A Note on Improved Results for One Round Distributed Clique Listing

In this note, we investigate listing cliques of arbitrary sizes in bandwidth-limited, dynamic networks. The problem of detecting and listing triangles and cliques was originally studied in great detail by Bonne and Censor-Hillel (ICALP 2019). We extend this study to dynamic graphs where more than one update may occur as well as resolve an open question posed by Bonne and Censor-Hillel (2019). Our algorithms and results are based on some simple observations about listing triangles under various settings and we show that we can list larger cliques using such facts. Specifically, we show that our techniques can be used to solve an open problem posed in the original paper: we show that detecting and listing cliques (of any size) can be done using $O(1)$-bandwidth after one round of communication under node insertions and node/edge deletions. We conclude with an extension of our techniques to obtain a small bandwidth $1$-round algorithm for listing cliques when more than one node insertion/deletion and/or edge deletion update occurs at any time.Comment: To appear in IP

The Predicted-Deletion Dynamic Model: Taking Advantage of ML Predictions, for Free

The main bottleneck in designing efficient dynamic algorithms is the unknown nature of the update sequence. In particular, there are some problems, like 3-vertex connectivity, planar digraph all pairs shortest paths, and others, where the separation in runtime between the best partially dynamic solutions and the best fully dynamic solutions is polynomial, sometimes even exponential. In this paper, we formulate the predicted-deletion dynamic model, motivated by a recent line of empirical work about predicting edge updates in dynamic graphs. In this model, edges are inserted and deleted online, and when an edge is inserted, it is accompanied by a "prediction" of its deletion time. This models real world settings where services may have access to historical data or other information about an input and can subsequently use such information make predictions about user behavior. The model is also of theoretical interest, as it interpolates between the partially dynamic and fully dynamic settings, and provides a natural extension of the algorithms with predictions paradigm to the dynamic setting. We give a novel framework for this model that "lifts" partially dynamic algorithms into the fully dynamic setting with little overhead. We use our framework to obtain improved efficiency bounds over the state-of-the-art dynamic algorithms for a variety of problems. In particular, we design algorithms that have amortized update time that scales with a partially dynamic algorithm, with high probability, when the predictions are of high quality. On the flip side, our algorithms do no worse than existing fully-dynamic algorithms when the predictions are of low quality. Furthermore, our algorithms exhibit a graceful trade-off between the two cases. Thus, we are able to take advantage of ML predictions asymptotically "for free.'

Scalable Auction Algorithms for Bipartite Maximum Matching Problems

In this paper, we give new auction algorithms for maximum weighted bipartite matching (MWM) and maximum cardinality bipartite $b$-matching (MCbM). Our algorithms run in $O\left(\log n/\varepsilon^8\right)$ and $O\left(\log n/\varepsilon^2\right)$ rounds, respectively, in the blackboard distributed setting. We show that our MWM algorithm can be implemented in the distributed, interactive setting using $O(\log^2 n)$ and $O(\log n)$ bit messages, respectively, directly answering the open question posed by Demange, Gale and Sotomayor [DNO14]. Furthermore, we implement our algorithms in a variety of other models including the the semi-streaming model, the shared-memory work-depth model, and the massively parallel computation model. Our semi-streaming MWM algorithm uses $O(1/\varepsilon^8)$ passes in $O(n \log n \cdot \log(1/\varepsilon))$ space and our MCbM algorithm runs in $O(1/\varepsilon^2)$ passes using $O\left(\left(\sum_{i \in L} b_i + |R|\right)\log(1/\varepsilon)\right)$ space (where parameters $b_i$ represent the degree constraints on the $b$-matching and $L$ and $R$ represent the left and right side of the bipartite graph, respectively). Both of these algorithms improves \emph{exponentially} the dependence on $\varepsilon$ in the space complexity in the semi-streaming model against the best-known algorithms for these problems, in addition to improvements in round complexity for MCbM. Finally, our algorithms eliminate the large polylogarithmic dependence on $n$ in depth and number of rounds in the work-depth and massively parallel computation models, respectively, improving on previous results which have large polylogarithmic dependence on $n$ (and exponential dependence on $\varepsilon$ in the MPC model).Comment: To appear in APPROX 202