Classification of Local Optimization Problems in Directed Cycles

TL;DR

This work delivers a complete, computable classification of the distributed round complexity for α-approximate local optimization problems on directed cycles in both deterministic and randomized LOCAL models. By modeling Π with a node-weighted de Bruijn graph and seven problem parameters β_opt, β_flex, δ_flex, β_coprime, β_gap, δ_gap, β_const, the authors show these parameters fully determine the optimal complexity class and enable automatic synthesis of asymptotically optimal distributed algorithms. The classification yields four principal time regimes, namely constant, Θ(log^* n), and Θ(n) in both models, with nuanced behavior in sum-max/min variants; the framework covers classical tasks such as MIS, MVC, MIN-DOMSET, and vertex coloring, among others. This work thereby resolves the landscape of local optimization on directed cycles, showing computable, automatic decision procedures and offering a foundation for extending the results to broader graph families and input models.

Abstract

We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem $Π$ (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any \emph{constant} approximation ratio $α$, the task of finding an $α$-approximation of $Π$ in directed cycles has one of the following complexities: 1. $O(1)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 2. $Θ(\log^* n)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 3. $Θ(\log^* n)$ rounds in deterministic LOCAL, $Θ(\log^* n)$ rounds in randomized LOCAL, 4. $Θ(n)$ rounds in deterministic LOCAL, $Θ(n)$ rounds in randomized LOCAL. Moreover, for any given $Π$ and $α$, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

Figures (2)

• Figure 1: De Bruijn digraphs for \ref{['ex:max-ind-set', 'ex:min-dom-set', 'ex:min-ver-col', 'ex:max-dom-par', 'ex:sloppy-col']}. In blue, the cost of each neighborhood. Nodes with cost $\bot$ can be considered as non-existing in the rest of the paper.
• Figure 2: Illustration of the construction of $W'$ using $W$ in \ref{['lem:zero_pot-minsum']}. Each time $W$ uses some part of $W_{k+1}$, we add at the end of it the unused part of $W_{k+1}$ (the green upper part) and any path going from $a^i_{k_i}$ to $a_1^{i+1}$ in $\cup_i^k W_{i}$. This way it is clear that $W'$ consists of possibly multiple instances of $W_{k+1}$ and some closed walk on $\cup_i^k W_{i}$, making the shifted cost of $W'$ necessarily $0$.

Theorems & Definitions (32)

• Remark
• Example 1: maximum independent set
• Example 2: minimum dominating set
• Example 3: minimum vertex coloring
• Example 4: maximum domatic partition
• Example 5: sloppy coloring
• Definition 6: De Bruijn Graph
• Definition 7: $(u,v)$-walk
• Definition 8: Cost of a walk
• Definition 9: Flexible nodes and components