Local Graph Partitions for Approximation and Testing

Hassidim, Avinatan; Kelner, Jonathan A.; Nguyen, Huy; Onak, Krzysztof

doi:10.1109/focs.2009.77

Cited by 113 publications

(161 citation statements)

References 24 publications

Supporting

Mentioning

160

Contrasting

Order By: Relevance

“…Partitioning oracles have numerous applications described in [HKNO09]. Let us describe some general applications of partitioning oracles, and the results yielded by our efficient partitioning oracle.…”

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

“…Via the reduction from [HKNO09], our new oracle yields a poly(1/ε)-time tester for any minor closed family of graphs that has bounded treewidth. Sample minor-closed families of graphs with this property are k-outerplanar graphs, series-parallel graphs, and pseudoforests.…”

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

“…Benjamini, Schramm, and Shapira [BSS08] prove that any minor-closed property can be tested in 2 2 2 poly(1/ε) time. Hassidim et al [HKNO09] introduce partitioning oracles and show how to use them to obtain a tester of running time 2 poly(1/ε) (see [Ona10] for a simplified full proof). Yoshida and Ito [YI10] show that outerplanarity and the property of being a cactus can be tested in poly(1/ε) time.…”

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

“…[HKNO09] shows how to design such an oracle for minor-free graphs (and also for some other hyperfinite graphs). However, their oracles have query complexity and running time of 2 poly(1/ǫ) (see [Ona10] for a simple oracle with this property), and the main question left open by their paper is whether one can design a partitioning oracle that runs in time poly(1/ǫ).…”

Section: Introductionmentioning

confidence: 99%

“…In [HKNO09], this intuition was used to design constant-time algorithms for various problems. Fix a partition P of the vertices in the input graph G such that the following properties hold:…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Edelman

Hassidim

Nguyen

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

Self Cite

View full text Add to dashboard Cite

Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any ε > 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/ε), and the number of edges removed is at most εn, where n is the number of vertices in the graph.However, the oracle of Hassidim et al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/ε)) queries to the input graph to answer each query about the partition.Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, seriesparallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/ε)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/ε)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive εn in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/ε) time whether the input bounded-treewidth graph is k-colorable or perfect.

show abstract

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Edelman

Hassidim

Nguyen

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

Self Cite

View full text Add to dashboard Cite

show abstract

Finding cycles and trees in sublinear time

Czumaj

Goldreich

Ron

et al. 2012

Random Struct Algorithms

View full text Add to dashboard Cite

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k ≥ 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being C k -minor free (resp., free from having the corresponding tree-minor). In particular, if the graph is Ω(1)-far from being cycle-free (i.e., a constant fraction of the edges must be deleted to make the graph cycle-free), then the algorithm finds a cycle of polylogarithmic length in time O( √ N ), where N denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors.The foregoing results are the outcome of our study of the complexity of one-sided error property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of N -vertex graphs can be tested with one-sided error within time complexity O(poly(1/ǫ) · √ N ), where ǫ denotes the proximity parameter. This matches the known Ω( √ N ) query lower bound for one-sided error cycle-freeness testing, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on ǫ. We show that the same upper bound holds for testing whether the input graph has a simple cycle of length at least k, for any k ≥ 3. On the other hand, for any fixed tree T , we show that T -minor freeness has a one-sided error tester of query complexity that only depends on the proximity parameter ǫ.Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in o( √ N ) complexity.

show abstract

Constructing near spanning trees with few local inspections

Levi

Moshkovitz

Ron

et al. 2016

Random Struct Algorithms

View full text Add to dashboard Cite

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory.Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1+ )n edges (for a prespecified constant > 0), where the decision for different edges should be consistent with the same subgraph G . Can this task be performed by inspecting only a constant number of edges in G? Our main results are:• We show that if every t-vertex subgraph of G has expansion 1/(log t) 1+o(1) then one can (deterministically) construct a sparse spanning subgraph G of G using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm.• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion 1/(log t) 1−o(1) . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

show abstract

Local Graph Partitions for Approximation and Testing

Cited by 113 publications

References 24 publications

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Finding cycles and trees in sublinear time

Constructing near spanning trees with few local inspections

Contact Info

Product

Resources

About