Dynamic programming and fast matrix multiplication
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
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Cybernetics and Systems Analysis
Minimal proper interval completions
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Theoretical Computer Science
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Theoretical Computer Science
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A note on minimal d-separation trees for structural learning
Artificial Intelligence
Fast minimal triangulation algorithm using minimum degree criterion
Theoretical Computer Science
Subexponential parameterized algorithm for minimum fill-in
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An O(n2)-time algorithm for the minimal interval completion problem
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Fully dynamic algorithm for chordal graphs with O(1) query-time and O(n 2) update-time
Theoretical Computer Science
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Triangulation and clique separator decomposition of claw-free graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
An O( n2)-time algorithm for the minimal interval completion problem
Theoretical Computer Science
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The problem of computing minimal triangulations of graphs, also called minimal fill, was introduced and solved in 1976 by Rose, Tarjan, and Lueker [SIAM J. Comput., 5 (1976), pp. 266-283] in time $O(nm)$ and thus $O(n^3)$ for dense graphs. Although the topic has received increasing attention since then and several new results on characterizing and computing minimal triangulations have been presented, this first time bound has remained the best. In this paper we introduce an $O(n^\alpha \log n)$ time algorithm for computing minimal triangulations, where $O(n^\alpha)$ is the time required to multiply two $n \times n$ matrices. The current best known $\alpha$ is less than $2.376$, and thus our result breaks the longstanding asymptotic time complexity bound for this problem. To achieve this result, we introduce and combine several techniques that are new to minimal triangulation algorithms, such as working on the complement of the input graph, graph search for a vertex set $A$ that bounds the size of the connected components when $A$ is removed, and matrix multiplication.