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Parallel Unconstrained Local Search for Partitioning Irregular Graphs

Nikolai Maas, Lars Gottesbüren, Daniel Seemaier

TL;DR

The paper tackles the NP-hard problem of balanced graph partitioning and proposes unconstrained refinement as a way to escape local minima, especially on irregular graphs. It develops two parallel unconstrained local-search solvers, unconstrained FM and unconstrained label propagation, augmented with a penalty-based rebalancing framework and a scalable rebalancing engine. Across extensive experiments, the approach yields up to 9.6% better edge cuts and achieves best-known results on irregular graphs for 75% of cases, with modest runtime overhead. This work demonstrates that allowing temporary balance violations, followed by careful rebalancing, can substantially improve partition quality and scalability, and it opens avenues for further refinements in penalty design and coarsening integration.

Abstract

We present new refinement heuristics for the balanced graph partitioning problem that break with an age-old rule. Traditionally, local search only permits moves that keep the block sizes balanced (below a size constraint). In this work, we demonstrate that admitting large temporary balance violations drastically improves solution quality. The effects are particularly strong on irregular instances such as social networks. Designing efficient implementations of this general idea involves both careful selection of candidates for unconstrained moves as well as algorithms for rebalancing the solution later on. We explore a wide array of design choices to achieve this, in addition to our third goal of high parallel scalability. We present compelling experimental results, demonstrating that our parallel unconstrained local search techniques outperform the prior state of the art by a substantial margin. Compared with four state-of-the-art solvers, our new technique finds 75\% of the best solutions on irregular graphs. We achieve a 9.6\% improvement in edge cut over the next best competitor, while being only 7.7\% slower in the geometric mean.

Parallel Unconstrained Local Search for Partitioning Irregular Graphs

TL;DR

The paper tackles the NP-hard problem of balanced graph partitioning and proposes unconstrained refinement as a way to escape local minima, especially on irregular graphs. It develops two parallel unconstrained local-search solvers, unconstrained FM and unconstrained label propagation, augmented with a penalty-based rebalancing framework and a scalable rebalancing engine. Across extensive experiments, the approach yields up to 9.6% better edge cuts and achieves best-known results on irregular graphs for 75% of cases, with modest runtime overhead. This work demonstrates that allowing temporary balance violations, followed by careful rebalancing, can substantially improve partition quality and scalability, and it opens avenues for further refinements in penalty design and coarsening integration.

Abstract

We present new refinement heuristics for the balanced graph partitioning problem that break with an age-old rule. Traditionally, local search only permits moves that keep the block sizes balanced (below a size constraint). In this work, we demonstrate that admitting large temporary balance violations drastically improves solution quality. The effects are particularly strong on irregular instances such as social networks. Designing efficient implementations of this general idea involves both careful selection of candidates for unconstrained moves as well as algorithms for rebalancing the solution later on. We explore a wide array of design choices to achieve this, in addition to our third goal of high parallel scalability. We present compelling experimental results, demonstrating that our parallel unconstrained local search techniques outperform the prior state of the art by a substantial margin. Compared with four state-of-the-art solvers, our new technique finds 75\% of the best solutions on irregular graphs. We achieve a 9.6\% improvement in edge cut over the next best competitor, while being only 7.7\% slower in the geometric mean.
Paper Structure (38 sections, 1 theorem, 7 figures, 2 tables)

This paper contains 38 sections, 1 theorem, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Results
  4. Preliminaries
  5. Unconstrained Refinement
  6. Related Work
  7. Refinement and Parallelization
  8. Prior Work on Unconstrained Refinement
  9. Unconstrained $k$-way FM Local Search
  10. Sequential FM
  11. Shared-memory Parallel FM
  12. Calculating Approximate Penalties
  13. Performance Guarantees
  14. Parallel Updates
  15. Rebalancing and Global Move Sequence
  16. ...and 23 more sections

Key Result

Lemma 5.1

Consider a graph $G$ with unit node weights, a balanced partition $\{V_1, \dots, V_k\}$ of $G$ and a set of nodes $U = \{u_1, \dots, u_j\}$ that are moved to $V_1$. Let $U^i \coloneqq \{u_1, \dots, u_i\}$ for $i \le j$. Then, the partition $\mathrm{\Pi} \coloneqq \{V_1 \cup U, V_2 \setminus U, \dots

Figures (7)

  • Figure 1: Illustration of unconstrained refinement. First, unconstrained moves are applied (left), which results in all high degree nodes being assigned to $V_1$ (center). Since the new partition is imbalanced, rebalancing is applied that moves low degree nodes to $V_2$. The resulting partition has a much smaller cut (right).
  • Figure 2: Unconstrained Parallel FM
  • Figure 3: Performance profiles comparing the solution quality for multiple variants of unconstrained local search on set I. This includes different combinations of unconstrained/constrained label propagation (LP) and FM (left) and alternative approaches to unconstrained FM (right).
  • Figure 4: Performance profile comparing the solution quality for different rebalancing approaches on set I.
  • Figure 5: Strong scalability results for our rebalancing algorithm on set I. Each point corresponds to an instance (sorted by sequential time), and the line is a rolling window geometric mean.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 5.1