We show that for every r 2 there exists 0"> r 0 such that any r -uniform hypergraph on m edges with bounded vertex degree has a set of at most ( 2 1 − r ) m edges the removal of which breaks the hypergraph into connected components with at most m 2 edges. We use this to give a satisfiability algorithm for n -variable ( d k ) -CSPs in which every variable appears in at most r constraints in time d (1 − r ) n where r depends only on r , provided that k is small enough as a function of m . We will also show that unsatisfiable (2 k ) -CSPs with variable frequency r can be refuted in tree-like resolution in size 2 (1 − r ) n . 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.