Pareto Sums of Pareto Sets: Lower Bounds and Algorithms

TL;DR

Several new algorithms for efficient Pareto sum computation are presented, including an output-sensitive successive algorithm with a running time of O(nlogn+nk) and an output-sensitive successive algorithm with a running time of O(nlogn+nk).

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 $M = \{a+b|a \in A, b\in B\}$. Since the Minkowski sum has a size of $n^2$, 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 $O(n \log n + nk)$ and a space consumption of $O(n+k)$ for $k=|C|$. If the elements of $C$ are streamed, the space consumption reduces to $O(n)$. For output sizes $k \geq 2n$, we prove a conditional lower bound for Pareto sum computation, which excludes running times in $O(n^{2-δ})$ for $δ> 0$ unless the (min,+)-convolution hardness conjecture fails. The successive algorithm matches this lower bound for $k \in Θ(n)$. However, for $k \in Θ(n^2)$, 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 $O(n^2 \log n)$, 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.

Figures (17)

• Figure 1: Example instance with input Pareto sets $A, B$ of size 10. The Minkowski sum has 100 elements. The Pareto sum $C$ consists of 27 elements (marked green in the plot as well as in the matrix representation).
• Figure 2: Left: Example road network (gray edges) with bi-criteria edge costs. The blue edges indicate shortcut edges. Each shortcut has a set of cost vectors that encodes the aggregated costs of all Pareto-optimal paths between its endpoints. Right: Inserting a new shortcut between the two red points requires to compute the Pareto sum of the two cost vector sets of the blue shortcuts. In this example, the Minkowski sum contains 9 elements out of which 6 form the Pareto sum (marked green in the table). Note that same computation is necessary in a bi-directional bi-criteria path search from the two red points at the moment at which the node in the middle is visited from both directions.
• Figure 3: Left: Pareto sets $A, B$ together with their convex hulls. Right: The Minkowski sum $CH(A) \oplus CH(B)$ of the convex hulls encloses all pairwise vector additions $a+b$ with $a \in CH(A)$ and $b \in CH(B)$. The respective vertices (green points) are a subset of the Pareto Sum $C$.
• Figure 4: Input Pareto sets following different kinds of distributions (top) and schematic depiction of the corresponding Minkowski matrix $M$ (middle). Green dots indicate entries in $M$ that are points on the Minkowski sum of the convex hulls of $A$ and $B$, black dots indicate entries that are dominated by the green ones, and orange dots encode the remaining elements of the Pareto sum which then together dominate the white dots. In the bottom row, the Minkowski sum and the Pareto sum are illustrated based on point coordinates.
• Figure 5: Initial search range (green rectangle) spanned by $M_{11}$ and $M_{nn}$. The range-minimum element $m$ then leads to a reduction of $y_{\max}$ and an increase of $x_{\min}$ (blue arrows) which tightens the search range for the next element of $C$.

Theorems & Definitions (17)

• Definition 1: Domination
• Definition 2: Pareto set
• Definition 3: Minkowski sum
• Definition 4: Pareto sum
• Theorem 1
• proof
• Conjecture 2
• Corollary 1
• Corollary 2
• Corollary 3