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Register a new userLinear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem
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Added by
Kook Jin Ahn
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniques to adapt and optimize linear programming based approaches in the semi-streaming model with an application to the maximum matching problem. As a consequence, we improve (almost) all previous results on this problem, and also prove new results on interesting variants.
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In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use $2^{frac1epsilon}$ passes or $n^{frac1epsilon}$ time to compute a $(1-epsilon)$ approximation. In this paper we provide the first FPTAS (polynomial in $n,frac1epsilon$) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the size of the matching in $O(frac1epsilon)$ passes using slightly superlinear space. To achieve both results, we use the structural properties of the matching polytope such as the laminarity of the tight sets and total dual integrality. The algorithms are iterative, and are based on the fractional packing and covering framework. However the formulations herein require exponentially many variables or constraints. We use laminarity, metric embeddings and graph sparsification to reduce the space required by the algorithms in between and across the iterations. This is the first use of these ideas in the semi-streaming model to solve a combinatorial optimization problem.
We present a deterministic $(1+varepsilon)$-approximate maximum matching algorithm in $mathsf{poly}(1/varepsilon)$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/varepsilon)^{O(1/varepsilon)}$-pass algorithm from the seminal work by McGregor [APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity [FSTTCS18], as well as the deterministic $log n cdot mathsf{poly}(1/varepsilon)$-pass algorithm by Ahn and Guha [ICALP11].
We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemeredi graphs: * Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least $(1- Omega(frac{log{RS(n)}}{log{n}}))$, where $RS(n)$ denotes the maximum number of induced matchings of size $Theta(n)$ in any $n$-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that $n^{Omega(1/!loglog{n})} leq RS(n) leq frac{n}{2^{O(log^*{!(n)})}}$ and closing this (large) gap between upper and lower bounds has remained a notoriously difficult problem in combinatorics. Under the plausible hypothesis that $RS(n) = n^{Omega(1)}$, our lower bound is the first to rule out small-constant approximation two-pass semi-streaming algorithms for the maximum matching problem, making progress on a longstanding open question in the graph streaming literature.
We study the problem of parameterized matching in a stream where we want to output matches between a pattern of length m and the last m symbols of the stream before the next symbol arrives. Parameterized matching is a natural generalisation of exact matching where an arbitrary one-to-one relabelling of pattern symbols is allowed. We show how this problem can be solved in constant time per arriving stream symbol and sublinear, near optimal space with high probability. Our results are surprising and important: it has been shown that almost no streaming pattern matching problems can be solved (not even randomised) in less than Theta(m) space, with exact matching as the only known problem to have a sublinear, near optimal space solution. Here we demonstrate that a similar sublinear, near optimal space solution is achievable for an even more challenging problem. The proof is considerably more complex than that for exact matching.
We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. Algorithms with low sensitivity are desirable because they are robust to edge failure or attack. In this work, we show a randomized $(1-epsilon)$-approximation algorithm with worst-case sensitivity $O_{epsilon}(1)$, which substantially improves upon the $(1-epsilon)$-approximation algorithm of Varma and Yoshida (arXiv 2020) that obtains average sensitivity $n^{O(1/(1+epsilon^2))}$ sensitivity algorithm, and show a deterministic $1/2$-approximation algorithm with sensitivity $exp(O(log^*n))$ for bounded-degree graphs. We show that any deterministic constant-factor approximation algorithm must have sensitivity $Omega(log^* n)$. Our results imply that randomized algorithms are strictly more powerful than deterministic ones in that the former can achieve sensitivity independent of $n$ whereas the latter cannot. We also show analogous results for vertex sensitivity, where we remove a vertex instead of an edge. As an application of our results, we give an algorithm for the online maximum matching with $O_{epsilon}(n)$ total replacements in the vertex-arrival model. By comparison, Bernstein et al. (J. ACM 2019) gave an online algorithm that always outputs the maximum matching, but only for bipartite graphs and with $O(nlog n)$ total replacements. Finally, we introduce the notion of normalized weighted sensitivity, a natural generalization of sensitivity that accounts for the weights of deleted edges. We show that if all edges in a graph have polynomially bounded weight, then given a trade-off parameter $alpha>2$, there exists an algorithm that outputs a $frac{1}{4alpha}$-approximation to the maximum weighted matching in $O(mlog_{alpha} n)$ time, with normalized weighted sensitivity $O(1)$. See paper for full abstract.
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