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