8 research outputs found
Faster space-efficient algorithms for Subset Sum, k -Sum, and related problems
We present randomized algorithms that solve subset sum and knapsack instances with n items in O∗ (20.86n) time, where the O∗ (∙ ) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve binary integer programming on n variables with few constraints in a similar running time. We also show that for any constant k ≥ 2, random instances of k-Sum can be solved using O(nk -0.5polylog(n)) time and O(log n) space, without the assumption of random access to random bits.Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.</p
Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems
We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with items using time and polynomial space, where the notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on variables with few constraints in a similar running time. We also show that for any constant , random instances of -Sum can be solved using time and space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length with integers bounded by a polynomial in share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using space significantly faster than the trivial time algorithm if no value occurs too often in the same list
Faster space-efficient algorithms for subset sum, k -sum, and related problems
\u3cp\u3eWe present randomized algorithms that solve subset sum and knapsack instances with n items in O\u3csup\u3e∗\u3c/sup\u3e (2\u3csup\u3e0.86n\u3c/sup\u3e) time, where the O\u3csup\u3e∗\u3c/sup\u3e (∙ ) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve binary integer programming on n variables with few constraints in a similar running time. We also show that for any constant k ≥ 2, random instances of k-sum can be solved using O(n\u3csup\u3ek\u3c/sup\u3e -\u3csup\u3e0.5\u3c/sup\u3epolylog(n)) time and O(log n) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n\u3csup\u3e2\u3c/sup\u3e) time algorithm if no value occurs too often in the same list.\u3c/p\u3
Faster space-efficient algorithms for Subset Sum, k-Sum and related problems
We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with items using time and polynomial space, where the notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on variables with few constraints in a similar running time. We also show that for any constant , random instances of -Sum can be solved using time and space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length with integers bounded by a polynomial in share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using space significantly faster than the trivial time algorithm if no value occurs too often in the same list
Weighted Shortest Common Supersequence problem revisited
A weighted string, also known as a position weight matrix, is a sequence of probability distributions over some alphabet. We revisit the Weighted Shortest Common Supersequence (WSCS) problem, introduced by Amir et al. [SPIRE 2011], that is, the SCS problem on weighted strings. In the WSCS problem, we are given two weighted strings (Formula presented) and (Formula presented) and a threshold (Formula presented) on probability, and we are asked to compute the shortest (standard) string S such that both (Formula presented) and (Formula presented) match subsequences of S (not necessarily the same
Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems
We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with items using time and polynomial space, where the notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on variables with few constraints in a similar running time. We also show that for any constant , random instances of -Sum can be solved using time and space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length with integers bounded by a polynomial in share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using space significantly faster than the trivial time algorithm if no value occurs too often in the same list
Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors
We present an time and
space randomized algorithm for solving
worst-case Subset Sum instances with integers. This is the first
improvement over the long-standing time and
space algorithm due to Schroeppel and Shamir (FOCS
1979).
We breach this gap in two steps: (1) We present a space efficient reduction
to the Orthogonal Vectors Problem (OV), one of the most central problem in
Fine-Grained Complexity. The reduction is established via an intricate
combination of the method of Schroeppel and Shamir, and the representation
technique introduced by Howgrave-Graham and Joux (EUROCRYPT 2010) for designing
Subset Sum algorithms for the average case regime. (2) We provide an algorithm
for OV that detects an orthogonal pair among given vectors in
with support size in time . Our
algorithm for OV is based on and refines the representative families framework
developed by Fomin, Lokshtanov, Panolan and Saurabh (J. ACM 2016).
Our reduction uncovers a curious tight relation between Subset Sum and OV,
because any improvement of our algorithm for OV would imply an improvement over
the runtime of Schroeppel and Shamir, which is also a long standing open
problem.Comment: STOC 2021, 38 pages, 3 figure