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In the field of security, multi-objective security games (MOSGs) allowdefenders to simultaneously protect targets from multiple heterogeneousattackers. MOSGs aim to simultaneously maximize all the heterogeneous payoffs,e.g., life, money, and crime rate, without merging heterogeneous attackers. Inreal-world scenarios, the number of heterogeneous attackers and targets to beprotected may exceed the capability of most existing state-of-the-art methods,i.e., MOSGs are limited by the issue of scalability. To this end, this paperproposes a general framework called SDES based on many-objective evolutionarysearch to scale up MOSGs to large-scale targets and heterogeneous attackers.SDES consists of four consecutive key components, i.e., discretization,optimization, evaluation, and refinement. Specifically, SDES first discretizesthe originally high-dimensional continuous solution space to thelow-dimensional discrete one by the maximal indifference property in gametheory. This property helps evolutionary algorithms (EAs) bypass thehigh-dimensional step function and ensure a well-convergent Pareto front. Then,a many-objective EA is used for optimization in the low-dimensional discretesolution space to obtain a well-spaced Pareto front. To evaluate solutions,SDES restores solutions back to the original space via greedily optimizing anovel divergence measurement. Finally, the refinement in SDES boosts theoptimization performance with acceptable cost. Theoretically, we prove theoptimization consistency and convergence of SDES. Experiment results show thatSDES is the first linear-time MOSG algorithm for both large-scale attackers andtargets. SDES is able to solve up to 20 attackers and 100 targets MOSGproblems, while the state-of-the-art (SOTA) methods can only solve up to 8attackers and 25 targets ones. Ablation study verifies the necessity of allcomponents in SDES.

Scaling Multi-Objective Security Games Provably via Space Discretization Based Evolutionary Search