Distributed Algorithms for Potential Problems

In this work, we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges. The distributed round complexity of the locally optimal cut problem has been wide open; the problem is known to require $Ω(\log n)$ rounds in the deterministic LOCAL model and $Ω(\log \log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in constant-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds. We show that in constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in $\log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $\log^{Θ(1)} n$. Our algorithms also apply to the general case of graphs of maximum degree $Δ$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(Δ^{2} \log^{6} n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $Δ$ is necessary: we prove a lower bound of $Ω(\min{Δ,\sqrt{n}})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

💡 Research Summary

The paper tackles a long‑standing gap in the distributed complexity of locally optimal cuts, a problem that asks each node in a graph to choose a label (+1 or –1) such that at least half of its incident edges are cut edges. Prior work only gave a trivial $O(n)$‑round algorithm and lower bounds of $\Omega(\log n)$ (deterministic) and $\Omega(\log\log n)$ (randomized) in the LOCAL model, leaving the true complexity unknown even for constant‑degree graphs.

The authors introduce a broader class called local potential problems. Each such problem has a finite set of node labels, a locally checkable “happy/unhappy” predicate, and a potential function that assigns a value to nodes or edges based only on a constant‑radius neighbourhood. Flipping the label of an unhappy node strictly increases the total potential, guaranteeing that a globally happy configuration always exists and can be reached by a centralized greedy process.

Leveraging this structure, the paper designs a distributed algorithm that repeatedly (i) identifies all unhappy nodes, (ii) extracts a maximal independent set (MIS) among them, and (iii) flips the labels of the MIS nodes in parallel. Because each flip raises the local potential by at least a constant $\lambda$ and the total potential is bounded by $\Lambda\cdot n$, the number of iterations is $O((\Lambda/\lambda)\log n)$. In constant‑degree graphs $\Lambda$ and $\lambda$ are constants, yielding an $O(\log^{6} n)$‑round randomized algorithm. For general maximum degree $\Delta$, the analysis shows a dependence of $O(\Delta^{2}\log^{6} n)$ rounds. The MIS step can be performed with standard $O(\log^{*} n)$‑round randomized techniques; derandomizing it adds only a polylogarithmic overhead, giving a deterministic runtime of $O(\Delta^{2}\log^{8} n\cdot\operatorname{polylog} n)$.

To complement the upper bounds, the authors prove a lower bound that holds even in the quantum‑LOCAL model: any algorithm for locally optimal cuts (and more generally for local potential problems) requires $\Omega(\min{\Delta,\sqrt{n}})$ rounds. This shows that for unbounded degree graphs a polylogarithmic‑round algorithm is impossible, and even for constant degree a linear dependence on $\Delta$ cannot be avoided.

The framework is not limited to cuts. The paper demonstrates how defective coloring, where each node may have at most $d$ neighbours of the same colour, fits the local potential model when $c(d+1)>\Delta$. The same algorithm solves such coloring problems in polylogarithmic time. Moreover, the authors argue that many LCL (locally checkable labeling) problems can be reinterpreted as local potential problems; conversely, problems that cannot be expressed this way (e.g., stable matchings) are known to require linear rounds, providing a new lens for classifying distributed problems.

Finally, the paper lists several open directions: whether randomness can universally reduce the deterministic $\log^{\Theta(1)} n$ bound to $O(\log\log n)$ for all local potential problems, whether quantum communication can give any advantage beyond the proven lower bound, and whether the existence of a suitable potential function for a given specification is decidable.

In summary, the work bridges a gap between LCL theory and distributed optimization by introducing local potential functions, delivering the first non‑trivial polylogarithmic‑round algorithms for locally optimal cuts, extending them to general degree graphs, proving tight degree‑dependent lower bounds, and opening a new research avenue for classifying distributed problems through potential‑based formulations.