No abstract

In this paper, we bring the techniques of the Laplacian paradigm to the congested clique, while further restricting ourselves to deterministic algorithms. In particular, we show how to solve a Laplacian system up to precision 𝜖 in 𝑛 𝑜 (1) log(1/𝜖) rounds. We show how to leverage this result within existing interior point methods for solving ow problems. We obtain an 𝑚 3/7+𝑜 (1) 𝑈 1/7 round algorithm for maximum ow on a weighted directed graph with maximum weight 𝑈 , and we obtain an Õ (𝑚 3/7 (𝑛 0.158 + 𝑛 𝑜 (1) poly log𝑊 )) round algorithm for unit capacity minimum cost ow on a directed graph with maximum cost 𝑊 . Hereto, we give a novel routine for computing Eulerian orientations in 𝑂 (log 𝑛 log * 𝑛) rounds, which we believe may be of separate interest.

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of $$(1\pm \epsilon )$$ ( 1 ± ϵ ) . This paper considers computing cut sparsifiers of weighted graphs of size $$O(n\log (n)/\epsilon ^2)$$ O ( n log ( n ) / ϵ 2 ) . Our algorithm computes such a sparsifier in time $$O(m\cdot \min (\alpha (n)\log (m/n),\log (n)))$$ O ( m · min ( α ( n ) log ( m / n ) , log ( n ) ) ) , both for graphs with polynomially bounded and unbounded integer weights, where $$\alpha (\cdot )$$ α ( · ) is the functional inverse of Ackermann’s function. This improves upon the state of the art by Benczúr and Karger (SICOMP, 2015), which takes $$O(m\log ^2 (n))$$ O ( m log 2 ( n ) ) time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP, 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi–Ibaraki forest packing. MSF packings have previously been used by Abraham et al. (FOCS, 2016) in the dynamic setting, and are defined as follows: an M-partial MSF packing of G is a set $$\mathcal {F}=\{F_1, \ldots , F_M\}$$ F = { F 1 , … , F M } , where $$F_i$$ F i is a maximum spanning forest in $$G{\setminus } \bigcup _{j=1}^{i-1}F_j$$ G \ ⋃ j = 1 i - 1 F j . Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given k ≥ 2, can be used to compute a spanner of stretch 2k − 1 and expected size O(n 1+1/k ) in k rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG'19]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the total number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given β ∈ (0, 1], we compute a low diameter decomposition with diameter bound O log n β such that each edge e ∈ E is an inter-cluster edge with probability at most β • w(e) in O log n β rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA'13]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results.