Approximating the Caro-Wei Bound for Independent Sets in Graph Streams

Cormode, Graham; Dark, Jacques; Konrad, Christian

doi:10.1007/978-3-319-96151-4_9

Cited by 9 publications

(10 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our prior work, it was shown how to return an estimate γ ∈ Ω β(G) log n with γ ≤ α(G) from an explicit vertex arrival stream using only O(log 3 n) space [8]. This result for example gives a O( log 2 n log log n )approximation on unit interval graphs (see Figure 1).…”

Section: Explicit Vertex Streamsmentioning

confidence: 99%

See 1 more Smart Citation

Independent Sets in Vertex-Arrival Streams

Cormode,

Dark,

Konrad

2018

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the classic maximal and maximum independent set problems in three models of graph streams:In the edge-arrival model we see a stream of edges which collectively define a graph; this model has been well-studied for a variety of problems. We first show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that the problem does not become much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution using only the space needed to store the independent set.In the "explicit" vertex stream model, the input stream is a sequence of vertices making up the graph, where every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than for edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires space Ω( n 2 c 7 ), where n is the number of vertices of the input graph, and it is already known that space Θ( n 2 c 2 ) is necessary and sufficient in the edge arrival model (Halldórsson et al. 2012). The MIS problem is thus not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction to a new multi-party communication problem closely related to pointer jumping.In the "implicit" vertex stream model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that can map a pair of objects to the presence or absence of an edge, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for ball intersection graphs, in both explicit and implicit streams.

show abstract

Section: Explicit Vertex Streamsmentioning

confidence: 99%

“…It is also known to be hard (requiring Θ( n 2 c 2 ) space to c-approximate on an n-vertex graph) in the edge streaming model, despite being allowed unlimited computation [14]. However, we can do much better for graphs of bounded independence, given as vertex streams [8].…”

Section: Introductionmentioning

confidence: 99%

Independent Sets in Vertex-Arrival Streams

Cormode,

Dark,

Konrad

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our work matches the lower bound for one-pass, and extends it to multiple-pass algorithms. The maximum independent set and maximum clique problems were also previously studied [24,36,50,50]; the known lower bounds also apply to these problems with a gap promise, so our construction is weaker in that sense. On the other hand, we improve upon the best results for maximum independent set [50] in a poly-logarithmic factor and in the simplicity of our construction, and on the results for maximum clique [24] in that we handle multiple-pass algorithms.…”

Section: Streaming Algorithmsmentioning

confidence: 99%

Smaller Cuts, Higher Lower Bounds

Abboud¹,

Censor-Hillel²,

Khoury³

et al. 2019

Preprint

View full text Add to dashboard Cite

This paper proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts.The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs.Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortestpaths problem is discussed.Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

show abstract

“…[26,13,21,27,18,24,31,12]), independent sets (e.g. [15,14,9,10]), and subgraph counting (e.g. [20,11,7]), have since been studied in this model.…”

Section: Introductionmentioning

confidence: 99%

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Dark,

Konrad

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and(3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n 2 )-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

show abstract

Approximating the Caro-Wei Bound for Independent Sets in Graph Streams

Abstract: Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

Cited by 9 publications

References 24 publications

Independent Sets in Vertex-Arrival Streams

Independent Sets in Vertex-Arrival Streams

Smaller Cuts, Higher Lower Bounds

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Contact Info

Product

Resources

About