Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs [article]

Vincent Cohen-Addad, Éric Colin de Verdière, Daniel Marx, Arnaud de Mesmay
2021 arXiv   pre-print
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We
more » ... rove a time lower bound for this problem of n^Ω(g/log g) conditionally to ETH. In other words, the first n^O(g)-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n^Ω(√(gt + g^2+t)/log(g+t)), conditionally to ETH, for any choice of the genus g≥0 of the graph and the number of terminals t≥4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.
arXiv:1903.08603v3 fatcat:uasnm56bvje37mnm2ociswceqa