Differentiable Quadratic Optimization For the Maximum Independent Set Problem

Ismail Alkhouri, Cedric Le Denmat, Yingjie Li, CUNXI YU, Jia Liu, Rongrong Wang, Alvaro Velasquez

Abstract: Combinatorial Optimization (CO) addresses many important problems, including the challenging Maximum Independent Set (MIS) problem. Alongside exact and heuristic solvers, differentiable approaches have emerged, often using continuous relaxations of quadratic objectives. Noting that an MIS in a graph is a Maximum Clique (MC) in its complement, we propose a new quadratic formulation for MIS by incorporating an MC term, improving convergence and exploration. We show that every maximal independent set corresponds to a local minimizer, derive conditions with respect to the MIS size, and characterize stationary points. To tackle the non-convexity of the objective, we propose optimizing several initializations in parallel using momentum-based gradient descent, complemented by an efficient MIS checking criterion derived from our theory. We dub our method as **p**arallelized **C**lique-Informed **Q**uadratic **O**ptimization for MIS (pCQO-MIS). Our experimental results demonstrate the effectiveness of the proposed method compared to exact, heuristic, sampling, and data-centric approaches. Notably, our method avoids the out-of-distribution tuning and reliance on (un)labeled data required by data-centric methods, while achieving superior MIS sizes and competitive run-time relative to their inference time. Additionally, a key advantage of pCQO-MIS is that, unlike exact and heuristic solvers, the run-time scales only with the number of nodes in the graph, not the number of edges. Our code is available at the GitHub repository: https://github.com/ledenmat/pCQO-mis-benchmark/tree/refactor.

Lay Summary:

Many important real-world problems, such as scheduling, managing resources, or designing networks, can be framed as combinatorial optimization problems. One such problem is the Maximum Independent Set (MIS) problem, which involves finding the largest group of non-connected nodes in a graph. This task is especially difficult as the size of the graph increases.
Traditional methods for solving MIS include exact algorithms and heuristics, but they can be slow or require expert tuning. Recently, new approaches have emerged that use smooth mathematical functions and gradient descent to search for solutions. In our work, we propose a new way to design such a function that makes it easier and more reliable to find large independent sets.
Our method, called parallelized Clique-inspired Quadratic Optimization (pCQO-MIS), runs multiple searches in parallel and uses mathematical insights to check whether each candidate solution is valid. pCQO-MIS doesn’t rely on any training data, making it more broadly applicable than many data-driven methods. We show that our approach outperforms many existing techniques in terms of both solution quality and speed—especially on large and dense graphs—while being more efficient in how it scales.

Link To Code: https://github.com/ledenmat/pCQO-mis-benchmark

Primary Area: Optimization->Discrete and Combinatorial Optimization

Keywords: Maximum Independent Set, Quadratic optimization, Gradient Descent, Combinatorial Optimization

Submission Number: 7460