Efficient Sampling Algorithms for Approximate Motif Counting in Temporal Graph Streams

A great variety of complex systems, from user interactions in communication networks to transactions in financial markets, can be modeled as temporal graphs consisting of a set of vertices and a series of timestamped and directed edges. Temporal motifs are generalized from subgraph patterns in static graphs which consider edge orderings and durations in addition to topologies. Counting the number of occurrences of temporal motifs is a fundamental problem for temporal network analysis. However, existing methods either cannot support temporal motifs or suffer from performance issues. Moreover, they cannot work in the streaming model where edges are observed incrementally over time. In this paper, we focus on approximate temporal motif counting via random sampling. We first propose two sampling algorithms for temporal motif counting in the offline setting. The first is an edge sampling (ES) algorithm for estimating the number of instances of any temporal motif. The second is an improved edge-wedge sampling (EWS) algorithm that hybridizes edge sampling with wedge sampling for counting temporal motifs with $3$ vertices and $3$ edges. Furthermore, we propose two algorithms to count temporal motifs incrementally in temporal graph streams by extending the ES and EWS algorithms referred to as SES and SEWS. We provide comprehensive analyses of the theoretical bounds and complexities of our proposed algorithms. Finally, we perform extensive experimental evaluations of our proposed algorithms on several real-world temporal graphs. The results show that ES and EWS have higher efficiency, better accuracy, and greater scalability than state-of-the-art sampling methods for temporal motif counting in the offline setting. Moreover, SES and SEWS achieve up to three orders of magnitude speedups over ES and EWS while having comparable estimation errors for temporal motif counting in the streaming setting.Comment: 27 pages, 11 figures; overlapped with arXiv:2007.1402

Efficient sampling algorithms for approximate temporal motif counting

Ministry of Education, Singapore under its Academic Research Funding Tier

Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ?}) update time and O(m^{1 -?}) query time, for any small ? > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching

A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree

A structural approach to matching problems with preferences

This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem. We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAX-SMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAX-SMTI, improving upon the previous best approximation algorithm, which gave a performance guarantee of 5/3. Next, we study the sex-equal stable marriage problem (SESM). We show that SESM is W[1]-hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, low-order exponential time algorithm. This is the first non-trivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem. Turning our attention to the hospitals / residents problem with couples (HRC), we show that HRC is NP-complete, even if very severe restrictions are placed on the input. By contrast, we give a linear-time algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical Gale-Shapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of. We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching for a 3DSM instance is NP-complete. These hardness results resolve some open questions in the literature. Finally, we study the popular matching problem (POP-M) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously best-known results for this problem