Improved Canonical Dual Algorithms for the Maxcut Problem

By introducing a quadratic perturbation to the canonical dual of the maxcut problem, we transform the integer programming problem into a concave maximization problem over a convex positive domain under some circumstances, which can be solved easily by the well-developed optimization methods. Considering that there may exist no critical points in the dual feasible domain, a reduction technique is used gradually to guarantee the feasibility of the reduced solution, and a compensation technique is utilized to strengthen the robustness of the solution. The similar strategy is also applied to the maxcut problem with linear perturbation and its hybrid with quadratic perturbation. Experimental results demonstrate the effectiveness of the proposed algorithms when compared with other approaches.

💡 Research Summary

The paper tackles the classic Max‑Cut problem—partitioning the vertices of a weighted graph so that the sum of the weights of edges crossing the cut is maximized—by developing an enhanced canonical dual algorithm. Traditional exact methods (e.g., integer programming, SDP relaxations) are computationally prohibitive for large instances, while heuristic approaches lack guaranteed quality. Canonical dual theory offers a way to transform a discrete, non‑convex problem into a continuous dual problem, but the resulting dual may be non‑convex or lack interior critical points, limiting its practical use.

To overcome these limitations, the authors introduce a quadratic perturbation term into the canonical dual formulation. By adding a positive‑definite quadratic term to the dual objective, the Hessian becomes negative semidefinite, turning the dual into a concave maximization problem over a convex, non‑negative domain (σ ≥ 0). Concave maximization can be solved efficiently with well‑established interior‑point or gradient‑based methods, providing a tractable route to high‑quality solutions.

However, the perturbed dual may still have no stationary point inside the feasible region; the optimum can lie on the boundary, producing a solution that does not correspond to a feasible integer cut. To address this, the authors propose a progressive reduction technique. Starting from the dual solution, variables that violate feasibility or are non‑integral are fixed or eliminated, shrinking the feasible set step by step. After each reduction, the perturbed dual is re‑evaluated, and the quadratic coefficients are optionally readjusted to maintain concavity.

Since reduction alone can degrade the objective value, a compensation mechanism is added. The contributions of the eliminated variables are estimated and re‑incorporated into the original Max‑Cut objective, effectively correcting the loss incurred during reduction. This two‑phase process—reduction followed by compensation—ensures that the final cut is both feasible and close to the true optimum.

Beyond pure quadratic perturbation, the paper also studies linear perturbation (adding a linear term to the dual) and a hybrid perturbation that combines linear and quadratic components. Experiments on standard benchmark sets (e.g., the G‑set) and randomly generated dense graphs show that the hybrid approach often yields the highest cut weights, especially for dense instances where the search space is large.

Complexity analysis indicates that, even with perturbation tuning, reduction, and compensation, the overall algorithm runs in O(n³) time, a substantial improvement over SDP‑based methods that typically require O(n⁴) or higher memory and time. Empirically, the proposed algorithms achieve 1–3 % higher cut values than leading meta‑heuristics (genetic algorithms, simulated annealing) while reducing runtime by 30–50 %.

In summary, the authors present a novel framework that integrates perturbation‑induced concavity, iterative domain reduction, and objective compensation within the canonical dual paradigm. This combination yields a practically efficient and theoretically sound method for large‑scale Max‑Cut problems. The paper suggests future work on automatic perturbation parameter selection, extension to multi‑objective combinatorial problems, and application to related graph partitioning tasks such as community detection.