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Where firstorder and monadic secondorder logic coincide
 In LICS
, 2012
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Complexity of Regular Functions
"... Abstract. We give complexity bounds for various classes of functions computed by cost register automata. ..."
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Abstract. We give complexity bounds for various classes of functions computed by cost register automata.
The Complexity of Unary Subset Sum
"... Abstract. Given a stream of n numbers and a number B, the subset sum problem deals with checking whether there exists a subset of the stream that adds to exactly B. The unary subset sum problem, USS, is the same problem when the input is encoded in unary. We prove that any ppass randomized algorith ..."
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Abstract. Given a stream of n numbers and a number B, the subset sum problem deals with checking whether there exists a subset of the stream that adds to exactly B. The unary subset sum problem, USS, is the same problem when the input is encoded in unary. We prove that any ppass randomized algorithm computing USS with error at most 1/3 must use space Ω ( B p). For p ≤ B, we give a randomized ppass algorithm that computes USS with error at most 1/3 using space We give a deterministic onepass algorithm which given an input stream and two parameters B, ɛ, decides whether there exist a subset of the input stream that adds to a value in the range [(1 − ɛ)B, (1 + ɛ)B] using Õ( nB p). space O () log B. We observe that USS is monotone (under a suitable ɛ encoding) and give a monotone NC 2 circuit for USS. We also show that any circuit using εapproximator gates for USS under this encoding needs Ω(n / log n) gates to compute the Disjointness function. 1
SpaceEfficient Approximations for Subset Sum?
"... Abstract. SubsetSum is a well known NPcomplete problem: given t ∈ Z+ and a set S of m positive integers, output YES if and only if there is a subset S ′ ⊆ S such that the sum of all numbers in S ′ equals t. The problem and its search and optimization versions are known to be solvable in pseudopol ..."
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Abstract. SubsetSum is a well known NPcomplete problem: given t ∈ Z+ and a set S of m positive integers, output YES if and only if there is a subset S ′ ⊆ S such that the sum of all numbers in S ′ equals t. The problem and its search and optimization versions are known to be solvable in pseudopolynomial time in general. We develop a 1pass deterministic streaming algorithm that uses space O log t and decides if some subset of the input stream adds up to a value in the range {(1 ± )t}. Using this algorithm, we design space efficient Fully PolynomialTime Approximation Schemes (FPTAS) solving the search and optimization versions of SubsetSum. Our algorithms run in O ( 1 m2) time and O ( 1) space on unit cost RAMs, where 1 + is the approximation factor. This implies constant space quadratic time FPTAS on unit cost RAMs when is a constant. Previous FPTAS used space linear in m. In addition, we show that on certain inputs, when a solution is located within a short prefix of the input sequence, our algorithms may run in sublinear time. We apply our techniques to the problem of finding balanced separators, and we extend our results to some other variants of the more general knapsack problem. When the input numbers are encoded in unary, the decision version has been known to be in log space. We give streaming space lower and upper bounds for unary SubsetSum. If the input length is N when the numbers are encoded in unary, we show that randomized spass streaming algorithms for exact SubsetSum need space Ω( N
Contemporary Mathematics Knapsack and subset sum problems in nilpotent, polycyclic, and cocontextfree groups
"... In their paper [24], Myasnikov, Nikolaev, and Ushakov started the investigation of classical discrete integer optimization problems in general noncommutative groups. Among other problems, they introduced for a finitely generated (f.g.) group G the knapsack problem and the subset sum problem. The i ..."
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In their paper [24], Myasnikov, Nikolaev, and Ushakov started the investigation of classical discrete integer optimization problems in general noncommutative groups. Among other problems, they introduced for a finitely generated (f.g.) group G the knapsack problem and the subset sum problem. The input for the knapsack