In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.
We study the problem $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property $$\varPhi $$ Φ . It is shown that, given any graph property $$\varPhi $$ Φ that distinguishes independent sets from bicliques, $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) is hard for the class $$\#\mathsf {W[1]}$$ # W [ 1 ] , i.e., the parameterized counting equivalent of $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P . Under additional suitable density conditions on $$\varPhi $$ Φ , satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen $$\#\mathsf {W[1]}$$ # W [ 1 ] -hardness by establishing that $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) cannot be solved in time $$f(k)\cdot n^{o(k)}$$ f ( k ) · n o ( k ) for any computable function f, unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.
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