Approximate Densest Subgraph

Problem Specification

Input

$G=(V, E)$, an undirected graph, and a parameter $\epsilon$.

Output

$U \subseteq V$, a set of vertices such that the density of $G_{U}$ is a $2(1+\epsilon)$ approximation of the density of the densest subgraph of $G$.

Definitions

The density of a subset of vertices $S$ is the number of edges in the subgraph $S$ divided by the number of vertices.

Algorithm Implementations

In GBBS we implement the elegant $(2+\epsilon)$-approximation algorithm of Bahmani et al. [1]. More details about our implementation can be found in Section 6.4 of [2].

The code for our implemenation is available here.

Cost Bounds

Our implementation of the algorithm runs in $O(m+n)$ work and $O(\log_{1+\epsilon} n \cdot \log n)$ depth. We provide a proof of this result in Section 6.4 of [2].

Compiling and Running

The benchmark can be compiled by running:

bazel build -c opt //benchmarks/ApproximateDensestSubgraph/ApproxPeelingBKV12/...

It can be run on a test input graph in the uncompressed format as follows:

numactl -i all ./bazel-bin/benchmarks/ApproximateDensestSubgraph/ApproxPeelingBKV12/DensestSubgraph_main -s -m -src 1 inputs/rMatGraph_J_5_100

It can be run on a test input graph in the compressed format as follows:

numactl -i all ./bazel-bin/benchmarks/ApproximateDensestSubgraph/ApproxPeelingBKV12/DensestSubgraph_main -s -c -m -src 1 inputs/rMatGraph_J_5_100.bytepda

References

[1] Bahman Bahmani, Ravi Kumar, and Sergei Vassilvitskii
Densest Subgraph in Streaming and MapReduce
Proc. {VLDB} Endow. 5(5), pp. 454-465, 2012

[2] Laxman Dhulipala, Guy Blelloch, and Julian Shun
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
Proceedings of the ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 393-404, 2018.