ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

Abstract

We present an algorithm for the extensively studied {\sc Long Path} and {\sc Long Cycle} problems on unit disk graphs that runs in time $2^{O(\sqrt{k})}(n+m)$. Under the Exponential Time Hypothesis, {\sc Long Path} and {\sc Long Cycle} on unit disk graphs cannot be solved in time $2^{o(\sqrt{k})}(n+m)^{O(1)}$ [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the $2^{O(\sqrt{k})}(n+m)^{O(1)}$-time algorithm for the (arguably) much simpler {\sc Vertex Cover} problem by de Berg et al.~[STOC 2018] (which easily follows from the existence of a $2k$-vertex kernel for the problem), {\em this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs}. Previously, {\sc Long Path} and {\sc Long Cycle} on unit disk graphs were only known to be solvable in time $2^{O(\sqrt{k}\log k)}(n+m)$. This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width $O(\sqrt{k})$.