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01 integer linear programming with a linear number of constraints. arXiv preprint arXiv:1401.5512
"... We give an exact algorithm for the 01 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time 2 (1−poly(1/c))n where n is the number of variables and cn is the number of constrain ..."
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We give an exact algorithm for the 01 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time 2 (1−poly(1/c))n where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algorithm for that subproblem.
More applications of the polynomial method to algorithm design
, 2015
"... In lowdepth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into lowdegree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute all ..."
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In lowdepth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into lowdegree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute allpairs shortest paths in n3/2Ω( logn) time on dense nnode graphs. In this paper, we extend this methodology to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods. First, we give an algorithm for BOOLEAN ORTHOGONAL DETECTION, which is to detect among two sets A,B ⊆ {0, 1}d of size n if there is an x ∈ A and y ∈ B such that 〈x, y 〉 = 0. For vectors of dimension d = c(n) logn, we solve BOOLEAN ORTHOGONAL DETECTION in n2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. We apply this as a subroutine in several other new algorithms: • In BATCH PARTIAL MATCH, we are given n query strings from from {0, 1,?}c(n) logn (? is a “don’t care”), n strings from {0, 1}c(n) logn, and wish to determine for each query whether or not there is a string matching the query. We solve this problem in n2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. • Let t ≤ v be integers. Given a DNF F on c log t variables with t terms, and v arbitrary assignments on the variables, F can be evaluated on all v assignments in v · t1−1/O(log c) time, with high probability. • There is a randomized algorithm that solves the Longest Common Substring with don’t cares problem on two strings of length n in n2/2Ω( logn) time. • Given two strings S, T of length n, there is a randomized algorithm that computes the length of the longest substring of S that has EditDistance less than k to a substring of T in k1.5n2/2Ω( logn