Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Algorithms for Potential Problems

Alkida Balliu, Thomas Boudier, Francesco d'Amore, Fabian Kuhn, Dennis Olivetti, Gustav Schmid, Jukka Suomela

TL;DR

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, and the deterministic round complexity of the locally optimal cut problem is settled to $\log^{\Theta(1)} n$.

Abstract

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.

Distributed Algorithms for Potential Problems

TL;DR

In constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in rounds, both in the deterministic and randomized LOCAL models, and the deterministic round complexity of the locally optimal cut problem is settled to .

Abstract

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 rounds in the deterministic LOCAL model and rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of 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 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 . Our algorithms also apply to the general case of graphs of maximum degree . For the special case of locally optimal cut, we obtain a randomized algorithm that runs in rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in is necessary: we prove a lower bound of rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

Paper Structure

This paper contains 42 sections, 31 theorems, 70 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Locally optimal cuts
  3. Distributed graph algorithms.
  4. Distributed algorithms for locally optimal cuts.
  5. Broader context: locally checkable labelings.
  6. Prior work.
  7. Our contribution.
  8. Open questions.
  9. Local potential problems
  10. Local potential problems.
  11. Locally optimal problems.
  12. Reductions and generalizations.
  13. Our contribution.
  14. An application: defective coloring.
  15. Future work and open questions.
  16. ...and 27 more sections

Key Result

Lemma 2.9

Consider a GLOP $(\Pi, \Psi)$ as defined in def:preliminaries:lop-alternative. Then, there exists an LOP $(\Pi', \Psi')$ as defined in def:preliminaries:lop such that any solution to $(\Pi', \Psi')$ can be converted into a solution of $(\Pi, \Psi)$ by an $O(1)$-round LOCAL algorithm.

Theorems & Definitions (73)

  • Definition 2.1: Labeled graph
  • Definition 2.2: Centered graph
  • Definition 2.3: Labeled centered graph
  • Definition 2.4: Set of constraints
  • Definition 2.5: Labeled graph satisfying a set of constraints
  • Definition 2.6: Locally checkable labeling (LCL) problems
  • Definition 2.7: Locally Optimal Problems
  • Definition 2.8: Generalized Locally Optimal Problems
  • Lemma 2.9
  • proof
  • ...and 63 more