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

Venue: | SIAM Journal on Computing |

Citations: | 69 - 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 ...

### Citations

10921 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
(Show Context)
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... |

2343 | Computational Complexity - Papadimitriou - 1994 |

536 | A Classical Introduction to Modern Number Theory, Second Edition - Ireland, Rosen - 1990 |

506 |
The Complexity of Computing the Permanent
- Valiant
- 1979
(Show Context)
Citation Context ... counting problems. It is easy to construct counting problems which are obviously equivalent in difficulty, but for which there can be no one-to-one correspondence between solution sets. Hence, as in =-=[Val79a]-=-, we consider a problem \Pi to be as hard as a problem \Gamma iff \Gamma can be solved by a polynomial-time Turing machine with a \Pi-oracle, and we denote this by \Gammas\Pi. Such a reduction is know... |

421 |
The Complexity of Enumeration and Reliability Problems
- Valiant
- 1979
(Show Context)
Citation Context ...by [DL92] have shown that counting perfect matchings remains #P-complete when restricted to 3-regular bipartite graphs, the standard reduction from counting perfect matchings to counting matchings in =-=[Val79b]-=- blows up the degree of the graph and does not enable us to conclude that counting matchings remains difficult in either regular or bounded-degree graphs. In this paper, we introduce a new reduction t... |

314 | Polynomial-time approximation algorithms for the ising model - Jerrum, Sinclair - 1993 |

297 | Approximating the permanent - Jerrum, Sinclair - 1989 |

281 |
Random generation of combinatorial structures from a uniform distribution
- Jerrum, Valiant, et al.
- 1986
(Show Context)
Citation Context ...ng point for achieving other such results. 2 Preliminaries Nearly all of the counting problems we will be considering are in Valiant's class #P, and the remainder are closely related to #P. Following =-=[JVV86]-=-, #P can be defined in terms of p-relations. Let \Sigma be a finite alphabet. A relation R ae \Sigma \Theta \Sigma is said to be a prelation iff it is polynomially-balanced, i.e. there exists a polyno... |

257 | Approximate counting, uniform generation and rapidly mixing Markov chains - Jerrum, Sinclair - 1989 |

218 | On the Hardness of Approximate Reasoning
- Roth
- 1996
(Show Context)
Citation Context ...ially equivalent to counting the number of ways that the edges could fail without losing connectivity. Another application of counting arises in Artificial Intelligence research. As explained by Roth =-=[Rot96]-=-, the Bayesian approach to reasoning requires evaluating the probability that a formula in the propositional calculus is true, and this is tantamount to counting the number of satisfying assignments. ... |

180 | Algorithms for Random Generation and Counting: A Markov Chain Approach - Sinclair - 1993 |

126 |
The complexity of counting cuts and of computing the probability that a graph is connected
- Provan, Ball
- 1983
(Show Context)
Citation Context ...: ; n. By Fact 4.1, we can recover the coefficients of f in polynomial time. A 0 is the number of perfect matchings in G. 5. #4\Delta-Matchings #5\Delta-Maximal Matchings The reduction we use is from =-=[PB83]-=-, though they use it to prove something different. Given a bipartite undirected graph G = (V; E) of degrees4, construct G 0 = (V 0 ; E 0 ), where V 0 = V [ fv 0 : v 2 V g and E 0 = E [ f(v; v 0 ) : v ... |

107 | On the computational complexity of the Jones and Tutte polynomials - Jaeger, Vertigan, et al. - 1990 |

84 |
Dimer statistics and phase transitions
- Kasteleyn
- 1963
(Show Context)
Citation Context ...lanar graphs [Val79b, Jer87, Jer90]. It may stand out that there is no hardness result for counting perfect matchings in planar graphs. However, efficient algorithms, due to Fisher [Fis61], Kasteleyn =-=[Kas63]-=-, and Temperley and Fisher [TF61], are known for this problem, so a #P-completeness result would be very unexpected. We also prove that counting minimum cardinality vertex covers within a ratio of 2 n... |

66 | The complexity of computing the volume of a polyhedron - Dyer, Frieze - 1988 |

66 |
Dimer problem in statistical mechanics – an exact result. Philosophical Magazine 6: 1061
- Temperley, Fisher
- 1961
(Show Context)
Citation Context ...]. It may stand out that there is no hardness result for counting perfect matchings in planar graphs. However, efficient algorithms, due to Fisher [Fis61], Kasteleyn [Kas63], and Temperley and Fisher =-=[TF61]-=-, are known for this problem, so a #P-completeness result would be very unexpected. We also prove that counting minimum cardinality vertex covers within a ratio of 2 n 1\Gammaffl remains NP-hard even ... |

47 | Markov chains and polynomial time algorithms
- Kannan
- 1994
(Show Context)
Citation Context ...overs could be found even in very restricted cases. Sinclair and Jerrum's Markov chain techniques [SJ89, Sin93] have yielded polynomial-time approximation algorithms for a number of counting problems =-=[Kan94]-=-; perhaps they can be applied to the problems here. Some progress has already been made in this direction. There are polynomial-time Markov chain algorithms for approximately counting the number of ma... |

46 | Counting linear extensions is #p-complete - Brightwell, Winkler - 1991 |

41 | Two-dimensional monomer-dimer systems are computationally intractable - Jerrum - 1987 |

36 |
Statistical mechanics of dimers on a plane lattices
- Fisher
- 1961
(Show Context)
Citation Context ... and unrestricted planar graphs [Val79b, Jer87, Jer90]. It may stand out that there is no hardness result for counting perfect matchings in planar graphs. However, efficient algorithms, due to Fisher =-=[Fis61]-=-, Kasteleyn [Kas63], and Temperley and Fisher [TF61], are known for this problem, so a #P-completeness result would be very unexpected. We also prove that counting minimum cardinality vertex covers wi... |

36 | enumeration problems in geometry and combinatorics - Hard - 1986 |

26 | Graph orientations with no sink and an approximation for a hard case - Bubley, Dyer - 1997 |

25 | Approximating the permanent of graphs with large factors
- Dagum, Luby
- 1992
(Show Context)
Citation Context ...stroy special properties of the original problem instance. This makes it difficult to deduce additional restricted-case results from known restrictedcase results. For example, although Dagum and Luby =-=[DL92]-=- have shown that counting perfect matchings remains #P-complete when restricted to 3-regular bipartite graphs, the standard reduction from counting perfect matchings to counting matchings in [Val79b] ... |

17 |
On the solutions of x d = 1 in symmetric groups
- Moser, Wyman
- 1955
(Show Context)
Citation Context ...1) is a bit harder to handle. Dividing by M 2 k\Gamma1 and applying the quadratic formula, we see that the condition can be reformulated as: M k M k\Gamma1 6= 1 \Sigma p 4k \Gamma 3 2 Moser and Wyman =-=[MW55]-=- have shown the following: 7 1 + p 4k \Gamma 3 2 M k M k\Gamma1 1 + p 4k + 1 2 7 Moser and Wyman discuss the number of solutions to x 2 = 1 in the symmetric group on k elements. It is easy to see that... |

17 | On the number of Eulerian orientations of a graph - Mihail, Winkler - 1992 |

11 |
Eric Vigoda. Approximately counting up to four (extended abstract
- Luby
- 1997
(Show Context)
Citation Context ...ov chain algorithms for approximately counting the number of matchings in general graphs and the number of perfect matchings in dense graphs or graphs with large factors [JS89, DL92]. Luby and Vigoda =-=[LV97]-=- have recently devised a Markov chain which approximates the number of independent sets in graphs of degree at most four [LV97]. These results, and that of [LV97] in particular, are even more surprisi... |

8 | Automating Polya theory: The computational complexity of the cycle index polynomial - Goldberg - 1993 |

5 | editor: Graph Theory and Theoretical - Harary - 1967 |

4 |
On closure properties of #P
- Ogihara, Thierauf, et al.
- 1996
(Show Context)
Citation Context ... cardinality is NP-hard [GJ79, Thm 3.3]. But we will reduce it to a #P problem, proving that it is #P-easy. The failure of #P to be closed under reductions and even simpler operations is discussed in =-=[OTTW96]-=-. Input: A regular graph G. Output: The number of minimal vertex covers in G. Corollary 1 The following problems are #P-complete: 1. #3\Delta-Planar Bipartite Maximum Cardinality Independent Set Input... |

1 |
The complexity of counting. Undergraduate thesis
- Vadhan
- 1995
(Show Context)
Citation Context ...puter Science, Massachusetts Institute of Technology, 545 Technology Square, Cambridge, MA 02139. Email: salil@math.mit.edu. Some of this work originally appeared in the author's undergraduate thesis =-=[Vad95]-=-, done at Harvard University under the supervision of Leslie Valiant. The author is currently supported by a National Defense Science and Engineering Graduate Fellowship. explanation for the apparent ... |