## Abstract combinatorial programs and efficient property testers (2005)

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Citations: | 15 - 6 self |

### BibTeX

@MISC{Czumaj05abstractcombinatorial,

author = {Artur Czumaj and Christian Sohler},

title = {Abstract combinatorial programs and efficient property testers },

year = {2005}

}

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### Abstract

Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester. We apply our framework to a variety of problems. We present efficient property testing algorithms for geometric clustering problems, for the reversal distance problem, and for graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size. Our framework allows us to analyze all our testers in a unified way, and the obtained complexity bounds either match or improve the previously known bounds. Furthermore, even if the asymptotic complexity of the testers is not improved, the obtained proofs are significantly simpler than the previous ones. We believe that our framework will help to understand the structure of efficiently testable properties.

### Citations

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Citation Context ...connection to program checking. This notion arises naturally in the context of program verification [7, 25], learning theory, and, in a more theoretical setting, in probabilistically checkable proofs =-=[6]-=-. In [17], the study of property testing for combinatorial objects was initiated. In this and other more recent papers (see, the excellent surveys in [13, 16, 24] and the references therein), various ... |

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Citation Context ... performed to transform the permutation into the identity permutation. Because of its applications in computational biology, sorting by reversals has been widely studied in the last years (see, e.g., =-=[22, 23]-=-). In this paper, we introduce the notion of property testing in the context of sorting by reversals. We design a property testing algorithm that verifies if a given permutation has reversal distance ... |

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Citation Context ...be performed to transform the permutation into the identity permutation. Because of its applications in computational biology, sorting by reversals has been widely studied in recent years (see, e.g., =-=[12, 13, 18, 54, 63, 64]-=-). For example, Pevzner and Waterman [64], in their collection of open problems in computational biology, mentioned algorithmic issues of the sorting by reversals problem as one of the most important ... |

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Citation Context ... performed to transform the permutation into the identity permutation. Because of its applications in computational biology, sorting by reversals has been widely studied in the last years (see, e.g., =-=[22, 23]-=-). In this paper, we introduce the notion of property testing in the context of sorting by reversals. We design a property testing algorithm that verifies if a given permutation has reversal distance ... |

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Citation Context ...2] and matrices [37]; for testing properties of regular languages and branching problems [6, 38, 61]; for testing monotonicity [29, 36, 44], etc. (see, e.g., the result about quantum property testing =-=[17]-=-). In all these algorithms the goal is to verify whether an input function (or an object) has a predetermined property or is “far” from having the property. Since the exact and deterministic solution ... |

16 | On some tighter inapproximability results, further improvements
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Citation Context ...f the sorting by reversals problem as one of the most important problems in genome rearrangements. It is known that sorting by reversals is NP-hard [18], that its optimization version is Max-SNP-hard =-=[14]-=-, and that there exits a polynomial-time 1.375-approximation algorithm [13] (see also [12, 54]). In this paper, we introduce the notion of property testing in the context of sorting by reversals. We d... |

16 | A new algorithm approach to the general Lovász local lemma with applications to scheduling and satisfiability problems (extended abstract
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Citation Context ...n a factor n1−ε for any constant ε>0. See also [48, 55] for further inapproximability results. The property of hypergraph 2-colorability has also been extensively studied in combinatorics (see, e.g., =-=[21, 22, 30, 66]-=-), and, for example, the study of this problem led to the discovery of the celebrated Lovász Local lemma [30]. In the context of property testing, Czumaj and Sohler [23] were the first to design effic... |

15 | Testing satisfiability
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Citation Context ...n R d this paper O(k � −1 d ɛ (1 + 2/β) ); any Lp metric this paper O(k � −1 d−1 d ɛ (2/β) ); any Lp metric sorting by reversals this paper O(k/ɛ) k-coloring [11] ( � O(k 2 ℓ 2 /ɛ 2 )) ℓ of ℓ-uniform =-=[5]-=- ( � O(k ℓ−1 /ɛ 2 )) ℓ hypergraphs this paper ( � O(k ℓ/ɛ 2 )) ℓ Table 1. Summary of selected specific results. Instead, we can deal with pure combinatorial arguments and hence, simplify the proof to ... |

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Citation Context |

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Citation Context ...n a factor n1−ε for any constant ε>0. See also [48, 55] for further inapproximability results. The property of hypergraph 2-colorability has also been extensively studied in combinatorics (see, e.g., =-=[21, 22, 30, 66]-=-), and, for example, the study of this problem led to the discovery of the celebrated Lovász Local lemma [30]. In the context of property testing, Czumaj and Sohler [23] were the first to design effic... |

10 |
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Citation Context ...rty Π too). A graph property Π is hereditary if it is closed under taking induced subgraphs, that is, if for every graph G having property Π every induced subgraph of G has property Π too (see, e.g., =-=[8]-=-). We call a graph property Π stronglytestable [1] if for every ɛ > 0 there exists a (one-sided error) ɛ-tester for Π whose query complexity is bounded only by a function of ɛ, which is independent of... |

9 | Testing Hypergraph Coloring
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Citation Context ... 2 −1 2 d d ɛ (2/β) ); only in L2 metric in R d this paper O(k � −1 d ɛ (1 + 2/β) ); any Lp metric this paper O(k � −1 d−1 d ɛ (2/β) ); any Lp metric sorting by reversals this paper O(k/ɛ) k-coloring =-=[11]-=- ( � O(k 2 ℓ 2 /ɛ 2 )) ℓ of ℓ-uniform [5] ( � O(k ℓ−1 /ɛ 2 )) ℓ hypergraphs this paper ( � O(k ℓ/ɛ 2 )) ℓ Table 1. Summary of selected specific results. Instead, we can deal with pure combinatorial ar... |

7 |
Efficient testing of hypergraphs
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(Show Context)
Citation Context ...luding graph coloring, clique, cut, and bisection. Another general approach, which uses the Szemerédi regularity lemma, has been proposed recently for studying graph problems and problems on matrices =-=[3, 14, 21]-=-. Even if this method is very powerful (and in particular, it allowed to prove that all first order graph properties without a quantifier alternation of type ∀∃ have property testers whose complexity ... |

3 |
The art of uniformed decisions. A primer to property testing
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(Show Context)
Citation Context ...setting, in probabilistically checkable proofs [6]. In [17], the study of property testing for combinatorial objects was initiated. In this and other more recent papers (see, the excellent surveys in =-=[13, 16, 24]-=- and the references therein), various algorithms have been proposed for testing graph and hypergraph properties, for testing geometric properties, for testing properties of metrics and matrices, for t... |

2 |
Property testing. In Handobook of Randomized Algorithms
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(Show Context)
Citation Context ...setting, in probabilistically checkable proofs [6]. In [17], the study of property testing for combinatorial objects was initiated. In this and other more recent papers (see, the excellent surveys in =-=[13, 16, 24]-=- and the references therein), various algorithms have been proposed for testing graph and hypergraph properties, for testing geometric properties, for testing properties of metrics and matrices, for t... |

1 |
Testing of clustering. 41st FOCS
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- 2000
(Show Context)
Citation Context ...plexity of the tester) so that the algorithm our analysis of clustering has a similar flavor as the previous is a correct property tester? It is easy to see that for prop- analysis of this problem in =-=[2]-=-. What distinguish our analerties closed under taking restrictions, if f has the required ysis, however, is that we do not have to deal with the probproperty then the algorithm always accepts f. Thus,... |