# Quick Approximation to Matrices and Applications

@article{Frieze1999QuickAT, title={Quick Approximation to Matrices and Applications}, author={Alan M. Frieze and Ravi Kannan}, journal={Combinatorica}, year={1999}, volume={19}, pages={175-220} }

m×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n).We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph…

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## References

SHOWING 1-10 OF 30 REFERENCES

### Approximating the Permanent

- MathematicsSIAM J. Comput.
- 1989

A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

### How hard is it to marry at random? (On the approximation of the permanent)

- Computer ScienceSTOC '86
- 1986

Al though finding a perfect matching is easy and finding a Hamil tonian circuit is hard, counting perfect matchings and counting Hamiltonian circuits is equally hard, as hard as computing the number of solutions of any problem in NP.

### The algorithmic aspects of the regularity lemma

- Mathematics, Computer ScienceProceedings., 33rd Annual Symposium on Foundations of Computer Science
- 1992

The authors first demonstrate the computational difficulty of finding a regular partition; they show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete, and prove that despite this difficulty theLemma can be made constructive.

### Polynomial time randomised approximation schemes for the Tutte polynomial of dense graphs

- Mathematics, Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

A general technique is developed that supplies fully polynomial randomised approximation schemes for approximating the valve of T(G; x,, y) for any dense graph G, that is, any graph on n vertices whose minimum degree is /spl Omega/(n).

### The regularity lemma and approximation schemes for dense problems

- MathematicsProceedings of 37th Conference on Foundations of Computer Science
- 1996

The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.

### Polynomial time approximation schemes for dense instances of NP-hard problems

- Computer ScienceSTOC '95
- 1995

We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph…

### A Fast Approximation Algorithm for Computing the Frequencies of Subgraphs in a Given Graph

- Mathematics, Computer ScienceSIAM J. Comput.
- 1995

In this paper we give an algorithm which, given a labeled graph on $n$ vertices and a list of all labeled graphs on $k$ vertices, provides for each graph $H$ of this list an approximation to the…

### A new rounding procedure for the assignment problem with applications to dense graph arrangement problems

- MathematicsProceedings of 37th Conference on Foundations of Computer Science
- 1996

Abstract.We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satisfies any linear inequality, then with high…

### Some Simplified NP-Complete Graph Problems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1976

### Excluding Induced Subgraphs III: A General Asymptotic

- Mathematics
- 1992

In this article we study asymptotic properties of the class of graphs not containing a fixed graph H as an induced subgraph. In particular we show that the number Forbn★(H) of such graphs on n…