Clique-counting is a fundamental problem that has application in many areas eg. dense subgraph discovery, community detection, spam detection, etc. The problem of k-clique-counting is difficult because as k increases, the number of k-cliques goes up exponentially. Enumeration algorithms (even parallel ones) fail to count k-cliques beyond a small k. Approximation algorithms, like TuránShadow have been shown to perform well upto k = 10, but are inefficient for larger cliques. The recently proposed Pivoter algorithm significantly improved the state-of-the-art and was able to give exact counts of all k-cliques in a large number of graphs. However, the clique counts of some graphs (for example, com-lj) are still out of reach of these algorithms.
We revisit the TuránShadow algorithm and propose a generalized framework called YACC that leverages several insights about real-world graphs to achieve faster clique-counting. The bottleneck in TuránShadow is a recursive subroutine whose stopping condition is based on a classic result from extremal combinatorics called Turán's theorem. This theorem gives a lower bound for the k-clique density in a subgraph in terms of its edge density. However, this stopping condition is based on a worst-case graph that does not reflect the nature of real-world graphs. Using techniques for quickly discovering dense subgraphs, we relax the stopping condition in a systematic way such that we get a smaller recursion tree while still maintaining the guarantees provided by TuránShadow. We deploy our algorithm on several real-world data sets and show that YACC reduces the size of the recursion tree and the running time by over an order of magnitude. Using YACC, we are able to obtain clique counts for several graphs for which clique-counting was infeasible before, including com-lj.
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This content will become publicly available on March 26, 2025
HERO: A Hierarchical Set Partitioning and Join Framework for Speeding up the Set Intersection Over Graphs
As one of the most primitive operators in graph algorithms, such as the triangle counting, maximal clique enumeration, and subgraph listing, a set intersection operator returns common vertices between any two given sets of vertices in data graphs. It is therefore very important to accelerate the set intersection, which will benefit a bunch of tasks that take it as a built-in block. Existing works on the set intersection usually followed the merge intersection or galloping-search framework, and most optimization research focused on how to leverage the SIMD hardware instructions. In this paper, we propose a novel multi-level set intersection framework, namely hierarchical set partitioning and join (HERO), by using our well-designed set intersection bitmap tree (SIB-tree) index, which is independent of SIMD instructions and completely orthogonal to the merge intersection framework. We recursively decompose the set intersection task into small-sized subtasks and solve each subtask using bitmap and boolean AND operations. To sufficiently achieve the acceleration brought by our proposed intersection approach, we formulate a graph reordering problem, prove its NP-hardness, and then develop a heuristic algorithm to tackle this problem. Extensive experiments on real-world graphs have been conducted to confirm the efficiency and effectiveness of our HERO approach. The speedup over classic merge intersection achieves up to 188x and 176x for triangle counting and maximal clique enumeration, respectively.
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- Award ID(s):
- 2217104
- PAR ID:
- 10533452
- Publisher / Repository:
- Association for Computing Machinery
- Date Published:
- Volume:
- 2
- Issue:
- 1
- Page Range / eLocation ID:
- 1-25
- Subject(s) / Keyword(s):
- Set intersection over graphs hierarchical set partitioning and join
- Format(s):
- Medium: X
- Location:
- Santiago, Chile
- Sponsoring Org:
- National Science Foundation
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