Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs   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 $n^{\Omega(g/\log g)}$ 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 $n^{\Omega(\sqrt{gt +
g^2+t}/\log(g+t))}$, conditionally to ETH, for any choice of the genus $g\ge0$
of the graph and the number of terminals $t\ge4$. 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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas