arXiv subjects

Sumedh Tirodkar

Publications and source records attributed to Sumedh Tirodkar.

3 recordsLinked to original sources

arXiv2018

Maximum Matching on Trees in the Online Preemptive and the Incremental Dynamic Graph Models

We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the {\em Online Preemptive} model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms. Epstein et al. gave a $5.356$-competitive randomized algorithm for MWM, and also proved a lower bound of $1.693$ for MCM. The same lower bound applies for MWM. In this paper we show that some of the results can be improved in the case of trees and some special classes of graphs. In the online preemptive model, we present a $64/33$-competitive (in expectation) randomized algorithm for MCM on trees. Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a "worst case" update time of $O(1)$, in the incremental dynamic graph model, which is $3/2$-approximate (in expectation) on trees, and $1.8$-approximate (in expectation) on general graphs with maximum degree $3$. Note that this algorithm works only against an oblivious adversary. Hence, we derandomize this algorithm, and give a $(3/2 + ε)$-approximate deterministic algorithm for MCM on trees, with an amortized update time of $O(1/ε)$. We also present a minor result for MWM in the online preemptive model, a $3$-competitive (in expectation) randomized algorithm on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).

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arXiv2017

Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams

We consider the maximum matching problem in the semi-streaming model formalized by Feigenbaum, Kannan, McGregor, Suri, and Zhang that is inspired by giant graphs of today. As our main result, we give a two-pass $(1/2 + 1/16)$-approximation algorithm for triangle-free graphs and a two-pass $(1/2 + 1/32)$-approximation algorithm for general graphs; these improve the approximation ratios of $1/2 + 1/52$ for bipartite graphs and $1/2 + 1/140$ for general graphs by Konrad, Magniez, and Mathieu. In three passes, we achieve approximation ratios of $1/2 + 1/10$ for triangle-free graphs and $1/2 + 1/19.753$ for general graphs. We also give a multi-pass algorithm where we bound the number of passes precisely---we give a $(2/3 -\varepsilon)$-approximation algorithm that uses $2/(3\varepsilon)$ passes for triangle-free graphs and $4/(3\varepsilon)$ passes for general graphs. Our algorithms are simple and combinatorial, use $O(n \log n)$ space, and have $O(1)$ update time per edge. For general graphs, our multi-pass algorithm improves the best known deterministic algorithms in terms of the number of passes: --Ahn and Guha give a $(2/3 - \varepsilon)$-approximation algorithm that uses $O(\log(1/\varepsilon)/\varepsilon^2)$ passes, whereas our $(2/3 - \varepsilon)$-approximation algorithm uses $4/(3\varepsilon)$ passes; --they also give a $(1-\varepsilon)$-approximation algorithm that uses $O(\log n \cdot \mathrm{poly}(1/\varepsilon))$ passes, where $n$ is the number of vertices of the input graph; although our algorithm is $(2/3 - \varepsilon)$-approximation, our number of passes do not depend on $n$. Earlier multi-pass algorithms either have a large constant inside big-$O$ notation for the number of passes or the constant cannot be determined due to the involved analysis, so our multi-pass algorithm should use much fewer passes for approximation ratios bounded slightly below $2/3$.

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arXiv2015

On Randomized Algorithms for Matching in the Online Preemptive Model

We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded. The complexity of the problem is settled for deterministic algorithms. Almost nothing is known for randomized algorithms. A lower bound of $1.693$ is known for MCM with a trivial upper bound of $2$. An upper bound of $5.356$ is known for MWM. We initiate a systematic study of the same in this paper with an aim to isolate and understand the difficulty. We begin with a primal-dual analysis of the deterministic algorithm due to McGregor. All deterministic lower bounds are on instances which are trees at every step. For this class of (unweighted) graphs we present a randomized algorithm which is $\frac{28}{15}$-competitive. The analysis is a considerable extension of the (simple) primal-dual analysis for the deterministic case. The key new technique is that the distribution of primal charge to dual variables depends on the "neighborhood" and needs to be done after having seen the entire input. The assignment is asymmetric: in that edges may assign different charges to the two end-points. Also the proof depends on a non-trivial structural statement on the performance of the algorithm on the input tree. The other main result of this paper is an extension of the deterministic lower bound of Varadaraja to a natural class of randomized algorithms which decide whether to accept a new edge or not using independent random choices.

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