Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

Vincent Cohen-Addad,Éric Colin De Verdière,Dániel Marx,Arnaud De Mesmay

doi:10.1145/3450704

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.

Highlights

There has been a flurry of works investigating the complexity of solving exactly optimization problems on planar graphs, leading to what was coined as the “square root phenomenon” by the third author [27]: many problems turn out to be easier on planar graphs, and the improvement compared to the general case is captured exactly by a square root
On the side of upper bounds, the improvement oft√en stems from the fact that planar graphs have planar separators of size O( n), and the theory of bidimensionality provides an elegant framework for a similar speedup in the parameterized setting for some problems [12]
Isomorphism, where some of the hardness results feature the genus of the graph, the lower bounds of Curticapean and the third author [8] on the problem of counting perfect matchings and the work of Chen et al [1]. We address this surprising gap by providing lower bounds conditioned on Exponential Time Hypothesis (ETH) for two fundamental yet seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem

Summary

Introduction

There has been a flurry of works investigating the complexity of solving exactly optimization problems on planar graphs, leading to what was coined as the “square root phenomenon” by the third author [27]: many problems turn out to be easier on planar graphs, and the improvement compared to the general case is captured exactly by a square root. 2. Assuming ETH, there exists a universal constant αCG such that for any fixed integer g ≥ 2, there is no algorithm solving all the Shortest Cut Graph instances of genus at most g in time O(nαCG·g/ log g). The Grid Tiling problem of the third author [24] has emerged as a convenient, almost universal, tool to establish parameterized hardness results and precise lower bounds based on ETH. A result of the third author [25] shows that, assuming the ETH, such CSP instances cannot be solved in time f (k)nΩ(k/ log k), giving a similar lower bound for 4-Regular Graph Tiling (Theorem 9). The same hardness result holds in the embedded case and the question is not about whether we are given the embedding or not

Preliminaries

The 4-regular graph tiling problem

Multiway cut with four terminals

Shortest cut graph

Multiway cut with a large number of terminals