## Distance approximation in bounded-degree and general sparse graphs (2006)

Venue: | In Proceedings of the Tenth International Workshop on Randomization and Computation (RANDOM |

Citations: | 12 - 4 self |

### BibTeX

@INPROCEEDINGS{Marko06distanceapproximation,

author = {Sharon Marko and Dana Ron},

title = {Distance approximation in bounded-degree and general sparse graphs},

booktitle = {In Proceedings of the Tenth International Workshop on Randomization and Computation (RANDOM},

year = {2006}

}

### OpenURL

### Abstract

We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P. This fraction is taken with respect to a given upper bound m on the number of edges. In particular, for graphs with degree bound d over n vertices, m = dn. To perform such an approximation the algorithm may ask for the degree of any vertex of its choice, and may ask for the neighbors of any vertex. The problem of estimating the distance to having a property was first explicitly addressed by Parnas et. al. (ECCC 2004). In the context of graphs this problem was studied by Fischer and Newman (FOCS 2005) in the dense-graphs model. In this model the fraction of edge modifications is taken with respect to n 2, and the algorithm may ask for the existence of an edge between any pair of vertices of its choice. Fischer and Newman showed that every graph property that has a testing algorithm in this model with query complexity that is independent of the size of the graph, also has a distance-approximation algorithm with query complexity that is independent of the size of the graph. In this work we focus on bounded-degree and general sparse graphs, and give algorithms for all properties that were shown to have efficient testing algorithms by Goldreich and Ron (Algorithmica, 2002). Specifically, these properties are k-edge connectivity, subgraph-freeness (for constant size subgraphs), being a Eulerian graph, and cycle-freeness. A variant of our subgraph-freeness algorithm approximates the size of a minimum vertex cover of a graph in sublinear time. This approximation improves on a recent result of Parnas and Ron (ECCC 2005).

### Citations

427 | Property testing and its connection to learning and approximation
- Goldreich, Goldwasser, et al.
- 1998
(Show Context)
Citation Context ...ximation in Dense Graphs. In particular, Fischer and Newman [FN05] proved a general result on the relation between distance approximation and property testing in the dense-graphs model (introduced in =-=[GGR98]-=-). In this model, the distance of a graph G = (V, E) to having a property is defined as the fraction of edges that should be added/removed in order to obtain the property, where the fraction is with r... |

367 | A Simple Parallel Algorithm for the Maximal Independent Set Problem
- Luby
- 1986
(Show Context)
Citation Context ...r can be obtained using modifications of the subgraph-freeness algorithm. The algorithm is similar in spirit to the O(log n)-rounds distributed approximation algorithm for the maximal independent set =-=[Lub86]-=-. In the case of k-connectivity (k > 1), a relatively direct adaptation of the algorithm in [GR02] would give a multiplicative error of k (in addition to the additive error). To get a purely additive ... |

235 | Multi-commodity network flows - Hu - 1963 |

119 | Property testing in bounded degree graphs
- Goldreich, Ron
(Show Context)
Citation Context ...l in which Fischer and Newman obtained their result is clearly appropriate for dense graphs but not for sparse graphs. When studying property testing of sparse graphs, two models were considered (see =-=[GR02]-=- and [PR02]). In both models the testing algorithm may perform degree queries and neighbor queries. 1 That is, for any vertex v the algorithm may ask for the degree of v, and for any index i it may as... |

58 | R.: The price of being nearsighted
- Kuhn, Moscibroda, et al.
- 2006
(Show Context)
Citation Context ...er in a graph with degree bound d, using O(log(d/δ)) rounds. This improves on the (2 + δ)-factor approximation using O(δ −3 · log d) rounds given in a recent paper of Kuhn, Moscibroda and Wattenhofer =-=[KMW06]-=-. Their algorithm, which is quite complex, uses Linear Programming and falls into a more general framework of approximation algorithms for covering and packing problems. As in [PR05] we can use our al... |

50 | Global Min-Cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm
- Karger
- 1993
(Show Context)
Citation Context ...in this case the probability that no cut edge is traversed before X (t) v is found is at least 2t −2 . Their analysis is based on Karger’s analysis of his algorithm for finding minimum cut in a graph =-=[Kar93]-=-. Lemma 3 [GR02] For an undirected graph G, let L be a set of at most t vertices such that the cut (L, L) is a minimum cut. Then, starting with some vertex v ∈ L, the random search process of Step 1.a... |

47 | Tolerant property testing and distance approximation, manuscrip - Parnas, Ron, et al. - 2004 |

42 | Approximating the minimum spanning tree weight in sublinear time
- Chazelle, Rubinfeld, et al.
- 2005
(Show Context)
Citation Context ...thms [GR02] (where our error parameter δ is replaced by the distance parameter ɛ, and ¯ d is replaced by d). • Approximating the distance to k-connectivity for the special case k = 1 was addressed in =-=[CRT05]-=- as a central part of their algorithm for estimating the weight of a minimum spanning tree. • Subgraph-freeness is the only result in which we have a multiplicative factor in addition to the additive ... |

36 | Tight bounds for testing bipartiteness in general graphs
- Kaufman, Krivelevich, et al.
- 2004
(Show Context)
Citation Context ...l properties shown to have efficient property testing algorithms in [GR02] also have efficient distance 1 A third model, appropriate for testing properties of graphs that are neither dense nor sparse =-=[KKR04]-=-, also allows vertex-pair queries. 2 If v has less than i neighbors then a special symbol is returned. 1sapproximation algorithms. We leave open the interesting question of whether there exists a gene... |

29 | Testing the diameter of graphs. Random Structures and Algorithms - Parnas, Ron - 2002 |

28 | Testing versus estimation of graph properties
- Fischer, Newman
(Show Context)
Citation Context ...roximation, both positive [ACCL04, GR05, FN05] and negative [FF05]. These works considered properties of functions and strings [PRR, ACCL04, FF05, GR05], ensembles of points [PRR], and (dense) graphs =-=[FN05]-=-. Distance Approximation in Dense Graphs. In particular, Fischer and Newman [FN05] proved a general result on the relation between distance approximation and property testing in the dense-graphs model... |

26 | Very Simple Methods for All Pairs Network Flow Analysis - Gusfield - 1990 |

21 | Testing triangle-freeness in general graphs
- Alon, Kaufman, et al.
- 2006
(Show Context)
Citation Context ... to a factor √ C approximation in the symmetric definition. However, we find that it is less natural in our context. 2s• Testing subgraph-freeness in the general sparse model requires Ω( √ n) queries =-=[AKKR06]-=-, and the same is true for cycle-freeness. Techniques. Among the aforementioned results, the more interesting techniques are applied in distance approximation of subgraph freeness and k-connectivity. ... |

21 |
Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms
- Parnas, Ron
- 2007
(Show Context)
Citation Context ... vertex cover. Specifically, an approximation with a multiplicative error of 2 and an additive error of δn, is achieved in time d O(log(d/δ)) . This algorithm improves on a recent result presented in =-=[PR05]-=- which achieve the same approximation in time d O(δ−3 log d) . A few notes are in place: • With the exception of subgraph-freeness, the complexity of our algorithms is polynomially related to the corr... |

19 |
A fast algorithm for optimally increasing the edge connectivity
- Naor, Gusfield, et al.
- 1997
(Show Context)
Citation Context ...he additive error). To get a purely additive error we need to take a different approach. Specifically, we use different combinatorial representations of the connectivity structure of graphs (based on =-=[NGM97]-=-), rather than those used in [GR02]. On top of this, we adapt and extend some of the ideas in [GR02]. We believe that the analysis we present for distance approximation is actually easier to follow th... |

18 | Estimating the distance to a monotone function - Ailon, Chazelle, et al. - 2004 |

14 | On sums of independent random variables with unbounded variance and estimating the average degree in a graph - Feige - 2001 |

14 | Tolerant versus intolerant testing for boolean properties
- Fischer, Fortnow
- 2004
(Show Context)
Citation Context ...1-close and ɛ2-far to having the property) were first studied in [PRR]. Following that work, there have been several results on distance approximation, both positive [ACCL04, GR05, FN05] and negative =-=[FF05]-=-. These works considered properties of functions and strings [PRR, ACCL04, FF05, GR05], ensembles of points [PRR], and (dense) graphs [FN05]. Distance Approximation in Dense Graphs. In particular, Fis... |

11 | Tolerant locally testable codes - Guruswami, Rudra - 2005 |

8 | Approximating average parameters of graphs - Goldreich, Ron - 2005 |