Weighted Maximum-Cut Problem Overview

• Weighted Max-Cut is a discrete optimization task that partitions an edge-weighted graph into two sets to maximize the sum of crossing edge weights, with applications in VLSI, statistical physics, and machine learning.
• Semidefinite programming relaxations, such as the Goemans–Williamson algorithm, and heuristics like the signless MBO scheme offer strong approximation guarantees and scalable performance.
• Emerging methods, including quantum reformulations and photonic solvers, are advancing solution quality and speed, addressing NP-hardness in large-scale and constrained instances.

The weighted Maximum-Cut (Max-Cut) problem is a central paradigm in discrete optimization, graph algorithms, and combinatorial scientific computing. Given an edge-weighted undirected graph, the task is to partition its vertex set into two parts so as to maximize the sum of the weights of edges crossing the partition. This problem encapsulates both structural graph analytics and diverse applications in statistical physics, machine learning, circuit design, and quantum optimization. The weighted Max-Cut problem is NP-hard; it admits rigorous continuous and discrete relaxations, with significant recent advances in algorithm design, lower bounds, and emerging physical and quantum computing methods.

1. Formal Definition and Problem Structure

Let $G = (V, E, w)$ denote an undirected simple graph with vertex set $V$, edge set $E \subseteq \binom{V}{2}$, and edge weight function $w: E \rightarrow \mathbb{R}$ (typically nonnegative but negative weights are possible). A cut is a partition of $V$ into two disjoint subsets $S$, $V \setminus S$. The weight of the cut is given by

$$w(S, V \setminus S) = \sum_{\{u,v\}\in E: u \in S, v \notin S} w_{\{u,v\}}.$$

The weighted Max-Cut problem is

$$\mathrm{MaxCut}(G) = \max_{S \subseteq V} w(S, V \setminus S).$$

Alternative equivalent formulations use indicator variables: assign to each vertex $i$ a label $x_i \in \{-1, +1\}$, with cut weight defined as

$\frac{1}{2}\sum_{i < j} w_{ij}(1 - x_i x_j).$

The weighted Max-Cut problem is NP-hard in general and APX-hard even in the unweighted case. Approximation schemes and relaxations are critical for algorithmic progress (Gutin et al., 2021, Proença et al., 2023).

2. Algorithmic Approaches: Relaxations, SDP, and Approximations

Semidefinite Programming (SDP) Relaxation and the Goemans–Williamson Bound

The seminal Goemans–Williamson framework solves the following SDP relaxation: $$\max \frac{1}{4} \sum_{\{i,j\} \in E} w_{ij} (1 - \langle v_i, v_j \rangle) \quad \text{subject to} \quad v_i \in \mathbb{R}^n,\; \|v_i\| = 1,$$ where $v_i$ are unit vectors. Rounding is performed by projecting onto a random hyperplane, yielding a randomized cut. For all $w \geq 0$,

$$\alpha_{\mathrm{GW}} \cdot \text{SDP} \leq \mathrm{MaxCut}(G) \leq \text{SDP},$$

with $\alpha_{\mathrm{GW}} \approx 0.878$, which is optimal assuming the Unique Games Conjecture (Proença et al., 2023).

Primal–Dual, Fractional Cut-Covering, and Certificates

There exists a primal-dual framework connecting weighted Max-Cut and the weighted fractional cut-covering number via antiblocker and gauge duality. The dual problem (fractional cut-covering) admits an SDP relaxation and randomized hyperplane rounding, leading to simultaneous combinatorial certificates (cuts and covers), all with the same 0.878 approximation guarantee (Proença et al., 2023).

Fast Approximation Methods

The signless Merriman–Bence–Osher (MBO) scheme is a variational, diffusion–thresholding framework for (weighted) Max-Cut. Given a weighted adjacency ω, the method iterates signless Laplacian diffusion, thresholding, and cut evaluation. On random, modular, and real-world graphs, signless MBO matches or outperforms GW's expected cut quality with 5–20× faster runtime; on large sparse graphs, it scales to millions of edges with near-linear iteration complexity (Keetch et al., 2017).

Table: Comparison of Key Classical Approximation Schemes

Approach	Theoretical Ratio	Worst-Case Complexity
Goemans–Williamson	0.878	SDP + n³
MBO (spectral)	Empirical ~0.99	O(
Greedy/metaheur.	Heuristic	O(n²)–O(n³) (turbo GA)

3. Lower Bounds, Exact Algorithms, and Parameterized Complexity

General Lower Bounds

Gutin and Yeo provide a systematic extension of classical and probabilistic bounds for the weighted case. For any B-subgraph (union of induced bipartite components) $R \subseteq G$,

$\mathrm{MaxCut}(G) \geq \frac{w(G) + w(R)}{2},$

where $w(G)$ is the total weight of $G$. Specialized lower bounds include those based on DFS-trees, minimum/maximum spanning trees, and matchings, e.g.,

$$\mathrm{MaxCut}(G) \geq \frac{w(G)}{2} + \frac{w(D)}{4}$$

for any DFS spanning tree $D$ (Gutin et al., 2021).

Parameterized Algorithms

The weighted Maximum-Cut problem is fixed-parameter tractable (FPT) with respect to the crossing number $k$ of a graph drawing: $T(n, k) = O(2^k \cdot p(n + k)),$ where $p(n)$ is the time to solve planar Max-Cut. The core procedure uses recursive elimination of crossings via bisubdivision and branching, preserving cut-value through a sequence of reductions and edge fixings. This result holds without requiring an explicit drawing, and generalizes to the minor crossing number (Chimani et al., 2019).

4. Extensions: Constraints, Quantum, and Physical Computing

Constrained and Generalized Max-Cut

Weighted Max-Cut with multiple cardinality constraints (Constrained Max-Cut) imposes blockwise size limitations on one side of the bipartition. For $c = O(1)$ constraints, a (0.858 − ε)-approximation is achieved via SDP kernelization and correlation rounding. This improves upon earlier 0.5-approximation results for constrained cases and generalizes hyperplane rounding to the block-constrained regime (Makarychev et al., 16 Jul 2025).

QUBO and Quantum Reformulations

The weighted Max-Cut can be cast as a Quadratic Unconstrained Binary Optimization (QUBO) problem: $$q(x) = \sum_{\{u,v\} \in E} w_{uv}(x_u + x_v - 2 x_u x_v),$$ where $x_u \in \{0, 1\}$. For QUBO encodings derived from the $(k=2)$-cut problem, tight penalty coefficients for enforcing one-hot constraints are given by the local weighted degree: $c_v = \frac{1}{2} \sum_{u \in N(v)} w_{uv}$ for each vertex $v$, minimizing the dynamic range of QUBO coefficients for embedding on quantum hardware (Harkness et al., 2 Nov 2025).

Quantum variational algorithms use block-encoded Ising Hamiltonian representations of Max-Cut, with shallow-depth circuits and normalized gradient descent producing high accuracy on small, dense graphs. Empirical results for $n \leq 10$ show approximation ratios above 0.9 and competitive performance with D-Wave annealers (Meli et al., 2023).

Large-Scale Photonic Solvers

A photonic spatial Ising machine maps the weighted Max-Cut Hamiltonian onto optical hardware via phase-only spatial light modulators, exploiting a quadrature encoding of weights. Record-scale experiments solve 20 736-node dense weighted Max-Cut instances with cut-values up to 49% above classical heuristics and a 122× speedup; the implementation supports arbitrary weights and is limited primarily by SLM resolution and refresh rates (Ye et al., 2023).

5. Empirical Performance and Paradigm Comparison

Recent systematic comparisons incorporate classical metaheuristics (Genetic Algorithm, GA), deep learning (Graph Neural Network, GNN), and quantum-inspired hardware-efficient algorithms such as Density Matrix Renormalization Group (DMRG). Each paradigm exhibits characteristic tradeoffs for Weighted Max-Cut on instances up to 250 nodes:

Solver	Approx. Ratio (Large)	Typical Runtime	Peak Memory
DMRG (χ=2)	~0.985–0.99	0.01 min ($n=250$)	170 MiB
GNN	0.83–0.97	0.4–0.6 min	<1 MiB
cGA-2000	0.95	~220–290 min	14–20 MiB
GA-OC	0.97–0.98	~200 min	3 MiB

DMRG is highly efficient and achieves near-optimality for large graphs. GNN is effective for medium-sized graphs with minimal memory; performance degrades for larger graphs. Classical metaheuristics (GA, GA-OC) offer flexibility but their runtime grows super-linearly with size and they are outperformed by tensor-based approaches on large graphs (Morais et al., 8 Apr 2025).

6. Open Problems, Lower Bound Theory, and Future Directions

Lower Bound Theory

While lower bounds for the unweighted case are classical, recent results extend and improve these for weighted graphs. E.g., for triangle-free subcubic graphs,

$\mathrm{MaxCut}(G) \geq 0.727\, w(G),$

with a conjectural lower bound of $4/5\, w(G)$ (Gutin et al., 2021). For arbitrary triangle-free graphs of maximum degree $\Delta$, explicit constants $s_\Delta, t_\Delta$ yield refined lower bounds via coloring and matching decomposition.

Open Problems

• Determining optimal lower bounds for weighted Max-Cut in various structural classes remains open, notably for triangle-free graphs and for cut-plus-tree inequalities.
• The gap between the 0.858-approximation for block-constrained Max-Cut and the unconstrained 0.878 remains open for $c > 0$, with integrality gap and Unique Games Conjecture barriers for further improvement (Makarychev et al., 16 Jul 2025).
• On the quantum side, characterizing the impact of penalty scaling and embedding on the feasibility of scalable QUBO optimization is an ongoing area of exploration (Harkness et al., 2 Nov 2025).
• Structural speedups beyond the $2^k$ FPT dependence on crossing number are likely to require fundamentally new insights in combinatorics and graph minor theory (Chimani et al., 2019).

7. Applications and Broader Context

Weighted Max-Cut optimization underpins applications in VLSI partitioning, statistical physics (spin glasses and Ising models), network design, clustering, quantum annealing, and combinatorial design. Emerging physical and quantum algorithms seek to address scalability bottlenecks posed by classical NP-hardness, with recent work demonstrating both theoretical and large-scale experimental advances.

The problem also serves as a prototypical testbed for the practical limits of continuous relaxations (SDP, Lasserre, QUBO), new approximation paradigms (such as MBO and DMRG), and the comparative benchmarking of classical, quantum-inspired, and analog computational architectures. In the presence of additional constraints and side information, the weighted Max-Cut framework generalizes naturally to support cardinality, matroid, or combinatorial restrictions, for which tight approximations and bounds are active research frontiers.

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References:

• (Chimani et al., 2019): Maximum Cut Parameterized by Crossing Number
• (Ye et al., 2023): 20736-node Weighted Max-Cut Problem Solving by Quadrature Photonic Spatial Ising Machine
• (Meli et al., 2023): A Universal Quantum Algorithm for Weighted Maximum Cut and Ising Problems
• (Makarychev et al., 16 Jul 2025): Max-Cut with Multiple Cardinality Constraints
• (Harkness et al., 2 Nov 2025): Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing
• (Morais et al., 8 Apr 2025): Comparative Analysis of Classical and Quantum-Inspired Solvers: A Preliminary Study on the Weighted Max-Cut Problem
• (Keetch et al., 2017): A Max-Cut approximation using a graph based MBO scheme
• (Gutin et al., 2021): Lower Bounds for Maximum Weighted Cut
• (Proença et al., 2023): A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem