Distributed Half-Integral Matching and Beyond

By prior work, it is known that any distributed graph algorithm finds a maximal matching requires $$\varOmega (\log ^* n)$$ communication rounds, while possible to find fractional in O(1) rounds bounded-degree graphs. However, all O(1)-round algorithms for use arbitrarily fine-grained values. In particular, none of them able half-integral solution, using only values from $$\{0, \frac{1}{2}, 1\}$$ . We show the necessary, and moreover we give complete characterization on exactly how small are needed: if consider graphs maximum degree $$\varDelta = 2d$$ , with round complexity $$T(\varDelta )$$ depends $$ independent n, has denominator at least $$2^d$$ new shows this also sufficient.