Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximating Traveling Salesman Problems Using a Bridge Lemma

Martin Böhm, Zachary Friggstad, Tobias Mömke, Joachim Spoerhase

TL;DR

This work addresses improved approximations for two metric traveling salesman variants, OTSP and k-TSPP, by leveraging an adapted Bridge Lemma within an iterative LP rounding framework. The authors introduce a $T$-rooted forest cost $c_T$ and decompose $s_i$-$t_i$ flows into branchings via Bang-Jensen et al., using randomized sampling to form partial paths and the Bridge Lemma to cheaply graft in the remaining nodes, followed by parity corrections. The main results are a $(3/2+e^{-1})<1.878$-approximation for OTSP and a $(1+2e^{-1/2})<2.214$-approximation for k-TSPP in polynomial time, both improvements over natural baselines. The approach generalizes the Bridge Lemma to vehicle routing and shows promise for broader routing improvements beyond the problems studied.

Abstract

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at some point after $o_i$ for each $1 \leq i < k$. This is the special case of Precedence-Constrained TSP ($PTSP$) in which the precedence constraints are given by a single chain on a subset of nodes. In $k$-Person TSP Path (k-TSPP), we are given pairs of nodes $(s_1,t_1), \ldots, (s_k,t_k)$. The goal is to find an $s_i$-$t_i$ path with minimum total cost such that every node is visited by at least one path. We obtain a $3/2 + e^{-1} < 1.878$ approximation for OTSP, the first improvement over a trivial $α+1$ approximation where $α$ is the current best TSP approximation. We also obtain a $1 + 2 \cdot e^{-1/2} < 2.214$ approximation for k-TSPP, the first improvement over a trivial $3$-approximation. These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.

Approximating Traveling Salesman Problems Using a Bridge Lemma

TL;DR

This work addresses improved approximations for two metric traveling salesman variants, OTSP and k-TSPP, by leveraging an adapted Bridge Lemma within an iterative LP rounding framework. The authors introduce a -rooted forest cost and decompose - flows into branchings via Bang-Jensen et al., using randomized sampling to form partial paths and the Bridge Lemma to cheaply graft in the remaining nodes, followed by parity corrections. The main results are a -approximation for OTSP and a -approximation for k-TSPP in polynomial time, both improvements over natural baselines. The approach generalizes the Bridge Lemma to vehicle routing and shows promise for broader routing improvements beyond the problems studied.

Abstract

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes . The TSP solution must have that is visited at some point after for each . This is the special case of Precedence-Constrained TSP () in which the precedence constraints are given by a single chain on a subset of nodes. In -Person TSP Path (k-TSPP), we are given pairs of nodes . The goal is to find an - path with minimum total cost such that every node is visited by at least one path. We obtain a approximation for OTSP, the first improvement over a trivial approximation where is the current best TSP approximation. We also obtain a approximation for k-TSPP, the first improvement over a trivial -approximation. These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.
Paper Structure (11 sections, 15 theorems, 11 equations, 1 figure, 3 algorithms)

This paper contains 11 sections, 15 theorems, 11 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our Results
  4. Overview of our Algorithm
  5. Notation and Preliminaries
  6. A Bridge Lemma
  7. Approximating OTSP
  8. Approximating $k$-Person TSP Path
  9. Warmup Algorithm
  10. An Improved $k$-TSPP Approximation
  11. Conclusion

Key Result

Theorem 1

There is a polynomial-time randomized algorithm for $k$-TSPP that finds a solution whose expected cost is at most $1 + 2 \cdot e^{-1/2} < 2.214$ times the optimum solution cost.

Figures (1)

  • Figure 1: Left: A component $C$ of the forest $(V,F)$. The square node is $t_C$ and the nodes in $C \cap S$ are shaded. Right: The corresponding directed tree $F'_S$. The $c'$-cost of each directed edge $(u,v)$ is the largest $c$-cost of edges on the $u$-$v$ path in $F$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Bang-Jensen, Frank, and Jackson BJ95; Swamy and Post Post15
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 16 more