Reducing path TSP to TSP
| Autor: | Rico Zenklusen, Vera Traub, Jens Vygen |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
FOS: Computer and information sciences
Mathematical optimization Discrete Mathematics (cs.DM) General Computer Science Generalization General Mathematics Computer Science::Neural and Evolutionary Computation MathematicsofComputing_NUMERICALANALYSIS 0211 other engineering and technologies 0102 computer and information sciences 02 engineering and technology Computer Science::Computational Complexity 01 natural sciences Travelling salesman problem Reduction (complexity) Set (abstract data type) Computer Science::Discrete Mathematics TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY Computer Science - Data Structures and Algorithms FOS: Mathematics Mathematics - Combinatorics Data Structures and Algorithms (cs.DS) Computer Science::Data Structures and Algorithms Mathematics Discrete mathematics 021103 operations research Approximation algorithm Dynamic programming 010201 computation theory & mathematics Core (graph theory) Path (graph theory) Combinatorics (math.CO) Computer Science - Discrete Mathematics MathematicsofComputing_DISCRETEMATHEMATICS |
| Zdroj: | STOC Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing |
| Popis: | We present a black-box reduction from the path version of the Traveling Salesman Problem (Path TSP) to the classical tour version (TSP). More precisely, we show that given an $\alpha$-approximation algorithm for TSP, then, for any $\epsilon >0$, there is an $(\alpha+\epsilon)$-approximation algorithm for the more general Path TSP. This reduction implies that the approximability of Path TSP is the same as for TSP, up to an arbitrarily small error. This avoids future discrepancies between the best known approximation factors achievable for these two problems, as they have existed until very recently. A well-studied special case of TSP, Graph TSP, asks for tours in unit-weight graphs. Our reduction shows that any $\alpha$-approximation algorithm for Graph TSP implies an $(\alpha+\epsilon)$-approximation algorithm for its path version. By applying our reduction to the $1.4$-approximation algorithm for Graph TSP by Seb\H{o} and Vygen, we obtain a polynomial-time $(1.4+\epsilon)$-approximation algorithm for Graph Path TSP, improving on a recent $1.497$-approximation algorithm of Traub and Vygen. We obtain our results through a variety of new techniques, including a novel way to set up a recursive dynamic program to guess significant parts of an optimal solution. At the core of our dynamic program we deal with instances of a new generalization of (Path) TSP which combines parity constraints with certain connectivity requirements. This problem, which we call $\Phi$-TSP, has a constant-factor approximation algorithm and can be reduced to TSP in certain cases when the dynamic program would not make sufficient progress. |
| Databáze: | OpenAIRE |
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