ETH-TIGHT ALGORITHMS FOR LONG PATH AND CYCLE ON UNIT DISK GRAPHS^∗

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{O(√ k)} (n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√ k)} (n + m)^O(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{O(√ k)} (n + m)^O(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^O(√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(^√ k).