Diversity Maximization Algorithm

• Diversity Maximization Algorithm is an evolutionary framework that generates highly diverse maximum matchings by maximizing aggregated Hamming distances.
• It employs evolutionary strategies like (μ+1)-EA₍D₎ and a two-phase 2P-EA₍D₎, using per-bit and vertex-level mutations to effectively balance solution validity with diversity.
• Empirical and theoretical analyses demonstrate its efficiency in reducing runtime bounds, with potential extensions to other combinatorial optimization problems.

A diversity maximization algorithm, in the context of combinatorial optimization, refers to any algorithmic framework that aims to produce a collection of solutions—typically subsets, structures, or assignments—such that the aggregate diversity within the collection, measured according to some well-defined objective (often based on pairwise distances, dissimilarity functions, or combinatorial separation), is as large as possible. This survey focuses on the rigorous study of diversity maximization for the maximum matching problem using evolutionary algorithms, as introduced by “Analysis of Evolutionary Diversity Optimisation for the Maximum Matching Problem” (Harder et al., 2024). The discussion encompasses the underlying framework, algorithmic mechanisms, runtime analysis, empirical findings, and broader implications.

1. Diversity Maximization Framework for Maximum Matching

For a graph $G = (V,E)$, let $|E|=m$ and fix a population size $\mu$. Each maximum matching $M \subseteq E$ is encoded as an $m$-bit string $x \in \{0,1\}^m$, with $x_i = 1$ if and only if edge $i\in M$. A solution $x$ is valid if no two edges selected share a vertex (i.e., $M$ is a matching), and is maximal if $|M|$ equals the size of a maximum matching for $G$.

The fitness function for a candidate $x$ is given by: $f(x)= \begin{cases} -\mathrm{col}(x) & \text{if %%%%13%%%% is invalid}\ |x| & \text{if %%%%14%%%% is a valid matching} \end{cases}$ where $\mathrm{col}(x)$ counts the number of edge–collision pairs in $x$. Only maximum matchings with $|x| = \nu(G)$ (the maximum matching number) are promoted.

The diversity measure is the total pairwise Hamming distance over the (distinct) population: $$D(P) = \sum_{(x,y)\in\tilde{P}\times\tilde{P}} H(x,y)$$ where $\tilde{P}$ is the set of distinct solutions in population $P$, and $H(x,y)$ is the Hamming distance. The contribution of a solution $x$ is defined as $c(x) = D(P) - D(P\setminus\{x\})$.

The optimization goal is to maximize $D(P)$ over all $\mu$-sized populations consisting of (distinct) maximum matchings.

2. Algorithmic Approaches: $(\mu+1)$-EA\textsubscript{D} and 2P-EA\textsubscript{D}

Two evolutionary algorithms are analyzed: a baseline $(\mu+1)$-EA\textsubscript{D} and a more structured two-phase $2P$-EA\textsubscript{D}. Both operate under a $(\mu+1)$, replacement-style selection scheme, driving the population $P$ towards increasing diversity $D(P)$ while maintaining validity and maximality.

(μ+1)-EA\textsubscript{D}

• Initialization: set $P$ as a population of $\mu$ random maximum matchings.
• Iteration:

1. Select $s \in P$ uniformly at random (u.a.r.).
2. Offspring $s'$ is generated by flipping each bit of $s$ independently with probability $1/m$ (standard bit mutation).
3. If $s'$ is a valid maximum matching, add it to $P$.
4. Remove $z \in \operatorname{argmin}_{x\in P} c(x)$ u.a.r. to maintain size $\mu$.

2P-EA\textsubscript{D}

• Initialization: as before.
• Iteration:

1. Select $s \in P$ u.a.r.; let $s' \gets s$.
2. Unmatching phase: For each $v\in V$, independently with probability $1/|V|$, clear all edges incident to $v$ in $s'$.
3. Rematching phase: For each $v$ selected above, if $v$ has unmatched neighbors, choose one u.a.r. and add the corresponding edge.
4. If $s'$ is a valid maximum matching, add $s'$ to $P$, remove $z \in \operatorname{argmin} c(x)$ u.a.r.

The two-phase mutation respects matching structure more closely, allowing efficient local changes toward higher diversity.

Pseudocode Table

Algorithm	Mutation Mechanism	Diversity Update
(μ+1)-EA\textsubscript{D}	Per-bit flip, $p=1/m$	Replace lowest-contribution member
2P-EA\textsubscript{D}	Unmatch/rematch per vertex, $p=1/|V|$	Replace lowest-contribution member

3. Proven Runtime Bounds for Complete Bipartite Graphs and Paths

Let $n=|V|$, $m=|E|$, $\mu$ as above. For $G$ as a complete bipartite graph $K_{L,R}$ ($|R|\le |L|$) or a path of $m$ edges:

Complete Bipartite Graphs

• Big-gap case ($\mu < |L| - |R|$):

| Algorithm | Expected Runtime | |-------------|---------------------------------------| | (μ+1)-EA\textsubscript{D} | $O(\mu^2 m^2 \log m)$ | | 2P-EA\textsubscript{D} | $O(\mu^2 n^2 \log n)$ |

• Small-gap case ($|L| - |R| \le \mu < |R|/2$):

| Algorithm | Expected Runtime | |-------------|---------------------------------------| | (μ+1)-EA\textsubscript{D} | $O(\mu^2 m^4 \log m)$ | | 2P-EA\textsubscript{D} | $O(\mu^2 m^2 \log m)$ |

The gap parameter determines the required “move type”: simple edge swaps in the big gap, more complex edge exchanges (4-bit flips) in the small gap.

Path Graphs

• | Algorithm | Expected Runtime | |-------------|---------------------------------------| | (μ+1)-EA\textsubscript{D} | $O(\mu^3 m^3)$ | | 2P-EA\textsubscript{D} | $O(\mu^3 m^2)$ |

The expected runtime bounds follow from a careful drift analysis, estimating the expected increase in $D(P)$ per step and then applying the additive/multiplicative drift theorems to upper bound the time to reach $D_\mathrm{opt}$. The sharper exponents for 2P-EA\textsubscript{D} reflect increased efficiency due to structure-aware mutations.

4. Empirical Observations and Scaling

Empirical studies confirmed the predicted polynomial scaling with $m$ and $\mu$, but observed substantial practical improvement over the theoretical upper bounds. For moderate values (e.g., $\mu=8$):

• Complete bipartite:
  • (μ+1)-EA\textsubscript{D}: Observed iterations scale as $\mu m^2$ in big gap, and $\mu m^{2.5}$ in small gap—better than the theoretical bounds by factors of $\sim\mu^{-1}/\log m$.
  • 2P-EA\textsubscript{D}: Both regimes scale as $\mu m$, much lower than $O(\mu^2 m^2 \log m)$.
• Paths:
  • (μ+1)-EA\textsubscript{D}: Empirically $\mu m^3$.
  • 2P-EA\textsubscript{D}: Empirically $\mu m^2$.

These improvements are attributed to conservative worst-case drift estimates that overestimate required steps; on typical instances, larger diversity gains per iteration are often realized.

5. Extensions and Generalizations

Several extensions follow from this foundational analysis:

• Tighter Drift Analysis: Conditioning on actual edge-sharing multiplicities or population statistics can further reduce runtime exponents.
• Alternative Diversity Metrics: One could replace total Hamming distance with other diversity objectives (e.g., entropy, discrepancy) and adapt the drift arguments accordingly.
• Other Combinatorial Structures: The binary-encoding, diversity-maximizing EA paradigm directly generalizes to TSP tours, spanning trees, vertex covers, knapsack configurations, and more. Two-phase mutation can be recast in terms of unfixed/refixed local structures in these problems.
• Algorithmic Practicalities: The 2P-EA\textsubscript{D} approach—mutating at the structure level (vertex, item, etc.) rather than at the bit level—repeatedly reduces the empirical runtime and increases the magnitude of diversity progress per step.

6. Interactions with Broader Research and Applications

This work situates the evolutionary diversity maximization paradigm firmly within the emerging area of evolutionary diversity optimization (EDO), addressing both theoretical run-time complexity and practical realization for the maximum matching problem. The drift-based runtime analysis establishes that maximal diversity of feasible combinatorial objects can be obtained in polynomial time (in both the matching case and generalized structurally similar problems), provided one leverages structurally informed mutation operators.

The results clarify that careful mutation strategy—local structure-aware design rather than naive per-bit flipping—yields both theoretical and practical performance gains for diversity-optimization goals. The methodology is compatible with other diversity objectives and combinatorial search models and provides a principled algorithmic foundation for ensemble selection, evolutionary sampling, and population-based optimization in the presence of diversity-critical requirements.