Improved FPT Approximation for Non-metric TSP

TL;DR

This work tackles the non-metric TSP by parameterizing the instance with $k$, the number of triangle-inequality-violating triangles, and aims for a fixed-parameter tractable (FPT) approximation. Building on Zhou, Li and Guo, the authors design a Christofides-inspired, $2.5$-approximation algorithm that runs in time $O((3k)! 8^k \cdot n^4)$, and prove the bound $c(ALG)\le 2.5\,c(OPT)$. The method partitions bad vertices, constructs a cycle on bad vertices, augments with a $t$-minimum spanning forest on good vertices, and adds a minimum-cost perfect matching on odd-degree vertices to obtain an Eulerian graph before careful shortcutting yields a Hamiltonian cycle. This result tightens the connection between non-metric and metric approximability and highlights directions for further improvement, including the potential of a constant-factor FPT algorithm parameterized by $k'$ (the vertex deletion distance to metric).

Abstract

In the Traveling Salesperson Problem (TSP) we are given a list of locations and the distances between each pair of them. The goal is to find the shortest possible tour that visits each location exactly once and returns to the starting location. Inspired by the fact that general TSP cannot be approximated in polynomial time within any constant factor, while metric TSP admits a (slightly better than) $1.5$-approximation in polynomial time, Zhou, Li and Guo [Zhou et al., ISAAC '22] introduced a parameter that measures the distance of a given TSP instance from the metric case. They gave an FPT $3$-approximation algorithm parameterized by $k$, where $k$ is the number of triangles in which the edge costs violate the triangle inequality. In this paper, we design a $2.5$-approximation algorithm that runs in FPT time, improving the result of [Zhou et al., ISAAC '22].

Figures (2)

• Figure 1: An illustration of Steps 2(a) and 2(b) of Algorithm \ref{['algo']}. $b_i$'s are bad vertices and $g_i$'s are good vertices. Top left: the cycle represents an optimal TSP solution. The algorithm “guesses” the occurrence order of bad vertices and the “gaps” between bad vertices in the optimal TSP solution, but the position of good vertices in the solution remains unknown. Top right: In Step 2(a), Algorithm \ref{['algo']} creates a simple cycle $C$ of only bad vertices following their order in the optimal solution. Bottom: The algorithm computes a $4$-minimum spanning forest ($t=4$, as the number of bad chains in the optimal TSP solution) of good vertices rooted at the bad end-vertices of each chain ($b_3, b_5, b_6, b_8$). Note that in this example, the $4$-minimum spanning forest does not connect any good vertex to $b_8$.
• Figure 2: An illustration of Step 1 of ShortCut (Algorithm \ref{['shortcut']}). $b_i$'s are bad vertices and $g_i$'s are good vertices. Note that before (and after) this step, all vertices have even degree. The purpose of this step is to remove all double edges between bad vertices from the created graph. Here, we would like to remove the double edge $(b_3,b_4)$ ($b_3, b_4$ are both end vertices, see Figure (a)). As you can see in Figure (b), we remove one edge $(b_3, b_4)$ and the edge $(b_3, g_1)$ (red dotted edges). Finally, we connect $g_1$ with $b_4$ (green edge) ensuring that the graph is connected and all vertices have even degree. It is trivial to see that this step does not increase the total cost of the edges of the graph due to the triangle inequality in $\Delta(b_3,g_1, b_4)$.

Theorems & Definitions (7)

• Lemma 1: Lemma 1 in zhou
• Theorem 2
• Lemma 3: Lemma 3 in zhou
• Lemma 4: Lemma 5 in zhou
• Lemma 5
• Lemma 6
• Theorem 7