Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Degrees and Gaps: Tight Complexity Results of General Factor Problems
  Parameterized by Treewidth and Cutwidth   For the General Factor problem we are given an undirected graph $G$ and for
each vertex $v\in V(G)$ a finite set $B_v$ of non-negative integers. The task
is to decide if there is a subset $S\subseteq E(G)$ such that $deg_S(v)\in B_v$
for all vertices $v$ of $G$. The maxgap of a finite integer set $B$ is the
largest $d\ge 0$ such that there is an $a\ge 0$ with $[a,a+d+1]\cap
B=\{a,a+d+1\}$. Cornu\'ejols (1988) showed that if the maxgap of all sets $B_v$
is at most 1, then the decision version of General Factor is poly-time
solvable. Dudycz and Paluch (2018) extended this result for the minimization
and maximization versions. Using convolution techniques from van Rooij (2020),
we improve upon the previous algorithm by Arulselvan et al. (2018) and present
an algorithm counting the number of solutions of a certain size in time
$O^*((M+1)^k)$, given a tree decomposition of width $k$, where $M=\max_v \max
B_v$.
  We prove that this algorithm is essentially optimal for all cases that are
not polynomial time solvable for the decision, minimization or maximization
versions. We prove that such improvements are not possible even for $B$-Factor,
which is General Factor on graphs where all sets $B_v$ agree with the fixed set
$B$. We show that for every fixed $B$ where the problem is NP-hard, our new
algorithm cannot be significantly improved: assuming the Strong Exponential
Time Hypothesis (SETH), no algorithm can solve $B$-Factor in time $O^*((\max
B+1-\epsilon)^k)$ for any $\epsilon>0$. We extend this bound to the counting
version of $B$-Factor for arbitrary, non-trivial sets $B$, assuming #SETH.
  We also investigate the parameterization of the problem by cutwidth. Unlike
for treewidth, a larger set $B$ does not make the problem harder: Given a
linear layout of width $k$ we give a $O^*(2^k)$ algorithm for any $B$ and
provide a matching lower bound that this is optimal for the NP-hard cases.

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The categories of this paper are: mtrl-th
2
The categories of this paper are: math
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph