Approximating Optimal Transport With Linear Programs

In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in faster additive approximation algorithms for optimal transport

Approximating optimal transport with linear programs

In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in faster additive approximation algorithms for optimal transport.Comment: To appear in SOSA 201

An Estimator for Matching Size in Low Arboricity Graphs with Two Applications

In this paper, we present a new simple degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by $\alpha$, the estimator gives a $\alpha+2$ factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a $3.5$ approximation of the matching size. Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized $O(\frac{\sqrt{n}}{\epsilon^2}\log n)$ space algorithm for approximating the matching size within $(3.5+\epsilon)$ factor in a planar graph on $n$ vertices. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within $(3.5+\epsilon)$ factor using $O(\frac{n^{2/3}}{\epsilon^2}\log n)$ communication from each player. In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor for the case of planar graphs

Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

We revisit the minimum dominating set problem on graphs with arboricity bounded by $\alpha$. In the (standard) centralized setting, Bansal and Umboh [BU17] gave an $O(\alpha)$-approximation LP rounding algorithm. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of $O(\alpha^2)$ in linear time. There is a similar situation in the distributed setting: While there are $\text{poly}\log n$-round LP-based $O(\alpha)$-approximation algorithms [KMW06, DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(\alpha^2)$-approximation within $O(\log n)$ rounds with high probability. We address the question of whether one can achieve a simple, elementary $O(\alpha)$-approximation algorithm not based on any LP-based methods, either in the centralized setting or in the distributed setting. We resolve these questions in the affirmative. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an $O(\alpha)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(\alpha)$-approximation algorithm in the $\mathsf{CONGEST}$ model that runs in $O(\alpha\log n )$ rounds with high probability. Our round complexity outperforms the best LP-based distributed algorithm for a wide range of parameters

Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

$\newcommand{\eps}{\varepsilon}$ We present an auction algorithm using multiplicative instead of constant weight updates to compute a (1-\eps)-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges in time O(m\eps^{-1}\log(\eps^{-1})), matching the running time of the linear-time approximation algorithm of Duan and Pettie [JACM '14]. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a (1-\eps)-approximate maximum weight matching under (1) edge deletions in amortized O(\eps^{-1}\log(\eps^{-1})) time and (2) one-sided vertex insertions. If all edges incident to an inserted vertex are given in sorted weight the amortized time is O(\eps^{-1}\log(\eps^{-1})) per inserted edge. If the inserted incident edges are not sorted, the amortized time per inserted edge increases by an additive term of $O(\log n)$. The fastest prior dynamic (1-\eps)-approximate algorithm in weighted graphs took time O(\sqrt{m}\eps^{-1}\log (w_{max})) per updated edge, where the edge weights lie in the range $[1,w_{max}]$.Comment: To appear in IPCO 202