A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Abstract

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity.Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph.An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G − Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is poly(δ) • log n.As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running inWe also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n o(δ) -time algorithm on planar δ-hyperbolic graphs, unless ETH fails.

Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours

“…Note that the aforementioned recent lower bound on Euclidean TSP [22] implies that our result for B N T S T is Gap-ETH-tight: there is no 2 o(1/ε) poly(n)time (1 + ε)-approximation scheme unless Gap-ETH fails.…”

“…However, it is not clear how to ensure that our solution is noncrossing when we restrict ourselves to solutions that respect only the edges of the spanner. To overcome this obstacle, we use a recently developed sparsity-sensitive patching procedure [22]. Roughly speaking, this allows us to reduce the number of portals from O(log(n)/ε) to O(1/ε 2 ) many portals without a need for spanners.…”

“…This is quite subtle and we need to resolve multiple technical issues. For example, even the initial perturbation step used in previous work [3,22] needs to be modi ed as we cannot simply place the points of the same color in the same point. We need to snap to the grid in such a way, that after looking at the original positions of points, the solution is still noncrossing.…”

“…The structure theorem shows that tours obeying certain restrictions are not much more expensive than unrestricted ones. Kisfaludi-Bak et al [22] called such restricted tours r-simple. Below, we extend this notion to our setting and introduce r-simple pairs of tours.…”

“…Namely, Eades and Rappaport [12] show a reduction of the red-blue separation problem to Euclidean TSP with an arbitrary small gap. This fact combined with a recent lower bound on Euclidean TSP [22] means that an 2 o(1/ε) poly(n)-time approximation for Euclidean Noncrossing R B G S would contradict Gap-ETH.…”