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This paper presents a method for achieving almost stable matching on general bipartite graphs in constant distributed rounds, utilizing only a few shared random bits. The authors demonstrate that their algorithm can compute a matching with an expected number of blocking pairs limited to ε|E| in O(log(1/ε)/ε⁴) rounds, independent of the number of vertices and maximum degree. Their innovative degree-guarded freezing rule allows for efficient handling of varying degrees, significantly improving upon previous algorithms that relied on polylogarithmic round complexity for general graphs.
Achieving almost stable matching in constant rounds on general bipartite graphs with minimal shared randomness could revolutionize distributed algorithms in large-scale networks.
We show that almost stable matching can be solved in constant distributed rounds on general bipartite graphs $G=(V,E)$ using only a few shared random bits. Specifically, in the $\congest$ model, we compute a matching whose expected number of blocking pairs is at most $\varepsilon |E|$ in $O\left(\frac{\log(1/\varepsilon)}{\varepsilon^4}\right)$ rounds using $O\left(\log(1/\varepsilon)\right)$ shared random bits. Thus, for every constant $\varepsilon>0$, the round complexity is $O(1)$, independent of the number of vertices and the maximum degree. Previous algorithms achieve constant round complexity only for bounded-degree or almost-regular graphs; on general graphs, their round complexity depends polylogarithmically on $n$. Our main technical idea is a degree-guarded freezing rule that allows widely varying degrees to be handled by a single global charging argument, avoiding the $\Theta(\log n)$ successive degree thresholds used in previous work. The shared random bits are used only to select a common random output iteration. As consequences, we obtain an $O\left( \frac{\log(1/\varepsilon)}{\varepsilon^4} + \frac{\log n}{\varepsilon} \right)$-round $\congest$ algorithm without pre-shared randomness, via a low-diameter decomposition, and an $O\left(\frac{\log(1/\varepsilon)}{\varepsilon^4}\right)$-round algorithm in the fully-scalable Massively Parallel Computation ($\mpc$) model with linear total memory.
