Candidate Crossover Algorithm

• Candidate crossover algorithms are a specialized class of evolutionary recombination techniques that prioritize complementarity and diversity in parent selection.
• They employ explicit co-fitness metrics, like Novel-2 and Novel-N, to ensure offspring inherit superior, non-overlapping subcomponents in complex optimization tasks.
• Hybrid strategies, especially Hybrid-2, effectively blend rank-based and candidate-based selection to balance exploration with the preservation of high-performing genetic material.

Candidate crossover algorithms constitute a specialized class of crossover strategies in evolutionary computation tailored to improve parental selection, offspring diversity, and the preservation of useful genetic material. Rather than pairing parents randomly or by fitness alone, candidate crossover methods employ explicit measures of parent diversity, complementarity, or solution coverage. This enhances the probability that offspring inherit diverse or collectively superior building blocks. The approach is especially pertinent in domains where solution fitness is a composite of subcomponents, as in decision tree evolution, feature selection, and multi-objective optimization.

1. Formal Algorithmic Foundations

Candidate crossover algorithms diverge from traditional crossover primarily in the parental pairing stage, employing a population-level analysis to quantify the complementarity or diversity among individuals. A representative formalization is as follows (Świechowski, 2021):

Let $P$ be the population of size $N$. For each unordered pair of individuals $(A,B)$:

• Compute a “co-fitness” or complementarity metric, $C[A,B]$, reflecting properties such as joint sample coverage or disjoint specialization.
• Sort all $(N^2-N)/2$ pairs descending by $C[A,B]$.
• Iteratively select the top-scoring pairs for crossover, ensuring that parental selection prioritizes complementarity rather than mere fitness rank.
• Offspring are then generated by a standard recombination operator (e.g., subtree crossover, uniform crossover).

The algorithm often provides hybrid variants that combine candidate-based and fitness-based parent selection to prevent loss of high-performing genetic material.

Exact Complementary-Fitness Metrics

Two co-fitness measures exemplify the approach:

• Novel-2: For each tree, split into left and right subtrees; compute per-part accuracy, and let

$$\mathrm{co\!-\!Fitness}_{AB} = \max(\mathrm{acc}(A.\mathrm{LEFT}), \mathrm{acc}(B.\mathrm{LEFT})) + \max(\mathrm{acc}(A.\mathrm{RIGHT}), \mathrm{acc}(B.\mathrm{RIGHT}))$$

• Novel-N: For a training set of size $TS$, count the samples correctly predicted by either parent:

$$\mathrm{co\!-\!Fitness}_{AB} = \sum_{k=1}^{TS} \bigl(\mathrm{correct}(Y^A_k, Y_k) \lor \mathrm{correct}(Y^B_k, Y_k)\bigr)$$

These metrics select for pairs whose union covers more of the search or data space than either would alone.

2. Computational Complexity and Implementation

The candidate crossover approach incurs a computational cost scaling as $O(N^2 \cdot TS)$ for Novel-N, or $O(N^2)$ when co-fitness calculation is independent of training set size. Sorting these pairs adds $O(N^2 \log N)$ overhead. The cost remains feasible for moderate $N$ ($\sim 100$–$400$), especially when $TS$ is not large.

A condensed pseudocode outline (Novel-N variant) (Świechowski, 2021):

K = floor(CR * N) for all unordered pairs (A,B): compute C[A,B] used, pairs = set(), [] for each (A,B) in sorted pairs: if A not in used or B not in used: pairs.append((A,B)) used.update({A,B}) if len(used) >= K: break return pairs

Hybrid schemes allocate $K/2$ parents via rank-based selection and $K/2$ via co-fitness maximization, randomly pairing within these selections for crossover.

3. Empirical Results and Comparative Analysis

Comprehensive experimental results on evolutionary decision tree construction demonstrate that the Hybrid-2 candidate crossover (50% rank-based, 50% complementarity-based parent selection) consistently outperforms both pure candidate-crossover variants and pure rank-based selection (Świechowski, 2021). The empirical findings across eight scenarios (varying population size, tree depth, test set size, and crossover rate) show:

• Hybrid-2 is superior to standard rank-based crossover in all cases, with statistically significant improvement in six out of eight conditions.
• Pure complementarity-based pairings (Novel-2, Novel-N) result in poorer performance or loss of diversity, indicating overexploitation of complementarity at the expense of global exploration.
• Hybrid-N did not deliver consistent gains and underperformed in certain high-crossover or large-test scenarios.

A summary table illustrates the comparative superiority of Hybrid-2, especially in early convergence and sustained improvements over generations.

4. Theoretical Rationale and Design Principles

Candidate crossover leverages the observation that fitness in many domains is aggregative or decomposable. Offspring resulting from “complementary” parents may inherit attributes covering disjoint or weakly overlapping facets of the objective. This is particularly advantageous in problems exhibiting:

• Submodular fitness, multitask, or ensemble structures (e.g., multi-view feature selection, regression/classification on heterogeneous datasets).
• Nonconvex fitness landscapes with diverse local optima.

However, exclusive reliance on complementarity can reduce genetic diversity because overemphasizing “coverage” may prioritize outliers or overly specialized individuals. Hybrid selection strategies mitigate this effect by insuring that high-fitness individuals are not neglected.

5. Broader Applicability and Generalizations

Candidate crossover algorithms generalize to any representation or optimization domain in which (a) partial or compositional solution quality is meaningful, and (b) co-fitness or coverage metrics can be computed efficiently. Recommended guidelines for deploying the approach (Świechowski, 2021):

• Identify domains where solutions decompose naturally, enabling a meaningful partial-fitness or “coverage” vector.
• Define a co-fitness metric that measures the union (rather than intersection) of parent strengths.
• Balance complementarity-based parental choice with rank-proportional selection to avoid premature convergence or loss of elite genotypes.
• Control quadratic evaluation costs via subsampling, restricting co-fitness computation to a top-M pool, or employing approximate neighbor searches in fitness-space.

Table: Candidate Crossover Variants

Variant	Parent selection	Empirical performance
Standard	Top-K by fitness, random pairing	Baseline
Novel-2	Max complementarity (left/right accuracy)	Worse than baseline
Novel-N	Max coverage per training sample	Worse than baseline
Hybrid-2	50% rank + 50% Novel-2, random pairing	Best, significant
Hybrid-N	50% rank + 50% Novel-N, random pairing	Inconclusive

6. Practical Guidelines and Recommendations

For effective usage of candidate crossover approaches:

1. Score parental complementarity, but avoid pure coverage maximization: Use metrics like partial-fitness union, but blend with fitness-based selection.
2. Hybrid weighting: A 50:50 blend of rank-based and candidate-complementarity selection is empirically effective, though tuning may be problem-dependent.
3. Scalability management: Restrict co-fitness computation by limiting candidate pool or leveraging approximate methods as $N$ or $TS$ grows.
4. Generalization: Candidate crossover can be applied wherever diverse subcomponents, tasks, or prediction targets exist within the broader optimization problem.

7. Position Within Evolutionary Algorithm Research

Candidate crossover algorithms represent a principled approach to recombination that exploits problem decomposition and solution diversity beyond the capabilities of random or strictly fitness-based pairing. By formalizing complementarity and integrating it into parental selection, these algorithms enhance the probability of constructing high-quality, robust, and diverse offspring (Świechowski, 2021). However, empirical results make clear that solely focusing on complementarity can reduce population diversity and degrade performance; balanced, hybrid strategies remain the practical choice for most applications.

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References

• “A Crossover That Matches Diverse Parents Together in Evolutionary Algorithms” (Świechowski, 2021).