All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) i…

Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current $\text{poly} (\log \log n)$-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number $n$ of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, w…

HybridMiner: Mining Maximal Frequent Itemsets Using Hybrid Database Representation Approach

In this paper we present a novel hybrid (arraybased layout and vertical bitmap layout) database representation approach for mining complete Maximal Frequent Itemset (MFI) on sparse and large datasets. Our work is novel in terms of scalability, item search order and two horizontal and vertical projection techniques. We also present a maximal algorithm using this hybrid database representation approach. Different experimental results on real and sparse benchmark datasets show that our approach is better than previous state of art maximal algorithms.

Massively Parallel Symmetry Breaking on Sparse Graphs: MIS and Maximal Matching

The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a significant attention to the Massively Parallel Computation (MPC) model over the past few years, especially on graph problems. In this work, we consider symmetry breaking problems of maximal independent set (MIS) and maximal matching (MM), which are among the most intensively studied problems in distributed/parallel computing, in MPC. These problems are known to admit efficient MPC algorithms if the space per machine is near-linear in $n$, the number of vertices in the graph. This space requirement ho…

通过图归约加速极大团枚举

Accelerating Maximal Clique Enumeration via Graph Reduction

针对极大团枚举中的冗余搜索问题，提出融合全局、动态与极大性检查归约的RMCE框架，在18个真实图上最高取得44.7倍加速。

结构稀疏图中最大团枚举的近最优算法

Near-Optimal Algorithms for Maximal Clique Enumeration in Structurally Sparse Graphs

针对排除团子式和排除团浸入的图，提出两种最大团枚举算法，分别将时间复杂度优化为n·4^(2t/5+o(t))和n·3^(t/3+o(t))，并证明其指数基为渐近最优。

Computing Maximal Layers Of Points in $E^{f(n)}$

In this paper we present a randomized algorithm for computing the collection of maximal layers for a point set in $E^{k}$ ($k = f(n)$). The input to our algorithm is a point set $P = \{p_1,...,p_n\}$ with $p_i \in E^{k}$. The proposed algorithm achieves a runtime of $O\left(kn^{2 - {1 \over \log{k}} + \log_k{\left(1 + {2 \over {k+1}}\right)}}\log{n}\right)$ when $P$ is a random order and a runtime of $O(k^2 n^{3/2 + (\log_{k}{(k-1)})/2}\log{n})$ for an arbitrary $P$. Both bounds hold in expectation. Additionally, the run time is bounded by $O(kn^2)$ in the worst case. This is the first non-tri…

BBK：一种更简单、更快速的大规模稀疏二分图中极大双团枚举算法

BBK: a simpler, faster algorithm for enumerating maximal bicliques in large sparse bipartite graphs

本文提出BBK算法，将Bron-Kerbosch扩展到二分图，用于快速枚举极大双团，理论分析复杂度并在大规模真实数据上验证其效率优于现有算法。