Euclidean TSP in Narrow Strips

TL;DR

The paper advances exact and subexponential-time algorithms for Euclidean TSP in narrow geometric regimes: it proves that in a plane strip of width $\delta$, a globally optimal tour can be bitonic for $\delta\leq 2\sqrt{2}$ when $x$-coordinates are distinct integers, and it provides fixed-parameter tractable dynamic programs that scale as $2^{O(\delta^{1-1/d})} n^2$ (and improved to $2^{O(\delta^{1-1/d})} n + O(\delta^2 n^2)$ for sparse point sets) in general $d$-dimensions. It also shows that a similar approach yields expected subexponential time $2^{O(\delta^{1-1/d})} n$ for random point sets within a narrow cylinder, with the analysis extended to the $d$-dimensional hypercylinder. Together, these results illuminate how near-1D structure (narrow strips/cylinders) yields significantly faster exact or near-exact algorithms for Euclidean TSP, and they open paths to higher-dimensional generalizations and refined constants.

Abstract

We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,δ]$ depends on the strip width $δ$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest tour overall when $δ\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $δ$. Our algorithm has running time $2^{O(\sqrtδ)} n + O(δ^2 n^2)$ for sparse point sets, where each $1\timesδ$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle $[0,n]\times [0,δ]$, it has an expected running time of $2^{O(\sqrtδ)} n$. These results generalise to point sets $P$ inside a hypercylinder of width $δ$. In this case, the factors $2^{O(\sqrtδ)}$ become $2^{O(δ^{1-1/d})}$.

Figures (17)

• Figure 1: Construction for $\delta > 2 \sqrt{2}$ for Theorem \ref{['thm:bitonic:main']}. The grey vertical segments are at distance 1 from each other. If $\delta > 2 \sqrt{2}$ then $T_1$, the shortest bitonic tour (in blue), is longer than $T_2$, the shortest non-bitonic tour (in red).
• Figure 2: Replacing the paths connecting endpoints of edges in $F$ by abstract connections. The copies of duplicated shared endpoints are slightly displaced in the figure to be able to distinguish them, but they are actually coinciding.
• Figure 3: The process of moving the points in $Q$. Grey vertical lines have integer $x$-coordinates. (i) Moving a point in $Q_{\mathrm{left}}$ so that it gets $x$-coordinate $z$. (ii) A possible configuration after $Q_{\mathrm{left}}$ and $Q_{\mathrm{right}}$ have been treated.
• Figure 4: The six different cases that result after applying Step 1 of the proof. Points indicated by filled disks have a fixed $x$-coordinate. The left-to-right order of points drawn inside a grey rectangle, on the other hand, is not known yet. The vertical order of the edges is also not fixed, as the points can have any $y$-coordinate in the range $[0,2\sqrt{2}]$.
• Figure 5: Two scenarios covering all subscenarios where the automated prover fails, up to symmetries. Each point has a fixed $x$-coordinate and a $y$-range specified by the array $Y$; the resulting possible locations are shown as small grey rectangles (drawn larger than they actually are for visibility). For all subscenarios, at least one of $\overline{F}'_1$ (in red) and $\overline{F}'_2$ (in blue) is at most as long as $\overline{F}$ (in black).

Theorems & Definitions (26)

• Theorem 1
• Lemma 2
• Corollary 4
• Lemma 7
• Lemma 8
• Theorem 9
• Theorem 10
• Theorem 11
• Theorem 12
• Lemma 13: De Berg et al. bbkk-ethtsp-2018