Analysis of Multitasking Pareto Optimization for Monotone Submodular Problems

Evolutionary multitasking is a recent approach that solves multiple related optimization problems within a single evolutionary run, rather than addressing each problem separately. We consider monotone submodular optimization problems with dynamic knapsack constraints and study a multitasking formulation in which all tasks share a common monotone submodular function $f$, but differ in their constraints. We focus on the case where elements within each constraint have uniform cost and show that this structure leads to small Pareto fronts in the multitasking formulation. This enables solution sharing across tasks and can improve performance compared to running standard evolutionary approaches independently, depending on the constraint regime. Using rigorous runtime analysis, we analyze the expected time until the proposed multitasking algorithms obtain a $(1 - 1/e)$-approximation for each task. Experimental results for the Maximum Coverage problem complement the theoretical analysis and provide further insight into the practical behavior of the approach across different budget settings.

Multi-task optimization is a powerful approach for solving a large number of tasks in parallel. However, existing algorithms face distinct limitations: Population-based methods scale poorly and remain underexplored for large task sets. Approaches that do scale beyond a thousand tasks are mostly MAP-Elites variants and rely on a fixed, discretized archive that disregards the topology of the task space. We introduce MONET (Multi-Task Optimization over Networks of Tasks), a multi-task optimization algorithm that models the task space as a graph: tasks are nodes, and edges connect tasks in the task parameter space. This representation enables knowledge transfer between tasks and remains tractable for high-dimensional problems while exploiting the topology of the task space. MONET combines social learning, which generates candidates from neighboring nodes via crossover, with individual learning, which refines a node's own solution independently via mutation. We evaluate MONET on four domains (archery, arm, and cartpole with 5,000 tasks each; hexapod with 2,000 tasks) and show that it matches or exceeds the performance of existing MAP-Elites-based baselines across all four domains.

Multi-party multi-objective optimization problems (MPMOPs) require consensus among autonomous decision makers and therefore differ from flattened many-objective formulations. Existing runtime theory for multi-objective evolutionary algorithms is largely tailored to single-party Pareto-front approximation and does not directly explain common-solution search in MPMOPs. We investigate cross-party recombination in two representative settings. On MP-JCG, a pseudo-Boolean benchmark with an explicit gap region, we prove that a payoff-guided mutation baseline faces a gap-crossing bottleneck requiring $Θ(n^2)$ expected fitness evaluations. In contrast, an analytical CPR-NSGA-II variant discovers both common Pareto-optimal solutions in $O(n\log n)$ expected evaluations by directly assembling complementary prefix and suffix templates distributed across party populations. Comparing this with the flattened four-objective formulation F-JCG, our full-front coverage analysis illustrates the additional coverage burden introduced by flattening. For BPBOMST, the bi-party, two-objective-per-party specialization of the multi-party multi-objective minimum spanning tree problem, we develop a layered support-cover analysis. For each common Pareto objective vector, the symmetric average projection induces an auxiliary bi-objective MST instance, and suitable support representatives yield a $2λ$-common approximation cover with $λ\in[1,2]$. We further derive an instance-parameterized expected runtime bound for a representative-pool CPR-NSGA-II variant using edge-union recombination and uniform repair. This bound separates the effects of local auxiliary-front filling, cross-party recombination shortcuts, and edge-union repair ambiguity.