A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding

Over three decades ago, Karp, Vazirani and Vazirani (STOC'90) introduced the online bipartite matching problem. They observed that deterministic algorithms' competitive ratio for this problem is no greater than $1/2$, and proved that randomized algorithms can do better. A natural question thus arises: \emph{how random is random}? i.e., how much randomness is needed to outperform deterministic algorithms? The \textsc{ranking} algorithm of Karp et al.~requires $\tilde{O}(n)$ random bits, which, ignoring polylog terms, remained unimproved. On the other hand, Pena and Borodin (TCS'19) established a lower bound of $(1-o(1))\log\log n$ random bits for any $1/2+\Omega(1)$ competitive ratio. We close this doubly-exponential gap, proving that, surprisingly, the lower bound is tight. In fact, we prove a \emph{sharp threshold} of $(1\pm o(1))\log\log n$ random bits for the randomness necessary and sufficient to outperform deterministic algorithms for this problem, as well as its vertex-weighted generalization. This implies the same threshold for the advice complexity (nondeterminism) of these problems. Similar to recent breakthroughs in the online matching literature, for edge-weighted matching (Fahrbach et al.~FOCS'20) and adwords (Huang et al.~FOCS'20), our algorithms break the barrier of $1/2$ by randomizing matching choices over two neighbors. Unlike these works, our approach does not rely on the recently-introduced OCS machinery, nor the more established randomized primal-dual method. Instead, our work revisits a highly-successful online design technique, which was nonetheless under-utilized in the area of online matching, namely (lossless) online rounding of fractional algorithms. While this technique is known to be hopeless for online matching in general, we show that it is nonetheless applicable to carefully designed fractional algorithms with additional (non-convex) constraints

Near-Optimal Schedules for Simultaneous Multicasts

We study the store-and-forward packet routing problem for simultaneous multicasts, in which multiple packets have to be forwarded along given trees as fast as possible. This is a natural generalization of the seminal work of Leighton, Maggs and Rao, which solved this problem for unicasts, i.e. the case where all trees are paths. They showed the existence of asymptotically optimal O(C +D)-length schedules, where the congestion C is the maximum number of packets sent over an edge and the dilation D is the maximum depth of a tree. This improves over the trivial O(CD) length schedules. We prove a lower bound for multicasts, which shows that there do not always exist schedules of non-trivial length, o(CD). On the positive side, we construct O(C + D + log2 n)-length schedules in any n-node network. These schedules are near-optimal, since our lower bound shows that this length cannot be improved to O(C + D) + o(log n).ISSN:1868-896

Round- and Message-Optimal Distributed Graph Algorithms

Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal $\tilde{O}(D+\sqrt n)$ round complexity and $\tilde{O}(m)$ message complexity. Furthermore, our algorithms require an optimal $\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.Comment: To appear in PODC 201

Beating the Folklore Algorithm for Dynamic Matching

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2-?(1))-approximate algorithm with O(m^{3/8}) = O(n^{3/4}) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ? > 0, no deterministic (2+?)-approximate algorithm with worst-case update time O(n^{0.99}) was known. Our second result is the first deterministic (2+?)-approximate weighted matching algorithm with O_?(1)? O(?{m}) = O_?(1)? O(?n) worst-case update time. Neither of our results were previously known to be achievable by a randomized algorithm against an adaptive adversary. Our main technical contributions are threefold: first, we characterize the tight cases for kernels, which are the well-studied matching sparsifiers underlying much of the (2+?)-approximate dynamic matching literature. This characterization, together with multiple ideas - old and new - underlies our result for breaking the approximation barrier of 2. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the recourse of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural 3/2 factor in the approximation ratio which such approaches naturally incur (reminiscent of the integrality gap of the fractional matching polytope in general graphs)