On the Extended TSP Problem

TL;DR

This paper introduces the Ext-TSP problem, analyzing its computational complexity, approximation algorithms, and exact solutions, especially focusing on special graph classes like trees and planar graphs.

Contribution

It formally defines Ext-TSP, proves its APX-hardness, provides approximation algorithms, and develops exact algorithms for trees, advancing understanding of this new problem.

Findings

Ext-TSP is APX-hard to approximate in general graphs.

A (k+1)-approximation algorithm is proposed for general graphs.

A PTAS exists for sparse graph classes such as planar graphs.

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 $w: E \rightarrow R^+$, and a non-increasing discount function $f(\cdot)$ 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 $\sum_{(u, v) \in E} f(|d_u - d_v|)\cdot w(u,v)$, where $d_v \in \{1, \ldots, |V| \}$ is the position of vertex~$v$ in the sequence. We show that \prob{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.