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Balanced TSP partitioning

Benjamin Aram Berendsohn, Hwi Kim, László Kozma

TL;DR

The paper investigates how much faster the Euclidean TSP can be solved with k ≥ 2 collaborating salespeople by formalizing the worst-case speedup ratio γ(k) = TSP_k(P)/TSP(P). It proves the exact value γ(2) = 1/2 + 1/π ≈ 0.818 via a circular point-set construction and a short-diagonal halving technique, and provides bounds for γ(k) with k ≥ 3, including γ(k) ≥ 1/k + (1/π) sin(π/k) and multiplicative/additive upper bounds such as γ(a·b) ≤ γ(a) γ(b) and γ(a+b) ≤ (1 + 2/π) γ(a) γ(b) / (γ(a) + γ(b)). The results yield worst-case decomposition guarantees for multi-vehicle routing in the planar Euclidean setting and outline open questions for tight bounds when k ≥ 3. The approach combines geometric arguments (width and short diagonals) with partitioning schemes to relate the k-partition TSP to the single-tour optimum, offering insights for broader high-dimensional generalizations and related partition problems.

Abstract

The traveling salesman problem (TSP) famously asks for a shortest tour that a salesperson can take to visit a given set of cities in any order. In this paper, we ask how much faster $k \ge 2$ salespeople can visit the cities if they divide the task among themselves. We show that, in the two-dimensional Euclidean setting, two salespeople can always achieve a speedup of at least $\frac12 + \frac1π\approx 0.818$, for any given input, and there are inputs where they cannot do better. We also give (non-matching) upper and lower bounds for $k \geq 3$.

Balanced TSP partitioning

TL;DR

The paper investigates how much faster the Euclidean TSP can be solved with k ≥ 2 collaborating salespeople by formalizing the worst-case speedup ratio γ(k) = TSP_k(P)/TSP(P). It proves the exact value γ(2) = 1/2 + 1/π ≈ 0.818 via a circular point-set construction and a short-diagonal halving technique, and provides bounds for γ(k) with k ≥ 3, including γ(k) ≥ 1/k + (1/π) sin(π/k) and multiplicative/additive upper bounds such as γ(a·b) ≤ γ(a) γ(b) and γ(a+b) ≤ (1 + 2/π) γ(a) γ(b) / (γ(a) + γ(b)). The results yield worst-case decomposition guarantees for multi-vehicle routing in the planar Euclidean setting and outline open questions for tight bounds when k ≥ 3. The approach combines geometric arguments (width and short diagonals) with partitioning schemes to relate the k-partition TSP to the single-tour optimum, offering insights for broader high-dimensional generalizations and related partition problems.

Abstract

The traveling salesman problem (TSP) famously asks for a shortest tour that a salesperson can take to visit a given set of cities in any order. In this paper, we ask how much faster salespeople can visit the cities if they divide the task among themselves. We show that, in the two-dimensional Euclidean setting, two salespeople can always achieve a speedup of at least , for any given input, and there are inputs where they cannot do better. We also give (non-matching) upper and lower bounds for .

Paper Structure

This paper contains 6 sections, 15 theorems, 12 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Width.
  5. Lower bounds: the circular point set
  6. Upper bounds via short diagonals

Key Result

Theorem 1

$\gamma(2) = \tfrac{1}{2} + \tfrac{1}{\pi} \approx 0.818$.

Figures (5)

  • Figure 1: (a) A curve $C$. (b) Curves $C(p,q)$ and $C(q,p)$, assuming a clockwise parametrization.
  • Figure 2: A polygon $P$ and its width with respect to directions $\theta_1$ and $\theta_2$. Here $w(P, \theta_1) < w(P, \theta_2)$.
  • Figure 3: (a) Point set $P_{16}$. (b) Balanced partition of $P_{16}$.
  • Figure 4: One step in the transformation of a tour $B$ into $A_m$.
  • Figure 5: Illustration of Lemma \ref{['lem:curve-vec']}.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 13 more