265,022 research outputs found
Improved Classical and Quantum Algorithms for Subset-Sum
We present new classical and quantum algorithms for solving random subset-sum
instances. First, we improve over the Becker-Coron-Joux algorithm (EUROCRYPT
2011) from downto
, using more general representations with
values in .
Next, we improve the state of the art of quantum algorithms for this problem
in several directions. By combining the Howgrave-Graham-Joux algorithm
(EUROCRYPT 2010) and quantum search, we devise an algorithm with asymptotic
cost , lower than the cost of the quantum
walk based on the same classical algorithm proposed by Bernstein, Jeffery,
Lange and Meurer (PQCRYPTO 2013). This algorithm has the advantage of using
\emph{classical} memory with quantum random access, while the previously known
algorithms used the quantum walk framework, and required \emph{quantum} memory
with quantum random access.
We also propose new quantum walks for subset-sum, performing better than the
previous best time complexity of given by
Helm and May (TQC 2018). We combine our new techniques to reach a time
. This time is dependent on a heuristic on
quantum walk updates, formalized by Helm and May, that is also required by the
previous algorithms. We show how to partially overcome this heuristic, and we
obtain an algorithm with quantum time
requiring only the standard classical subset-sum heuristics
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time O?(2^{n/2}) and O?(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based 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 p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time O?(2^{n/2}) and O?(2^{2n/5}), respectively
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time O?(2^{n/2}) and O?(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based 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 p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time O?(2^{n/2}) and O?(2^{2n/5}), respectively
Faster Minimization of Tardy Processing Time on a Single Machine
This paper is concerned with the problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in time, where is the total processing time of all jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for , each improving on Lawler and Moore's algorithm in a different scenario. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of -convolution which is interesting in its own right
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