# An algorithm for counting maximum weighted independent sets and its applications

@inproceedings{Dahllf2002AnAF, title={An algorithm for counting maximum weighted independent sets and its applications}, author={Vilhelm Dahll{\"o}f and P. Jonsson}, booktitle={SODA '02}, year={2002} }

We present an O(1.3247n) algorithm for counting the number of independent sets with maximum weight in graphs. We show how this algorithm can be used solving a number of different counting problems: counting exact covers, exact hitting sets, weighted set packing and satisfying assignments in 1-in-k SAT.

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#### References

SHOWING 1-10 OF 23 REFERENCES

Finding a Maximum Independent Set

- Mathematics, Computer Science
- SIAM J. Comput.
- 1977

An algorithm is presented which finds a maximum independent set in an n-vertex graph in 0($2^{n/3}$) time and can thus handle graphs roughly three times as large as could be analyzed using a naive algorithm. Expand

The complexity of counting colourings and independent sets in sparse graphs and hypergraphs

- Mathematics, Computer Science
- computational complexity
- 2000

Using polynomial interpolation techniques, it is shown that certain counting problems involving colourings of graphs and independent sets in hypergraphs are #P-complete and efficient approximate counting is the most one can realistically expect to achieve. Expand

An O(20.304n) Algorithm for Solving Maximum Independent Set Problem

- Mathematics, Computer Science
- IEEE Trans. Computers
- 1986

A faster algorithm for finding a maximum independent set in a graph is presented. The algorithm is an improved version of the one by Tarjan and Trojanowski [7]. A technique to further accelerate this… Expand

Algorithms for Maximum Independent Sets

- Mathematics, Computer Science
- J. Algorithms
- 1986

Abstract An algorithm is presented which finds (the size of) a maximum independent set of an n vertex graph in time O (2 0.276 n ) improving on a previous bound of O(2 n 3 ) . The improvement comes… Expand

The Complexity of Counting in Sparse, Regular, and Planar Graphs

- Computer Science, Mathematics
- SIAM J. Comput.
- 2001

It is proved that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. Expand

Algorithms for Sat and Upper Bounds on Their Complexity

- Computer Science, Mathematics
- Electron. Colloquium Comput. Complex.
- 2001

We survey recent algorithms for the propositional satisfiability problem. In particular, we consider algorithms having the best current worst-case upper bounds on their complexity. We also discuss… Expand

A Separator Theorem for Planar Graphs

- Mathematics
- 1977

Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more… Expand

Greedy local improvement and weighted set packing approximation

- Mathematics, Computer Science
- SODA '99
- 1999

An approximation algorithm for the weighted k-set packing problem is presented that combines the two paradigms by starting with an initial greedy solution and then repeatedly choosing the best possible local improvement, which is the first asymptotic improvement over the straightforward ratio of k. Expand

Finding maximum independent sets in sparse and general graphs

- Mathematics, Computer Science
- SODA '99
- 1999

The fastest MIS algorithms for sparse graphs and for graphs whose degree is bounded by 3 or by 4 are obtained, and the fastest MIS algorithm for bounded-degree graphs is obtained. Expand

Approximation algorithms for NP-complete problems on planar graphs

- Mathematics, Computer Science
- 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand