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.
Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs
