The Discrepancy of Shortest Paths

TL;DR

This work establishes tightish bounds for the hereditary discrepancy of shorted path systems, showing that systems of unique shortest paths in weighted graphs have hereditary vertex and edge discrepancy bounded by $ ilde{O}(n^{1/4})$, matching a corresponding lower bound up to polylog factors. The authors derive two complementary approaches: an existential constructive upper bound for consistent path systems (hence for unique shortest paths) via the primal shatter function, and a lower bound via a trace bound applied to a hopset-inspired graph construction, extended to planar and bipartite graphs. They also provide explicit colorings achieving the upper bounds and translate the discrepancy results into differential privacy lower bounds for APSD and ASRQ tasks, tightening the additive error landscape to $ ilde{ olinebreak} ilde{ olinebreak Omega}(n^{1/4})$. The results reveal a structural separation between shortest-path systems and arbitrary path systems, with implications for matrix factorization norms and DP-labeled queries. Open problems include improving the directed-edge bound and closing gaps for dense graph regimes, as well as refining the planar/bipartite lower bounds and their DP consequences.

Abstract

The hereditary discrepancy of a set system is a certain quantitative measure of the pseudorandom properties of the system. Roughly, hereditary discrepancy measures how well one can $2$-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has well-studied applications e.g. in communication complexity and derandomization. More recently, the hereditary discrepancy of set systems of shortest paths has found applications in differential privacy [Chen et al.~SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy $\widetilde{O}(n^{1/4})$, and we construct lower bound examples demonstrating that this bound is tight up to hidden $\text{polylog } n$ factors. Our lower bounds apply even in the planar and bipartite settings, and they improve on a previous lower bound of $Ω(n^{1/6})$ obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from $Ω(n^{1/6})$ [Chen et al.~SODA 23] to $Ω(n^{1/4})$, and we improve the lower bound on additive error for the differentially-private all sets range queries problem to $Ω(n^{1/4})$, which is tight up to hidden $\text{polylog } n$ factors [Deng et al.~WADS 23].

Figures (6)

• Figure 1: If we color the nodes of a unique shortest path with alternating colors, then its nodes will contribute discrepancy $0$, $+1$, or $-1$ to all unique shortest paths that intersect it.
• Figure 2: In this figure, paths $\pi_1, \pi_2, \pi_3 \in \Pi'$ from the path cover are intersecting a path $\pi \in \Pi$. Paths in the path cover are pairwise vertex-disjoint, and each path in the cover contributes discrepancy $0$, $-1$, or $+1$ to $\pi$.
• Figure 3: In this figure, paths $\pi_1, \pi_2 \in \Pi'$ are intersecting a path $\pi \in \Pi$. This arrangement of paths is forbidden by the No Repeats Property of Proposition \ref{['prop:edge_path_cover']}.
• Figure 4: An example in directed graphs that demonstrates how coloring unique shortest paths with alternating colors can fail to imply low discrepancy. The nodes of path $\pi'$ contribute discrepancy $\pm 3$ to path $\pi$; this should be contrasted with what we observe in \ref{['fig:intro-coloring']}.
• Figure 5: A $n\times 2$ grid graph (with vertices shown in hollow and edges in black) $G$ with one path (in blue) starting from the top left corner go the right. The solid vertices and edges in dashed red define the companion graph $G'$. The corresponding path in $G'$ is shown in pink.

Theorems & Definitions (66)

• Theorem 1: Main Result, Informal
• Definition 1: Edge and Vertex Incidence Matrices
• Definition 2: Discrepancy and Hereditary Discrepancy
• Theorem 2: Main Result
• Theorem 3
• Theorem 4: Informal version of Corollaries \ref{['cor:dp-apsd-lb']} and \ref{['cor:dp-asrq-lb']}
• Definition 3
• Theorem 4
• proof
• Theorem 5