The Pareto sum of two-dimensional point sets P and Q in ℝ² is defined as the skyline of the points in their Minkowski sum. The problem of efficiently computing the Pareto sum arises frequently in bi-criteria optimization algorithms. Prior work establishes that computing the Pareto sum of sets P and Q of size n suffers from conditional lower bounds that rule out strongly subquadratic O(n^{2-ε})-time algorithms, even when the output size is Θ(n). Naturally, we ask: How efficiently can we approximate Pareto sums, both in theory and practice? Can we beat the near-quadratic-time state of the art for exact algorithms? On the theoretical side, we formulate a notion of additively approximate Pareto sets and show that computing an approximate Pareto set is fine-grained equivalent to Bounded Monotone Min-Plus Convolution. Leveraging a remarkable Õ(n^{1.5})-time algorithm for the latter problem (Chi, Duan, Xie, Zhang; STOC '22), we thus obtain a strongly subquadratic (and conditionally optimal) approximation algorithm for computing Pareto sums. On the practical side, we engineer different algorithmic approaches for approximating Pareto sets on realistic instances. Our implementations enable a granular trade-off between approximation quality and running time/output size compared to the state of the art for exact algorithms established in (Funke, Hespe, Sanders, Storandt, Truschel; Algorithmica '25). Perhaps surprisingly, the (theoretical) connection to Bounded Monotone Min-Plus Convolution remains beneficial even for our implementations: in particular, we implement a simplified, yet still subquadratic version of an algorithm due to Chi, Duan, Xie and Zhang, which on some sufficiently large instances outperforms the competing quadratic-time approaches.

Abstract In bi-criteria optimization problems, the goal is typically to compute the set of Pareto-optimal solutions. Many algorithms for these types of problems rely on efficient merging or combining of partial solutions and filtering of dominated solutions in the resulting sets. In this article, we consider the task of computing the Pareto sum of two given Pareto sets A, B of size n. The Pareto sum C contains all non-dominated points of the Minkowski sum $$M = \{a+b|a \in A, b\in B\}$$ M = { a + b | a ∈ A , b ∈ B } . Since the Minkowski sum has a size of $$n^2$$ n 2 , but the Pareto sum C can be much smaller, the goal is to compute C without having to compute and store all of M. We present several new algorithms for efficient Pareto sum computation, including an output-sensitive successive algorithm with a running time of $$\mathcal {O}(n \log n + nk)$$ O ( n log n + n k ) and a space consumption of $$\mathcal {O}(n+k)$$ O ( n + k ) for $$k=|C|$$ k = | C | . If the elements of C are streamed, the space consumption reduces to $$\mathcal {O}(n)$$ O ( n ) . For output sizes $$k \ge 2n$$ k ≥ 2 n , we prove a conditional lower bound for Pareto sum computation, which excludes running times in $$\mathcal {O}(n^{2-\delta })$$ O ( n 2 - δ ) for $$\delta > 0$$ δ > 0 unless the (min,+)-convolution hardness conjecture fails. The successive algorithm matches this lower bound for $$k \in \Theta (n)$$ k ∈ Θ ( n ) . However, for $$k \in \Theta (n^2)$$ k ∈ Θ ( n 2 ) , the successive algorithm exhibits a cubic running time. But we also present an algorithm with an output-sensitive space consumption and a running time of $$\mathcal {O}(n^2 \log n)$$ O ( n 2 log n ) , which matches the lower bound up to a logarithmic factor even for large k. Furthermore, we describe suitable engineering techniques to improve the practical running times of our algorithms. Finally, we provide an extensive comparative experimental study on generated and real-world data. As a showcase application, we consider preprocessing-based bi-criteria route planning in road networks. Pareto sum computation is the bottleneck task in the preprocessing phase and in the query phase. We show that using our algorithms with an output-sensitive space consumption allows to tackle larger instances and reduces the preprocessing and query time compared to algorithms that fully store M.