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In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n²) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε²). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε²k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors. 

Understanding the characteristics of air-traffic delays and disruptions is critical for developing ways to mitigate their significant economic and environmental impacts. Conventional delay-performance metrics reflect only the magnitude of incurred flight delays at airports; in this work, we show that it is also important to characterize the spatial distribution of delays across a network of airports. We analyze graph-supported signals, leveraging techniques from spectral theory and graph-signal processing to compute analytical and simulation-driven bounds for identifying outliers in spatial distribution. We then apply these methods to the case of airport-delay networks and demonstrate the applicability of our methods by analyzing U.S. airport delays from 2008 through 2017. We also perform an airline-specific analysis, deriving insights into the delay dynamics of individual airline subnetworks. Through our analysis, we highlight key differences in delay dynamics between different types of disruptions, ranging from nor’easters and hurricanes to airport outages. We also examine delay interactions between airline subnetworks and the system-wide network and compile an inventory of outlier days that could guide future aviation operations and research. In doing so, we demonstrate how our approach can provide operational insights in an air-transportation setting. Our analysis provides a complementary metric to conventional aviation-delay benchmarks and aids airlines, traffic-flow managers, and transportation-system planners in quantifying off-nominal system performance.