Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Computing dominances inEn (short communication)
Information Processing Letters
On the all-pairs-shortest-path problem
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Journal of the ACM (JACM)
All pairs shortest paths for graphs with small integer length edges
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Fast rectangular matrix multiplication and applications
Journal of Complexity
Communications of the ACM
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
Finding Minimally Weighted Subgraphs
WG '90 Proceedings of the 16rd International Workshop on Graph-Theoretic Concepts in Computer Science
All Pairs Shortest Paths in Undirected Graphs with Integer Weights
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The effect of algebraic structure on the computational complexity of matrix multiplication
The effect of algebraic structure on the computational complexity of matrix multiplication
All-pairs shortest paths with real weights in O(n3/ log n) time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
All-pairs bottleneck paths for general graphs in truly sub-cubic time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Dynamic programming and fast matrix multiplication
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
All-pairs bottleneck paths in vertex weighted graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Finding a heaviest triangle is not harder than matrix multiplication
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Nondecreasing paths in a weighted graph or: how to optimally read a train schedule
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Efficient algorithms on sets of permutations, dominance, and real-weighted APSP
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Efficient approximation algorithms for shortest cycles in undirected graphs
Information Processing Letters
Finding, minimizing, and counting weighted subgraphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Efficient approximation algorithms for shortest cycles in undirected graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Finding heaviest H-subgraphs in real weighted graphs, with applications
ACM Transactions on Algorithms (TALG)
Nondecreasing paths in a weighted graph or: How to optimally read a train schedule
ACM Transactions on Algorithms (TALG)
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Finding the smallest H-Subgraph in real weighted graphs and related problems
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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We present the first truly sub-cubic algorithms for finding a maximum node-weighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B • n3+ω/2) = O(B • n2.688) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponent, and B is the number of bits of precision. The second is a strongly polynomial randomized algorithm that runs in O(n3+ω/2 log n) expected worst-case time. To achieve this, we show how to efficiently sample a weighted triangle uniformly at random, out of just those triangles whose total weight falls in some prescribed interval (W1,W2) for arbitrary weights W1 and W2. Previous approaches to the problem resulted in time bounds with either an exponential dependence on B, or a runtime of the form Ω(n3/(log n)c). The algorithms are easily extended to finding a maximum node-weighted induced subgraph on 3k nodes in Õ(n(3+ω)k/2) = O(n2.688 k) time.We give applications to a variety of problems, including a stable matching problem between buyers and sellers in computational economics, and discuss the possibility of extending our approach to a truly sub-cubic algorithm for computing all-pairs shortest paths on directed graphs with arbitrary weights.