 3 ) time so that any α-approximate solution for I gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on wide-spread benchmark data sets as well as on two real snow plowing instances from Berlin. On instances with few connected components, the number of vertices and required edges is reduced to about 50 % at a 1 % solution quality loss. We also make first steps towards a PSAKS for the parameter c. Problem 1.1 (Rural Postman Problem, RPP).Input: An undirected graph G = (V, E) with n vertices, edge weights ω : E → N ∪ {0}, and a multiset R of required edges of G. Task: Find a closed walk W * in G containing each edge of R and minimizing the total weight ω(W * ) of the edges on W * .We call any closed walk containing each edge of R an RPP tour. We will also consider the decision variant k-RPP, where one additionally gets a non-negative integer k ∈ N in the input and the task is to decide whether there is an RPP tour W of cost ω(W) ≤ k. RPP has direct applications in snow plowing, street sweeping, meter reading [14,22], vehicle depot location [29], drilling, and plotting [28,31]. The undirected version occurs especially in rural areas, where service vehicles can operate in both directions even on one-way roads [19]. Moreover, RPP is a special case of the Capacitated Arc Routing Problem (CARP) [30] and used in all "route first, cluster second" algorithms for CARP [1,10,49], which are notably the only ones with proven approximation guarantees [6,36,50]. Improved approximations for RPP automatically lead to better approximations for CARP.There is a folklore polynomial-time 3/2-approximation for RPP based on the Christofides-Serdyukov algorithm for the metric Traveling Salesman Problem [11,46] (we refer to Eiselt et al. [22] or van Bevern et al. [7] for a detailed algorithm description). We aim for (1 + ε)-approximations for all ε > 0. Unfortunately, containing the metric Traveling Salesman Problem as a special case, RPP is APX-hard [37]. Thus, finding such approximations typically requires exponential time, we present data reduction rules for this task. Their effectivity depends on the desired approximation factor. Graph theory. We generally consider multigraphs G = (V, E) wit...

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New inapproximability bounds for TSP

Cited by 86 publications

References 18 publications

The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

No-Wait Flowshop Scheduling Is as Hard as Asymmetric Traveling Salesman Problem

On $$(1+\varepsilon )$$ -approximate Data Reduction for the Rural Postman Problem

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