Subset-Sum is an NP-complete problem where one must decide if a multiset of integers contains a subset whose elements sum to a target value . The best known classical and quantum algorithms run in time and , respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel dynamic programming data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus , our data structure can be constructed in time , after which queries can be made in time to the lists of subsets summing to a same value modulo . We use this data structure to give new and classical and quantum algorithms for Subset-Sum, not based on the meet-in-the-middle method. We then use the data structure in combination with variable time amplitude amplification and a quantum pair finding algorithm, extending quantum element distinctness and claw finding algorithms to the multiple solutions case, to give an quantum algorithm for Shifted-Sums, an improvement on the best known classical running time. We also study Pigeonhole Equal-Sums and Pigeonhole Modular Equal-Sums, where the existence of a solution is guaranteed by the pigeonhole principle. For the former problem we give classical and quantum algorithms with running time and , respectively. For the more general modular problem we give a classical algorithm which also runs in time .
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