## The Complexity of Counting in Sparse, Regular, and Planar Graphs (1997)

Venue: | SIAM Journal on Computing |

Citations: | 71 - 0 self |

### BibTeX

@ARTICLE{Vadhan97thecomplexity,

author = {Salil P. Vadhan},

title = {The Complexity of Counting in Sparse, Regular, and Planar Graphs},

journal = {SIAM Journal on Computing},

year = {1997},

volume = {31},

pages = {398--427}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, in very restricted classes of graphs. In particular, it is shown 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. To achieve these results, a new interpolationbased reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NP-hard even when restricted to graphs of maximal degree 3. Previously, restrictedcase complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems. 1 Introduction Ever since the introduction of NP-completeness in the early 1970's, the primary focus of complexity theory has been on decision ...

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Citation Context ...do not know whether hard counting problems remain hard when additional restrictions are placed on the problem instances. A quick glance at Garey and Johnson's famous catalogue of NP-complete problems =-=[GJ79]-=- reveals that the restricted-case complexity of most difficult decision problems is understood in detail. This information is useful, because a complexity-theoretic hardness result often leads us to a... |

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