Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

Abstract

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 n Ω( g log g ) conditionally to the 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 the 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 gridlike 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.

Planar Multiway Cut with Terminals on Few Faces

“…, where g is the genus of the surface. This bound is almost tight assuming ETH holds, as was recently shown by Cohen-Added et al [19].…”

Modern Lower Bound Techniques in Database Theory and Constraint Satisfaction

“…However, it later turned out that it is possible to state it in a slightly more robust and expressive way that shows the precise complexity of individual primal graphs. T 6.7 ( [25]). Assuming the ETH, there exists a universal constant > 0 such that for any fixed primal graph with treewidth ≥ 2, there is no algorithm deciding the binary CSP instances = ( , , ) whose primal graph is in time (| | • /log ).…”