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.

中文翻译：

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度和间隙：由树宽和割宽参数化的一般因子问题的紧密复杂性结果

对于一般因子问题，我们得到一个无向图$ G $，并且对于V（G）$中的每个顶点$ v \ $一个非负整数的有限集$ B_v $。任务是确定是否存在子集$ S \ subseteq E（G）$，使得对于$ G $的所有顶点$ v $，$ deg_S（v）\在B_v $中。有限整数集$ B $的最大间隙是最大的$ d \ ge 0 $，因此存在一个$ a \ ge 0 $，其中$ [a，a + d + 1] \ cap B = \ {a，a + d + 1 \} $。Cornu'ejols（1988）指出，如果所有集合$ B_v $的最大间隙最大为1，则General Factor的决策版本是可乘次解的。Dudycz和Paluch（2018）将此结果扩展到最小化和最大化版本。使用van Rooij（2020）的卷积技术，我们对Arulselvan等人的先前算法进行了改进。（2018）提出了一种算法，该算法在时间$ O ^ *（（（M + 1）^ k）$）中计算一定大小的解的数量，给定宽度为$ k $的树分解，其中$ M = \ max_v \ max B_v $。我们证明了该算法对于决策，最小化或最大化版本无法通过多项式时间求解的所有情况而言，本质上都是最优的。我们证明，即使对于$ B $ -Factor，也无法实现这种改进，这是所有集合$ B_v $与固定集合$ B $一致的图形上的General Factor。我们表明，对于问题为NP困难的每个固定$ B $，我们的新算法都无法得到显着改善：假设强指数时间假设（SETH），没有算法可以及时解决$ B $ -Factor的问题。 （（\ max B + 1- \ epsilon）^ k）$对于任何$ \ epsilon> 0 $。对于任意的，非平凡的集合$ B $，我们将其范围扩展到$ B $ -Factor的计数版本（假设#SETH）。我们还通过cutwidth研究了问题的参数化。与树宽不同，