Combinatorial Mathematics

Ryser, H. J.

doi:10.5948/upo9781614440147

Cited by 835 publications

(281 citation statements)

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“…The best known algorithm for computing the permanent is very old, due to Ryser [37]. He gives a formula based on inclusion-exclusion that computes the permanent of an n×n matrix over a ring in O(2 n poly(n)) time and O(poly(n)) space.…”

Section: Computing Permanents Of Rectangular Matricesmentioning

confidence: 99%

Finding, minimizing, and counting weighted subgraphs

Vassilevska

Williams

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

102

156

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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edge-weighted graphs. Our results include:• The number of copies of an H with an independent set of size s can be computed exactly in O * (2 s n k−s+3 ) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found inThe O * notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings.• The number of copies of any H having minimum (or maximum) node-weight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity.• Finding an edge-weighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε ) time for all ε > 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε ) time. This suggests that the edge-weighted problem is much harder than its node-weighted version.

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Section: Computing Permanents Of Rectangular Matricesmentioning

confidence: 99%

Finding, minimizing, and counting weighted subgraphs

Vassilevska

Williams

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

102

156

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“…If m is odd, then by Lemma 1.4 we know 5 + 1 is a square. By considering the three cases 5 = 0, 3, 7 (mod 8) and using the fact that 5 + 1 is a square, we can show that both the Bruck-Ryser-Chowla theorem [9] applied to A* and the first part of Theorem 1.5 are equivalent in this case to the following: if 5 = 3 (mod 8) and if p is a prime dividing the square free part of 5 -1, then p^l (mod 8).…”

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confidence: 99%

Generalized relative difference sets

Payne¹

1970

Proc. Amer. Math. Soc.

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Abstract.A Bruck-Ryser type nonexistence theorem is given for a class of generalized relative difference sets. Some well-known results on (v, k, X)-designs are generalized and a new class of relative difference sets is given.1. A Bruck-Ryser type nonexistence theorem. Let m and n be positive integers with m>l.Let P0 be the identity matrix Imn of order mn; Pi the matrix Jmn of order mn each entry of which is 1; P2 the direct sum of /" taken m times, where J" is the matrix of order n each entry of which is 1. Then P0, Pj, and P2 form a basis for a commutative, linear associative algebra A* over the rationals Q.Let B = Ei-o CiPi, where CiEQ-Then put

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“…The Gale-Ryser Theorem [5], [6] Let µ and v be two partitions of n. Then there is a matrix of zeros and ones whose columns sum to µ and whose rows sum to v i ff v ~ µ*. There µ* is the dual partition of µ defined by µ~ = #{jlµj ~ i}.…”

Section: Introductionmentioning

confidence: 99%