No abstract

Triangle counting is a fundamental problem in the analysis of large graphs. There is a rich body of work on this problem, in varying streaming and distributed models, yet all these algorithms require reading the whole input graph. In many scenarios, we do not have access to the whole graph, and can only sample a small portion of the graph (typically through crawling). In such a setting, how can we accurately estimate the triangle count of the graph?We formally study triangle counting in the random walk access model introduced by Dasgupta et al (WWW '14) and Chierichetti et al (WWW '16). We have access to an arbitrary seed vertex of the graph, and can only perform random walks. This model is restrictive in access and captures the challenges of collecting realworld graphs. Even sampling a uniform random vertex is a hard task in this model.Despite these challenges, we design a provable and practical algorithm, TETRIS, for triangle counting in this model. TETRIS is the first provably sublinear algorithm (for most natural parameter settings) that approximates the triangle count in the random walk model, for graphs with low mixing time. Our result builds on recent advances in the theory of sublinear algorithms. The final sample built by TETRIS is a careful mix of random walks and degree-biased sampling of neighborhoods. Empirically, TETRIS accurately counts triangles on a variety of large graphs, getting estimates within 5% relative error by looking at 3% of the number of edges. CCS CONCEPTS• Mathematics of computing → Graph algorithms; Probabilistic algorithms; • Theory of computation → Sketching and sampling; Random walks and Markov chains.

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V, E), a cost function c : V → Z + , a partition P1, . . . , Pr of the edge set E, and a parameter ki for each partition Pi. The goal is to find a minimum cost set of vertices which cover at least ki edges from the partition Pi. We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(log r), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings.

The unparalleled growth and popularity of the Internet coupled with the advent of diverse modern applications such as search engines, on-line transactions, climate warning systems, etc., has catered to an unprecedented expanse in the volume of data stored world-wide. Efficient storage, management, and processing of such massively exponential amount of data has emerged as a central theme of research in this direction. Detection and removal of redundancies and duplicates in real-time from such multi-trillion record-set to bolster resource and compute efficiency constitutes a challenging area of study. The infeasibility of storing the entire data from potentially unbounded data streams, with the need for precise elimination of duplicates calls for intelligent approximate duplicate detection algorithms. The literature hosts numerous works based on the well-known probabilistic bitmap structure, Bloom Filter and its variants. In this paper we propose a novel data structure, Streaming Quotient Filter, (SQF) for efficient detection and removal of duplicates in data streams. SQF intelligently stores the signatures of elements arriving on a data stream, and along with an eviction policy provides near zero false positive and false negative rates. We show that the near optimal performance of SQF is achieved with a very low memory requirement, making it ideal for real-time memory-efficient de-duplication applications having an extremely low false positive and false negative tolerance rates. We present detailed theoretical analysis of the working of SQF, providing a guarantee on its performance. Empirically, we compare SQF to alternate methods and show that the proposed method is superior in terms of memory and accuracy compared to the existing solutions. We also discuss Dynamic SQF for evolving streams and the parallel implementation of SQF.

Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input G and the pattern H. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of H, H-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs. Can we precisely characterize the patterns H for which near-linear time algorithms are possible?We completely resolve this problem, discovering a clean dichotomy using fine-grained complexity. Let m denote the number of edges in G. We prove the following: if the largest induced cycle in H has length at most 5, then there is an O(m log m) algorithm for counting H-homomorphisms in bounded degeneracy graphs. If the largest induced cycle in H has length at least 6, then (assuming standard fine-grained complexity conjectures) there is a constant γ > 0, such that there is no o(m 1+γ ) time algorithm for counting H-homomorphisms.