The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between $\omega(\log^* n)$ and $o(\log n)$. This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where $O(1)$, $\Theta(\log^* n)$, and $\Theta(n)$ are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes

Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications

In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition. In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost $\text{poly}(\log n, 1/\epsilon)$ rounds deterministically. For bounded-degree graphs, our algorithms finish in $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A $(1-\epsilon)$-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in $O(\epsilon^{-1}\log^\ast n) + \epsilon^{-O(1)}$ rounds, nearly matching the $\Omega(\epsilon^{-1}\log^\ast n)$ lower bound of Lenzen and Wattenhofer [DISC 2008]. 2. Property testing of any additive minor-closed property can be done deterministically in $O(\log n)$ rounds if $\epsilon$ is a constant or $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ rounds if the maximum degree $\Delta$ is a constant, nearly matching the $\Omega(\epsilon^{-1}\log n)$ lower bound of Levi, Medina, and Ron [PODC 2018].Comment: To appear in PODC 202

The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-\epsilon)$-approximate solution in $O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+\epsilon)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}{\epsilon}\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm \epsilon)$-approximate solutions require $\Omega\left(\frac{\log n}{\epsilon}\right)$ rounds to compute.Comment: To appear in PODC 202