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On the Extended TSP Problem

Authors Julián Mestre , Sergey Pupyrev , Seeun William Umboh

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Julián Mestre
  • Facebook Inc., Menlo Park, CA, USA
  • The University of Sydney, Australia
Sergey Pupyrev
  • Facebook Inc., Menlo Park, CA, USA
Seeun William Umboh
  • The University of Sydney, Australia

Acknowledgements

We would like to thank Vahid Liaghat for fruitful discussions on the Ext-TSP problem.

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Julián Mestre, Sergey Pupyrev, and Seeun William Umboh. On the Extended TSP Problem. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.42

Abstract

We initiate the theoretical study of Ext-TSP, a problem that originates in the area of profile-guided binary optimization. Given a graph G = (V, E) with positive edge weights w: E → R^+, and a non-increasing discount function f(⋅) such that f(1) = 1 and f(i) = 0 for i > k, for some parameter k that is part of the problem definition. The problem is to sequence the vertices V so as to maximize ∑_{(u, v) ∈ E} f(|d_u - d_v|)⋅ w(u,v), where d_v ∈ {1, …, |V|} is the position of vertex v in the sequence.
We show that Ext-TSP is APX-hard to approximate in general and we give a (k+1)-approximation algorithm for general graphs and a PTAS for some sparse graph classes such as planar or treewidth-bounded graphs.
Interestingly, the problem remains challenging even on very simple graph classes; indeed, there is no exact n^o(k) time algorithm for trees unless the ETH fails. We complement this negative result with an exact n^O(k) time algorithm for trees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • profile-guided optimization
  • approximation algorithms
  • bandwidth
  • TSP

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