Graph Coloring with Heat Diffusion Model

• Heat Diffusion Model (HDM) is a continuous optimization method that relaxes graph coloring into a smooth, differentiable framework.
• It employs softmax-based relaxation and iterative projected updates to transform continuous score matrices into discrete vertex colorings.
• Empirical results show HDM improves clash rates over greedy methods while remaining competitive with classical heuristics like TabuCol.

In the context of graph coloring, the Heat Diffusion Model (HDM) is a formulation that casts coloring as a continuous optimization problem and solves it with a heat-diffusion-based gradient iterative method. In “Graph Coloring Using Heat Diffusion” (Chaudhary, 2024), an undirected graph is represented by an adjacency matrix, vertex colors are relaxed into continuous score vectors, clashes between adjacent vertices are penalized through a smooth objective, and the final discrete coloring is recovered by row-wise $\arg\max$ decoding. The reported results place HDM between a greedy baseline and TabuCol on benchmark graphs with known chromatic number, indicating competitiveness without claiming superiority over strong classical heuristics (Chaudhary, 2024).

1. Problem setting and graph-coloring objective

The paper considers an undirected graph

$G=(V,E), \qquad V=\{1,2,\dots,n\},$

and seeks a coloring function

$c:V\to \{1,2,\dots,q\}$

such that adjacent vertices do not share the same color:

$c(i)\neq c(j), \qquad \forall (i,j)\in E.$

The goal is to use at most $q$ colors, where the smallest feasible $q$ is the chromatic number. A coloring is valid if there are no clashes on edges (Chaudhary, 2024).

This formulation places HDM within the standard graph-coloring regime used in scheduling, resource allocation, and circuit design. In the experimental setting, the number of colors is not searched over adaptively; instead, the benchmark evaluation sets $k$ equal to the known chromatic number of each graph. That choice isolates the quality of the solver under a fixed color budget, but it also simplifies the problem relative to settings in which $k$ must be inferred.

2. Heat diffusion as a gradient-based solver framework

The method builds on the heat diffusion framework from Ma et al., described in the paper as a gradient-based iterative solver for combinatorial optimization. Rather than searching directly over a discrete combinatorial space, the framework embeds the problem into a continuous relaxation, defines a smooth objective, iteratively updates variables using gradients, projects the variables back into a feasible interval, and decodes the continuous solution into a discrete assignment (Chaudhary, 2024).

At time $t$ and location $\theta$, the temperature distribution is denoted by $G=(V,E), \qquad V=\{1,2,\dots,n\},$0, and the gradient of the temperature is approximated by

$G=(V,E), \qquad V=\{1,2,\dots,n\},$1

where $G=(V,E), \qquad V=\{1,2,\dots,n\},$2 is the target function to be minimized, $G=(V,E), \qquad V=\{1,2,\dots,n\},$3 are random samples, $G=(V,E), \qquad V=\{1,2,\dots,n\},$4 is the number of samples, and $G=(V,E), \qquad V=\{1,2,\dots,n\},$5 is the error function. The paper notes that $G=(V,E), \qquad V=\{1,2,\dots,n\},$6 in its setup. The role of the error function is to provide a smooth transform from continuous variables toward binary or discrete-like decisions.

The iterative procedure described in the paper is to sample $G=(V,E), \qquad V=\{1,2,\dots,n\},$7 from a uniform distribution over $G=(V,E), \qquad V=\{1,2,\dots,n\},$8, compute the gradient using the diffusion estimator, update $G=(V,E), \qquad V=\{1,2,\dots,n\},$9, project $c:V\to \{1,2,\dots,q\}$0 back into the valid interval $c:V\to \{1,2,\dots,q\}$1, and repeat for $c:V\to \{1,2,\dots,q\}$2 iterations. A generic projected update can be written as

$c:V\to \{1,2,\dots,q\}$3

where $c:V\to \{1,2,\dots,q\}$4 is projection onto the box constraint and $c:V\to \{1,2,\dots,q\}$5 is a step size. The “heat” terminology refers to the framework’s internal optimization picture: each parameter is treated as a location with an associated temperature, and diffusion through the problem landscape guides the solver toward an optimum.

3. HDM formulation for graph coloring

For graph coloring, the decision variable is a score matrix

$c:V\to \{1,2,\dots,q\}$6

where $c:V\to \{1,2,\dots,q\}$7 is the number of vertices and $c:V\to \{1,2,\dots,q\}$8 is the number of colors. Each row $c:V\to \{1,2,\dots,q\}$9 represents the color-assignment scores for vertex $c(i)\neq c(j), \qquad \forall (i,j)\in E.$0 (Chaudhary, 2024).

The paper uses a softmax-based relaxation. During optimization, each row is interpreted through a smooth, probability-like color distribution, and after convergence the coloring is decoded by taking the row-wise $c(i)\neq c(j), \qquad \forall (i,j)\in E.$1. In effect, each vertex is assigned the color corresponding to the largest entry in its score vector.

The target function is defined using the adjacency matrix $c(i)\neq c(j), \qquad \forall (i,j)\in E.$2. The paper gives the graph-coloring target function in compressed form as

$c(i)\neq c(j), \qquad \forall (i,j)\in E.$3

with the stated intent of using the adjacency structure together with softmax-relaxed assignments to penalize adjacent vertices that are assigned the same color. The exact expression is presented only briefly, but the functional role is clear: the objective measures clashes among neighboring vertices under soft assignments, and HDM minimizes this clash-oriented surrogate (Chaudhary, 2024).

The constraint handling is implicit. Variables are optimized in the continuous box $c(i)\neq c(j), \qquad \forall (i,j)\in E.$4, the softmax supplies a differentiable relaxation of color selection, and the final discrete assignment is obtained only at the decoding stage. This places the method squarely in the standard continuous-to-discrete paradigm: optimization is continuous, feasibility is approximated during training, and combinatorial structure is recovered at the end.

4. Optimization pipeline and decoding

The algorithmic workflow in the paper is concise but explicit. Given a graph $c(i)\neq c(j), \qquad \forall (i,j)\in E.$5 and adjacency matrix $c(i)\neq c(j), \qquad \forall (i,j)\in E.$6, the procedure sets $c(i)\neq c(j), \qquad \forall (i,j)\in E.$7 equal to the known chromatic number for benchmark evaluation, initializes $c(i)\neq c(j), \qquad \forall (i,j)\in E.$8 by uniform sampling, constructs a differentiable loss based on adjacency and softmax-based color assignment, and then applies the heat-diffusion gradient estimator inside an iterative projected update loop (Chaudhary, 2024).

The gradient step uses

$c(i)\neq c(j), \qquad \forall (i,j)\in E.$9

followed by projection back to $q$0. After $q$1 iterations, each vertex is assigned a color by row-wise $q$2, and the output is evaluated by counting clashing edges.

The evaluation metric is the percentage of edges that are clashing:

$q$3

For the greedy baseline, an edge is also considered clashing if either endpoint is assigned a dummy color. The use of clash rate rather than exact feasibility is consequential: the experiments assess how close the solver comes to a valid $q$4-coloring under a fixed budget, not whether it certifiably produces an optimal coloring in every instance.

5. Baselines and empirical performance

The empirical study uses 33 graphs from Leighton, all with known chromatic number. The paper compares HDM against two baselines: a largest-first greedy coloring method and TabuCol, a Tabu search heuristic known to perform well on coloring benchmarks (Chaudhary, 2024).

The greedy method processes vertices with a largest-first strategy, assigns colors greedily, and introduces a dummy color when the chromatic limit $q$5 is reached; dummy-colored vertices are then counted as clashes for evaluation. TabuCol serves as the stronger classical baseline.

Method	Characterization	Mean clash percentage
Greedy	Largest-first; dummy color if $q$6 is exhausted	$q$7
Heat Diffusion	Continuous relaxation with projected diffusion-based updates	$q$8
TabuCol	Tabu search heuristic	$q$9

The paper reports a consistent ranking: TabuCol performs best, Heat Diffusion is second best, and Greedy performs worst (Chaudhary, 2024). On this benchmark, HDM is therefore competitive in the narrow sense that it substantially improves on greedy coloring and remains reasonably close to TabuCol in mean clash percentage, but it does not surpass the strongest baseline.

This comparative outcome is important for situating HDM methodologically. The result does not present HDM as a replacement for high-performing combinatorial heuristics; instead, it shows that a diffusion-based continuous solver can operate credibly on a classical discrete problem and achieve nontrivial performance under a fixed chromatic budget.

6. Interpretation, scope, and limitations

Several points delimit the scope of the model. First, the paper does not claim exact optimal coloring; evaluation is based on clash rate rather than proof of feasibility or chromatic optimality (Chaudhary, 2024). A common misunderstanding would be to read HDM as an exact $q$0-coloring method whenever $q$1 equals the chromatic number. The paper does not make that claim.

Second, the experiments assume the known chromatic number of each benchmark graph. This improves comparability across methods, but it removes the need to solve the outer problem of determining the smallest feasible color count. A plausible implication is that the reported performance isolates the optimization quality of the continuous solver while leaving color-budget selection outside the experimental scope.

Third, the exact target function is described only briefly and somewhat tersely, so implementation details are not fully spelled out in the text. That limitation matters because the effectiveness of diffusion-based optimization depends strongly on the smooth surrogate, its gradient properties, and the interaction between softmax relaxation and final decoding.

Finally, HDM here is not a direct discrete local-search procedure. Its essential mechanism is continuous relaxation over an $q$2 score matrix, followed by projection and discrete readout. This distinguishes it from purely combinatorial methods such as greedy coloring and from neighborhood-search heuristics such as TabuCol. The paper’s principal contribution is therefore not a new coloring theorem, but a specific encoding of graph coloring inside a heat-diffusion optimization framework and an empirical demonstration that this encoding is viable on benchmark graphs (Chaudhary, 2024).