2019/03/20 by Vincent Cohen-Addad, Éric Colin de Verdière, Cohen-Addad, Vincent +5 · 1 citation
Computer Science · Engineering · #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Optimization and Packing Problems
paper · doi:10.48550/arxiv.1903.08603
openalex publication_date 2019/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 prove a time lower bound for this problem of nΩ(g/log g) conditionally to ETH. In other words, the first nO(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 nΩ(√(gt + g2+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.