Optimal distributed covering algorithms

Abstract

AbstractWe present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is $$(f+\varepsilon )$$ ( f + ε ) . Let $$\varDelta $$ Δ denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires $$O(\log {\varDelta } / \log \log \varDelta )$$ O ( log Δ / log log Δ ) rounds, for constants $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] and $$f\in {\mathbb {N}}^+$$ f ∈ N + . This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and $$\varepsilon $$ ε , our algorithm improves over the $$(f+\varepsilon )$$ ( f + ε ) -approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is $$O(\log \varDelta + \log W)$$ O ( log Δ + log W ) , where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in $$O(f\log n)$$ O ( f log n ) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of $$O(f\log ^2 n)$$ O ( f log 2 n ) . Finally, for weighted vertex cover ($$f=2$$ f = 2 ) our algorithm achieves a deterministic running time of $$O(\log n)$$ O ( log n ) , matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an $$(f\lceil \log _2(M)+1 \rceil +\varepsilon )$$ ( f ⌈ log 2 ( M ) + 1 ⌉ + ε ) -approximate integral solution in $$\begin{aligned} O\left( (1+f/\log n)\cdot \left( {\frac{\log \varDelta }{ \log \log \varDelta } + ({f\cdot \log M})^{1.01}\cdot \log \varepsilon ^{-1}\cdot (\log \varDelta )^{0.01}}\right) \right) \end{aligned}$$ O ( 1 + f / log n ) · log Δ log log Δ + ( f · log M ) 1.01 · log ε - 1 · ( log Δ ) 0.01 rounds, where f bounds the number of variables in a constraint, $$\varDelta $$ Δ bounds the number of constraints a variable appears in, and $$M=\max \left\{ 1, \lceil 1/a_{\min } \rceil \right\} $$ M = max 1 , ⌈ 1 / a min ⌉ , where $$a_{\min }$$ a min is the smallest normalized constraint coefficient.