Slobodan Mitrović

Slobodan Mitrović, Srikkanth Ramachandran, Ronitt Rubinfeld et al.

In this work, we focus on designing an efficient Local Computation Algorithm (LCA) for the set cover problem, which is a core optimization task. The state-of-the-art LCA for computing $O(\log Δ)$-approximate set cover, developed by Grunau, Mitrović, Rubinfeld, and Vakilian [SODA '20], achieves query complexity of $Δ^{O(\log Δ)} \cdot f^{O(\log Δ\cdot (\log \log Δ+ \log \log f))}$, where $Δ$ is the maximum set size, and $f$ is the maximum frequency of any element in sets. We present a new LCA that solves this problem using $f^{O(\log Δ)}$ queries. Specifically, for instances where $f = \text{poly} \log Δ$, our algorithm improves the query complexity from $Δ^{O(\log Δ)}$ to $Δ^{O(\log \log Δ)}$. Our central technical contribution in designing LCAs is to aggressively sparsify the input instance but to allow for \emph{retroactive updates}. Namely, our main LCA sometimes ``corrects'' decisions it made in the previous recursive LCA calls. It enables us to achieve stronger concentration guarantees, which in turn allows for more efficient and ``sparser'' LCA execution. We believe that this technique will be of independent interest.

Mina Dalirrooyfard, Konstantin Makarychev, Slobodan Mitrović

We revisit the complexity analysis of the recursive version of the randomized greedy algorithm for computing a maximal independent set (MIS), originally analyzed by Yoshida, Yamamoto, and Ito (2009). They showed that, on average per vertex, the expected number of recursive calls made by this algorithm is upper bounded by the average degree of the input graph. While their analysis is clever and intricate, we provide a significantly simpler alternative that achieves the same guarantee. Our analysis is inspired by the recent work of Dalirrooyfard, Makarychev, and Mitrović (2024), who developed a potential-function-based argument to analyze a new algorithm for correlation clustering. We adapt this approach to the MIS setting, yielding a more direct and arguably more transparent analysis of the recursive randomized greedy MIS algorithm.