Closing the Gap between Weighted and Unweighted Matching in the Sliding Window Model

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set $V$ of size $n$. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the $L$ most recent edges, for some integer $L$, using space $\tilde{O}(n)$. Crouch et al. [ESA'13] gave a $(3+\varepsilon)$-approximation for (unweighted) Maximum Matching (MM), and, very recently, Biabani et al. [ISAAC'21] gave a $(3.5+\varepsilon)$-approximation for MWM. In this paper, we give a $(3 + \varepsilon)$-approximation for MWM, thereby closing the gap between MWM and MM. Biabani et al.'s work makes use of the smooth histogram technique introduced by Braverman and Ostrovsky [FOCS'07]. Rather than designing sliding window algorithms directly, this technique reduces the problem to designing so-called lookahead algorithms that have certain smoothness properties. Biabani et al. showed that the one-pass MWM streaming algorithm by Paz and Schwartzman [SODA'17] constitutes a lookahead algorithm with approximation factor $3.5 + \varepsilon$, which yields their result. We first give a hard instance, showing that Paz and Schwartzman's algorithm is indeed no better than a $3.5$-approximation lookahead algorithm, which implies that Biabani et al.'s analysis is tight. To obtain our improvement, we give an alternative and more complex definition of lookahead algorithms that still maintains the connection to the sliding window model. Our new definition, however, reflects further smoothness properties of Paz and Schwartzman's algorithm, which we exploit in order to improve upon Biabani et al.'s analysis, thereby establishing our result.