Online Algorithms for Graph Partitioning into Cliques

Clique clustering is the problem of partitioning a graph into cliques so that some objective function is optimised. In online clustering the input graph is given one vertex at a time, and vertices that have been previously clustered are not allowed to be separated. The objective is to maintain a clustering that never deviates too far from the optimal offline solution. We give a constant competitive upper bound and a strategy (Lazy) for online clique clustering, where the objective function is to maximise the number of edges inside the clusters (Max-ECP). We also give almost matching upper and lower bounds on the competitive ratio for online clique clustering, where we want to minimise the number of edges between clusters (Min-ECP). In addition, we prove that the greedy method only gives linear competitive ratio for these problems. The research result shows that the proposed constant competitive strategy performs significantly better on bigger graphs than the greedy method

Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

In this paper, we study the $k$-forest problem in the model of resource augmentation. In the $k$-forest problem, given an edge-weighted graph $G(V,E)$, a parameter $k$, and a set of $m$ demand pairs $\subseteq V \times V$, the objective is to construct a minimum-cost subgraph that connects at least $k$ demands. The problem is hard to approximate---the best-known approximation ratio is $O(\min\{\sqrt{n}, \sqrt{k}\})$. Furthermore, $k$-forest is as hard to approximate as the notoriously-hard densest $k$-subgraph problem. While the $k$-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the $k$-forest problem can be viewed as to remove at most $m-k$ demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the $k$-forest problem that, for every $\epsilon>0$, removes at most $m-k$ demands and has cost no more than $O(1/\epsilon^{2})$ times the cost of an optimal algorithm that removes at most $(1-\epsilon)(m-k)$ demands

Adaptive Partitioning for Large-Scale Dynamic Graphs

Abstract—In the last years, large-scale graph processing has gained increasing attention, with most recent systems placing particular emphasis on latency. One possible technique to improve runtime performance in a distributed graph processing system is to reduce network communication. The most notable way to achieve this goal is to partition the graph by minimizing the num-ber of edges that connect vertices assigned to different machines, while keeping the load balanced. However, real-world graphs are highly dynamic, with vertices and edges being constantly added and removed. Carefully updating the partitioning of the graph to reflect these changes is necessary to avoid the introduction of an extensive number of cut edges, which would gradually worsen computation performance. In this paper we show that performance degradation in dynamic graph processing systems can be avoided by adapting continuously the graph partitions as the graph changes. We present a novel highly scalable adaptive partitioning strategy, and show a number of refinements that make it work under the constraints of a large-scale distributed system. The partitioning strategy is based on iterative vertex migrations, relying only on local information. We have implemented the technique in a graph processing system, and we show through three real-world scenarios how adapting graph partitioning reduces execution time by over 50 % when compared to commonly used hash-partitioning. I