# Efficient diagonalization of symmetric matrices associated with graphs of small treewidth

Let M=(m_ij) be a symmetric matrix of order n whose elements lie in an arbitrary field π½, and let G be the graph with vertex set {1,β¦,n} such that distinct vertices i and j are adjacent if and only if m_ijβ  0. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to M. If G is given with a tree decomposition π― of width k, then this can be done in time O(k|π―| + k^2 n), where |π―| denotes the number of nodes in π―. Among other things, this allows one to compute the determinant, the rank and the inertia of a symmetric matrix in time O(k|π―| + k^2 n).

## Authors

• 5 publications
• 3 publications
• 1 publication
• ### A heuristic use of dynamic programming to upperbound treewidth

For a graph G, let Ξ©(G) denote the set of all potential maximal cliques ...
09/17/2019 β by Hisao Tamaki, et al. β 0 β

• ### A polynomial time algorithm to compute the connected tree-width of a series-parallel graph

It is well known that the treewidth of a graph G corresponds to the node...
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• ### Fast FPT-Approximation of Branchwidth

Branchwidth determines how graphs, and more generally, arbitrary connect...
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• ### Spectral and Combinatorial Properties of Some Algebraically Defined Graphs

Let k> 3 be an integer, q be a prime power, and F_q denote the field of ...
08/25/2017 β by Sebastian M. CioabΔ, et al. β 0 β

• ### Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs

For graphs G and H, a homomorphism from G to H is an edge-preserving map...
06/19/2019 β by Karolina Okrasa, et al. β 0 β

• ### Fast Algorithms for Join Operations on Tree Decompositions

Treewidth is a measure of how tree-like a graph is. It has many importan...
06/02/2020 β by Johan M. M. van Rooij, et al. β 0 β