Probabilistic Relaxations of Graph Cuts

• The paper introduces continuous soft-assignment frameworks to relax classical graph cut problems, enabling differentiable and scalable optimization.
• It details quantum relaxations that convert combinatorial objectives into local Hamiltonians, achieving memory compression and robust empirical performance.
• The work unifies classical, probabilistic, and quantum methods by providing analytic surrogates with tight bounds for end-to-end deep learning integration.

Probabilistic relaxations of graph cuts provide a spectrum of methodologies for formulating, approximating, and optimizing discrete graph partitioning objectives using continuous, differentiable, and sometimes quantum-parametric frameworks. These approaches generalize classical combinatorial cut optimization by admitting probabilistic, soft, or quantum representations of cluster assignments, enabling scalable, end-to-end learning and optimized implementations on modern computational platforms.

1. Discrete Graph-Cut Problems and Classical Relaxations

Classically, graph-cut problems such as Max-Cut, Min-Cut, RatioCut, and Normalized Cut are posed as combinatorial optimization tasks: given an affinity matrix $A\in\mathbb{R}_+^{n\times n}$ of a graph $G=(V,E)$, one seeks a partition into subsets—often encoded via discrete indicator vectors $s\in\{0,1\}^n$ or $m\in\{-1,+1\}^n$—that extremize objectives of the form: $$\text{Cut}(S, \bar S) = \sum_{i\in S}\sum_{j\in\bar S} A_{ij}, \quad \text{or} \quad \mathrm{cut}(m) = \sum_{(i,j)\in E} \frac{1}{2}(1-m_im_j).$$ For normalized objectives, cluster volumes may enter denominators, as in

$$\mathrm{vCut}(S) = \frac{\mathrm{Cut}(S,\bar S)}{\mathrm{vol}(S)},$$

where $\mathrm{vol}(S) = \sum_{i\in S} s_i$.

Spectral and semi-definite programming (SDP) relaxations have long dominated the field, introducing continuous or vectorial surrogates for the hard discrete optimization variables and leading to well-understood theoretical guarantees in some regimes.

2. Probabilistic and Quantum Relaxation Methodologies

Recent advancements have introduced probabilistic relaxations, in which hard indicator variables are replaced by probability distributions or soft assignments, and quantum relaxations, which encode combinatorial structure into local Hamiltonians acting on quantum states.

2.1 Probabilistic Assignment Matrix Relaxations

The core idea is to substitute indicator variables in $\{0,1\}^n$ (for k-way cuts, assignment matrices in $\{0,1\}^{n\times k}$) by a soft probabilistic matrix $P \in [0,1]^{n\times k}$, where each row sums to one. Thus, each node is treated as belonging to each cluster with a certain probability, and the expected cut cost can be explicitly computed: $\E\bigl[\mathrm{Cut}(S_\ell,\bar S_\ell)\bigr] = \sum_{i,j} A_{ij} p_{i\ell}(1-p_{j\ell}),$ where $p_{i\ell}$ is the assignment probability of node $i$ to cluster $\ell$.

A crucial challenge arises when partition objectives involve denominators (volumes): the relaxation requires evaluating $\E[1/\sum s_i s_{i\ell}]$ where $s_{i\ell} \sim \mathrm{Bern}(p_{i\ell})$. Integral representations and tight analytic surrogates for expectations of reciprocals have therefore become key (Ghriss, 4 Nov 2025).

2.2 Quantum Relaxations via Local Hamiltonians

Quantum relaxations (Fuller et al., 2021) re-encode the cut objective as the ground-state energy of a local 2-local Hamiltonian $H$ constructed via a quantum random access code (QRAC) embedding. For Max-Cut on a graph $G=(V,E)$: $$H = \sum_{(i,j)\in E} \frac{1}{2}\left( I - dP_iP_j \right),$$ with $d$ determined by the QRAC variant, and $P_i$ denoting single-qubit Pauli operators assigned so adjacent vertices have distinct Paulis. An encoding map $F$ sends classical assignments to quantum density matrices. Exact commutation is achieved: $\operatorname{Tr}[H F(m)] = \mathrm{cut}(m).$ Memory compression emerges by encoding multiple classical bits per qubit (QRAC), with graph colorings dictating the block structure.

3. Analytic Surrogates and Integral Representations

A central technical advance in probabilistic frameworks is the construction of analytic upper bounds for the expected normalized cut objectives. Specifically, for random variables of the form $X=\sum_{i=1}^m\beta_ir_i$ with $r_i\sim\mathrm{Bern}(\alpha_i)$, one requires tight control of $\E[1/(q+X)]$.

The following integral identity is employed: $\frac{1}{x} = \int_0^1 t^{x-1} dt,$ which, when linearly extended to random $X$, yields: $\E\left[\frac{1}{q+X}\right] = \int_0^1 t^{q-1} \prod_{i=1}^m (1-\alpha_i+\alpha_i t^{\beta_i}) dt.$ For homogeneous exponents ($\beta_i \equiv \beta$), Jensen's inequality and logarithmic concavity enable further simplification to a closed-form via truncated Gauss hypergeometric polynomials: $$(q;\alpha,\beta) \leq \frac{1}{q} {}_2F_1(-m,1;c;\bar \alpha),$$ where $c=q/\beta+1$, and $\bar \alpha$ is the mean success probability.

For heterogeneous degree distributions, grouping into bins and applying Hӧlder's inequality produces a product envelope over bins. The gap between the upper bound and the true expectation can be explicitly bounded in terms of variance and a "zero-aware" penalty that vanishes when assignment probabilities are zero.

4. Practical Algorithms and Differentiable Frameworks

Probabilistic relaxations admit full differentiability, allowing coupling to optimization of deep pipelines:

1. Graph construction and embeddings: Nodes or input data are mapped to latent representations; similarities are computed to form an affinity matrix (for example, via RBF kernels or inner products).
2. Soft assignment parameterization: Cluster memberships are represented by the assignment probability matrix $P$, for instance via a softmax/sigmoid over unconstrained logits.
3. Surrogate computation: Surrogates for the expected partition cost are computed analytically, using the above envelope formulas for denominators and AM–GM gap penalties.
4. Gradient-based learning: Because all elements reduce to explicit polynomials, gradients (including derivatives of hypergeometric blocks) can be efficiently evaluated and backpropagated, supporting large-batch/minibatch and online learning contexts.
5. Annealing and sharpening of assignments: As training progresses, temperature parameters in the softmax may be lowered to drive assignments closer to binary, thereby tightening surrogates to the underlying discrete objectives.

Numerical stability is maintained by Kahan summation and early exit conditions in the evaluation of the polynomial terms. Spectral decompositions are not required at any stage, leading to significant computational savings compared to classical spectral clustering (Ghriss, 4 Nov 2025).

In the quantum relaxation setting, classical cut solutions are extracted from quantum states via either randomized magic-state rounding or deterministic Pauli measurement. The trade-off centers on statistical guarantees versus observed cut values, with the randomized protocol admitting a provable worst-case ratio, and deterministic rounding often delivering superior empirical outcomes.

5. Unified Frameworks and Connections to Classical and Quantum Methods

Probabilistic relaxations unify diverse graph partitioning objectives—Min-Cut, Max-Cut, RatioCut, Normalized Cut, and k-way generalizations—within a single formalism. Structural matrices (e.g., block design for bipartitions or multipartitions) can be imposed to encode arbitrary block-structures, accommodating both homophilous and heterophilous graphs.

By contrast to spectral relaxations, which operate via eigendecomposition of Laplacians and optimize over vector embeddings, probabilistic and quantum relaxations admit direct and monotonic parameterization over assignment matrices, avoid spectral gaps or ambiguity in hard thresholding, and are amenable to composition with modern deep learning encoders (Ghriss, 4 Nov 2025, Chanpuriya et al., 2023).

Quantum relaxations exploit the structure of QRACs for genuine memory compression; for 3-regular graphs, block size and graph coloring yield compression rates of approximately 2.6–2.7, mapping classical assignments to a smaller Hilbert space while retaining commutation with the combinatorial objective (Fuller et al., 2021).

6. Performance Bounds, Empirical Results, and Limitations

6.1 Quantum Relaxation Results

For Max-Cut and weighted Max-Cut problems encoded as local quantum Hamiltonians:

• The randomized (magic-state) rounding achieves a worst-case ratio $\gamma \geq 5/9 \approx 0.555$, enhanced to $0.625$ with two-bit-per-qubit QRAC.
• Empirical results on 3-regular random graphs of size up to 40 nodes demonstrate that magic-state rounding almost always exceeds the theoretical lower bound, while deterministic Pauli rounding frequently achieves near-unity approximations.
• In practical hardware runs on superconducting quantum processors, cuts of size $\gamma_{\mathrm{hw}} \approx 0.905$ and weighted cut ratios comparable with the combinatorial optimum have been achieved even for graphs up to 40 nodes.

6.2 Probabilistic Framework Performance

In the probabilistic surrogate setting (Ghriss, 4 Nov 2025):

• Tight analytic upper bounds on expected cuts are constructed, controlling the surrogate–true objective gap and enabling explicit scheduling (via coefficient $\rho$) of penalty terms.
• Empirical experiments demonstrate parity with or improvement over spectral clustering methods, especially for large or online settings where eigendecompositions are infeasible.
• The absence of hard theoretical ratios (in the sense of Goemans–Williamson bounds) is offset by flexibility, exact differentiability, minibatch scalability, and the capacity for explicit control over assignment dispersion and zero-awareness.

Limitations include the gap between the surrogate upper bound and the true expected objective, which must be monitored via variance-based penalties; in the quantum domain, the compression comes with the cost of working in a non-diagonal basis and potential overhead in physical implementation.

7. Extensions, Outlook, and Open Problems

The probabilistic relaxation paradigm extends directly to arbitrary block-model graphs, multiway cut objectives, and contrastive/self-supervised learning regimes where assignment probabilities serve both clustering and discriminative purposes. For instance, SimCLR or CLIP objectives emerge as surrogates for instance discrimination or bipartite cuts under specializations of the framework.

Potential refinements include searching for underlying graph families with large spectral gaps (in quantum relaxations), development of higher-rate QRACs to further compress classical information, and the synthesis of hybrid models that optimize the compression–approximation trade-off.

Advancements in quantum state preparation (e.g., QAOA, adiabatic evolution, VQE enhancements, phase estimation) may further close the gap with classical SDP relaxations, increasing the practicality of quantum approaches.

No evidence currently supports universally superior performance (in approximation ratio) for quantum or probabilistic relaxations relative to classical SDP methods (e.g., Goemans–Williamson achieves $\approx 0.878$), but empirical and architectural advantages in terms of scalability, differentiability, and memory compression are substantial. The frameworks accommodate ongoing trends toward differentiable, end-to-end graph partitioning in modern machine learning systems.

Relaxation Type	Theoretical Ratio (for Max-Cut)	Scalability	Differentiability	Compression
Classical SDP	0.878 (Goemans–Williamson)	Moderate	No	No
Quantum QRAC+Magic	0.555 or 0.625	Moderate	N/A (quantum)	Yes
Probabilistic (PGC)	Tight envelope, no universal	High	Yes	Yes

This convergence of probabilistic, spectral, and quantum relaxations represents a versatile toolkit for principled, scalable graph partitioning extending across combinatorial optimization, quantum computing, and large-scale unsupervised learning.