Improved Space Bounds for Subset Sum
CoRR(2024)
摘要
More than 40 years ago, Schroeppel and Shamir presented an algorithm that
solves the Subset Sum problem for n integers in time O^*(2^0.5n) and
space O^*(2^0.25n). The time upper bound remains unbeaten, but the space
upper bound has been improved to O^*(2^0.249999n) in a recent breakthrough
paper by Nederlof and Węgrzycki (STOC 2021). Their algorithm is a clever
combination of a number of previously known techniques with a new reduction and
a new algorithm for the Orthogonal Vectors problem.
In this paper, we give two new algorithms for Subset Sum. We start by
presenting an Arthur–Merlin algorithm: upon receiving the verifier's
randomness, the prover sends an n/4-bit long proof to the verifier who checks
it in (deterministic) time and space O^*(2^n/4). The simplicity of this
algorithm has a number of interesting consequences: it can be parallelized
easily; also, by enumerating all possible proofs, one recovers upper bounds on
time and space for Subset Sum proved by Schroeppel and Shamir in 1979. As it is
the case with the previously known algorithms for Subset Sum, our algorithm
follows from an algorithm for 4-SUM: we prove that, using verifier's coin
tosses, the prover can prepare a log_2 n-bit long proof verifiable in time
Õ(n). Another interesting consequence of this result is the following
fine-grained lower bound: assuming that 4-SUM cannot be solved in time
O(n^2-ε) for all ε>0, Circuit SAT cannot be solved in
time O(g2^(1-ε)n), for all ε>0.
Then, we improve the space bound by Nederlof and Węgrzycki to
O^*(2^0.246n) and also simplify their algorithm and its analysis. We
achieve this space bound by further filtering sets of subsets using a random
prime number. This allows us to reduce an instance of Subset Sum to a larger
number of instances of smaller size.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要