On exactness of SDP relaxation for the maximum cut problem

Semidefinite programming (SDP) provides a powerful relaxation for the maximum cut problem. For a graph with rational weights, the decision problem of whether the SDP relaxation for the maximum cut problem is exact is known to be $NP$-hard; however its complexity was unresolved for unweighted graphs. In this work, we extend the $NP$-hardness result to unweighted graphs. We characterize a few classes of graphs for which the SDP relaxation is exact. For each of these graph classes, we establish conditions for uniqueness of the SDP optimum. We complement these findings by identifying two graph operations that preserve the solution rank, and in turn exactness. These results reveal how the SDP relaxation for the maximum cut problem can remain exact in arbitrarily large graphs, owing to the presence of a small structural core that governs exactness. We further address two open problems posed by Mirka and Williamson (2024), by demonstrating that uniqueness of the maximum cut partition in exact relaxation does not imply uniqueness of the SDP optimum, and that exact relaxation with multiple optimal partitions may admit optimal SDP solutions lying outside the convex hull of rank-1 reference solutions.

💡 Research Summary

The paper investigates the exactness of the semidefinite programming (SDP) relaxation for the Maximum‑Cut (Max‑Cut) problem, focusing on unweighted graphs, computational complexity, structural graph classes, and open questions raised by Mirka and Williamson (2024).
First, the authors extend the known NP‑hardness result for weighted graphs to the unweighted case. By reducing from the Exact‑Sum problem (partition a set of integers into two subsets of equal sum, known to be NP‑complete), they construct an unweighted complete multipartite graph K(a₁,…,aₙ) whose SDP exactness is equivalent to the existence of a balanced subset. Lemma 3.1 shows that taking a split graph preserves exactness, allowing the reduction to work without edge weights. Consequently, deciding whether a given unweighted graph has an exact SDP relaxation is NP‑hard (Theorem 3.2).

Second, the paper characterizes several new families of unweighted graphs for which the GW‑relaxation is exact and provides sufficient conditions for the uniqueness of the optimal primal SDP solution. Building on earlier work (Delorme‑Poljak, Karloff, Hong et al.), the authors prove exactness for graphs such as Cₘ∨Cₘ, K₂ₘ∨C₂ₙ, Cₘ∨Kₙ, and fan graphs Pₘ∨Kₙ, among others. For each family they derive a rank‑equality relation that links the primal and dual optimal solutions; when this equality holds, the optimal solution must be rank‑1 and unique. They also give a complete uniqueness criterion for exact complete k‑partite graphs, showing that the ordering of part sizes determines whether the rank‑1 solution is unique or higher‑rank solutions inevitably appear.

Third, the authors study how graph operations affect exactness and solution rank. They prove that the join of two graphs with the same number of vertices preserves exactness, and that in a lexicographic product G₁·G₂ the rank of the optimal SDP solution is preserved whenever the first factor G₁ is exact. Moreover, the split operation (replacing each vertex by several copies) leaves exactness unchanged, a fact that underlies the NP‑hardness reduction. These results imply that the exactness of a large graph can often be inferred from a small structural core that remains unchanged under these operations.

Finally, the paper resolves two open problems from Mirka‑Williamson. (1) They exhibit exact graphs where the maximum‑cut partition is unique but the SDP optimum is not unique, demonstrating that uniqueness of the combinatorial solution does not guarantee uniqueness of the SDP solution. (2) They construct exact graphs with multiple optimal cuts whose SDP optimal points lie outside the convex hull of all rank‑1 (cut) matrices, showing that the convex hull of rank‑1 solutions does not always contain every optimal SDP point. These counterexamples highlight the presence of higher‑rank optimal solutions even in exact instances and suggest a richer geometry of the SDP feasible set than previously understood.

In summary, the paper makes four major contributions: (i) proving NP‑hardness of recognizing exactness for unweighted Max‑Cut, (ii) identifying new exact graph families together with uniqueness conditions, (iii) establishing graph‑operation preservation results that link large graphs to small exact cores, and (iv) answering two open questions by demonstrating higher‑rank phenomena in exact relaxations. The work deepens our theoretical understanding of SDP relaxations for combinatorial optimization and opens avenues for further study of the geometry of exact SDP solution sets.