Computing the Minimum Fill-in is NP^Complete

@article{Yannakakis1981ComputingTM,
  title={Computing the Minimum Fill-in is NP^Complete},
  author={Mihalis Yannakakis},
  journal={Siam Journal on Algebraic and Discrete Methods},
  year={1981},
  volume={2},
  pages={77-79}
}
  • M. Yannakakis
  • Published 1 March 1981
  • Computer Science, Mathematics
  • Siam Journal on Algebraic and Discrete Methods
We show that the following problem is NP-complete. Given a graph, find the minimum number of edges (fill-in) whose addition makes the graph chordal. This problem arises in the solution of sparse symmetric positive definite systems of linear equations by Gaussian elimination. 
View via Publisher
768 Citations
Highly Influential Citations
51
Background Citations
352
Methods Citations
63
Results Citations
3

768 Citations

Citation Type

Citation Type

Minimal fill in O(n2.69) time
Algorithms for Maximum Matching and Minimum Fill-in on Chordal Bipartite Graphs
TLDR
A linear time algorithm is given for the maximum matching problem and an O(n4) time algorithm for the minimum fill-in problem on chordal bipartite graphs improving previous results.
A Linear Time Algorithm for Minimum Fill-in and Treewidth for Distance Hereditary Graphs
Minimum fill-in for chordal bipartite graphs
Chordal bipartite graph are exactly those bipartite graph in which every cycle of length at least six has a chord. The MINIMUM FILL-IN problem is the problem of finding a chordal embedding of the…
Minimal Elimination of Planar Graphs
TLDR
It is proved that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs with the use of breadth-first search.
Pathwidth is NP-Hard for Weighted Trees
TLDR
It is proved in this paper that the pathwidth problem is NP-hard for particular subclasses of chordal graphs, and it is deduced that the problem remains hard for weighted trees.
Minimum Fill-In on Circle and Circular-Arc Graphs
TLDR
Two algorithms are presented solving the minimum fill-in problem on circle graphs and on circular-arc graphs in time O(n3) in order to solve the inequality of the following type: circle to circular graph.
On the Minimum Chordal Completion Polytope
A graph is chordal if every cycle of length at least four contains a chord—that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear syst...
Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In
We show that for a graph G on n vertices its treewidth and minimum fill-in can be computed roughly in 1.9601 n time. Our result is based on a combinatorial proof that the number of minimal separators…
How to Use the Minimal Separators of a Graph for its Chordal Triangulation
We prove a 1-1 correspondence between maximal sets of pairwise parallel minimal separators of a graph and its minimal chordal triangulations. This yields polynomial-time algorithms to determine the…
...
...

References

SHOWING 1-6 OF 6 REFERENCES
Algorithmic Aspects of Vertex Elimination on Graphs
TLDR
A graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations is considered, and it is conjecture that the problem of finding a minimum ordering is NP-complete.
Algorithmic aspects of vertex elimination on directed graphs.
We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse systems of linear eauations. We give efficient algorithms to: (1) calculate the fill-in…
Some Simplified NP-Complete Graph Problems
A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS
Node-Deletion Problems on Bipartite Graphs
TLDR
Those properties for which the bipartite restriction of the node-deletion problem is polynomial and those for which it remains NP-complete are characterized.
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge…