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A Bregman-Sinkhorn Algorithm for the Maximum Weight Independent Set Problem

Stefan Haller, Bogdan Savchynskyy

TL;DR

This work tackles the NP-hard maximum-weight independent set problem by solving a smoothed clique-cover LP relaxation with a dual coordinate descent method inspired by Bregman and Sinkhorn techniques. It couples this with a projection-based mechanism to extract feasible primal estimates and uses a duality-gap based temperature schedule to adapt smoothing, achieving fast convergence on large graphs. A sparsity-aware dual algorithm is paired with a simple primal heuristic that leverages reduced costs to generate high-quality integer solutions rapidly. Empirical results on real-world datasets demonstrate strong performance against state-of-the-art LP/ILP solvers and specialized MWIS heuristics, highlighting the method's scalability and practical impact for large-scale MWIS instances.

Abstract

We propose a scalable approximate algorithm for the NP-hard maximum-weight independent set problem. The core component of our algorithm is a dual coordinate descent applied to a smoothed LP relaxation of the problem. This technique is commonly known by the names Bregman method and Sinkhorn algorithm in the literature. Our algorithm addresses a family of clique cover LP relaxations, where the constraints are determined by the set of cliques covering the underlying graph. The objective function of the relaxation is smoothed with an entropy term. A crucial aspect determining efficiency of our approach is controlling the smoothing level during the optimization process. While several dedicated techniques have been considered in the literature to this end, we propose a new one based on estimation of the relaxed duality gap. To make this estimation possible, we developed a new projection method to the feasible set of the considered LP relaxation. We experimentally show that our smoothing scheduling significantly outperforms the standard one based on feasibility estimation. Additionally to solving the relaxed dual, we utilize a simple and very efficient primal heuristic to obtain feasible integer solutions of the original non-relaxed problem. Our heuristic is a combination of randomized greedy generation and optimal recombination applied to the reduced costs computed by the dual optimization. Our experimental validation considers two datasets rooted in real-world applications, where our method demonstrates the ability to discover high-quality approximate solutions within 10 seconds for graphs with up to 882 thousand nodes and 344 million edges.

A Bregman-Sinkhorn Algorithm for the Maximum Weight Independent Set Problem

TL;DR

This work tackles the NP-hard maximum-weight independent set problem by solving a smoothed clique-cover LP relaxation with a dual coordinate descent method inspired by Bregman and Sinkhorn techniques. It couples this with a projection-based mechanism to extract feasible primal estimates and uses a duality-gap based temperature schedule to adapt smoothing, achieving fast convergence on large graphs. A sparsity-aware dual algorithm is paired with a simple primal heuristic that leverages reduced costs to generate high-quality integer solutions rapidly. Empirical results on real-world datasets demonstrate strong performance against state-of-the-art LP/ILP solvers and specialized MWIS heuristics, highlighting the method's scalability and practical impact for large-scale MWIS instances.

Abstract

We propose a scalable approximate algorithm for the NP-hard maximum-weight independent set problem. The core component of our algorithm is a dual coordinate descent applied to a smoothed LP relaxation of the problem. This technique is commonly known by the names Bregman method and Sinkhorn algorithm in the literature. Our algorithm addresses a family of clique cover LP relaxations, where the constraints are determined by the set of cliques covering the underlying graph. The objective function of the relaxation is smoothed with an entropy term. A crucial aspect determining efficiency of our approach is controlling the smoothing level during the optimization process. While several dedicated techniques have been considered in the literature to this end, we propose a new one based on estimation of the relaxed duality gap. To make this estimation possible, we developed a new projection method to the feasible set of the considered LP relaxation. We experimentally show that our smoothing scheduling significantly outperforms the standard one based on feasibility estimation. Additionally to solving the relaxed dual, we utilize a simple and very efficient primal heuristic to obtain feasible integer solutions of the original non-relaxed problem. Our heuristic is a combination of randomized greedy generation and optimal recombination applied to the reduced costs computed by the dual optimization. Our experimental validation considers two datasets rooted in real-world applications, where our method demonstrates the ability to discover high-quality approximate solutions within 10 seconds for graphs with up to 882 thousand nodes and 344 million edges.
Paper Structure (38 sections, 14 theorems, 37 equations, 3 figures, 2 tables, 6 algorithms)

This paper contains 38 sections, 14 theorems, 37 equations, 3 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation.
  3. Overview of the known results and related work
  4. ILP formulation and edge relaxation.
  5. Clique cover relaxation family.
  6. Equality formulation.
  7. Relation to pseudo-boolean optimization.
  8. Solvability and approximation.
  9. Relation to linear assignment.
  10. Contribution and paper content
  11. Paper content.
  12. Dual of the clique cover relaxation
  13. Dual problem.
  14. Bregman method for MWIS
  15. Numerics
  16. ...and 23 more sections

Key Result

Proposition 1

Let $G$ be fully-connected, all costs positive, $c_j > 0$, $j\in\mathcal{V}$, and $c_i < \sum_{j\in\mathcal{V}\backslash\{i\}}c_j$ for $i=\mathop{\mathrm{arg\,max}}\limits_{j\in \mathcal{V}}c_j$. Then the edge relaxation has a unique solution $(\underbrace{1/2,1/2,\dots,1/2}_{n})$.

Figures (3)

  • Figure 1: Optimization of the problem relaxation. These plot track the relative gap to the optimum of the LP relaxation of MWIS problem \ref{['equ:MWIS-clique']} across run time. Dashed lines refer to the dual objective (upper bound) and solid lines refer to the primal (relaxed) objective (lower bound). Instead of individual models, we show the mean objective across all models of the dataset group, providing a consolidated view of algorithm performance. Our algorithms is able to provide high-quality solutions ($<$ 1 % rel. gap) for the majority of the instances within 1--10 seconds.
  • Figure 2: Primal heuristic ablation. Exemplary comparison on the problem instance AVR_004 from the AVR-large dataset. As the comparison suggests, optimized recombination leads to significant improvement of the resulting solution. Even more improvement is obtained due to usage of the reduced costs: Even the best recombination result obtained for the original costs is worse than each single greedily generated solution based on reduced costs.
  • Figure 3: Optimization of the non-relaxed problem. This plot presents the absolute primal objective values over run time, highlighting the effectiveness of our best-performing dual optimizer in conjunction with the proposed primal heuristics. Dashed lines refer to the dual objective (upper bound) and solid lines refer to the primal objective (lower bound). We compare our solver to Gurobi ILP and KaMIS. Similarly to \ref{['fig:plots-relaxed']} we show the mean objective across all models of the different groups. Our solver yields high-quality MWIS solutions after only a few seconds for the majority of instances. Several variants of the KaMIS algorithms are missing in some plots as their attained values lie outside the visible area of the respective plots.

Theorems & Definitions (32)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Proposition 4
  • Example 1
  • Lemma 1
  • Proposition 5
  • Lemma 2
  • Lemma 3
  • ...and 22 more