We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph on $m$ edges with bounded vertex degree has a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

中文翻译：

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超图的分离定理和CSP-SAT算法

我们证明，对于每个$ r \ ge 2 $，都存在$ \ epsilon_r> 0 $，这样，在$ m $边上具有受限顶点度的任何$ r $-均匀超图最多具有一组$（\ frac {1} {2}-\ epsilon_r）m $边的去除将超图分解为具有最多$ m / 2 $边的连接的分量。我们使用它来给出运行时间为$ d ^ {（1-\ epsilon_r）m} $的算法，该算法确定$ m $变量$（d，k）$-CSP的可满足性，其中每个变量最多出现在$ r $约束，其中$ \ epsilon_r $仅取决于o（\ sqrt {m}）$中的$ r $和$ k \。此外，我们的算法可解决这些CSP的相应＃CSP-SAT和Max-CSP-SAT。我们还表明，具有不定频率的$ r $的$（2，k）$-CSP的CNF表示可以用树状分辨率$ 2 ^ {（（-\ epsilon_r）m} $来驳斥。