We study the following version of cut sparsification. Given a large edge-weighted network G with k terminal vertices, compress it into a smaller network H with the same terminals, such that every minimum terminal cut in H (i.e., the minimum cut separating a given subset of terminals from all other terminals) approximates the corresponding one in G, up to a factor q ≥ 1 that is called the quality. (The case q = 1 is known also as a mimicking network). We provide new insights about the structure of minimum terminal cuts, leading to new results for cut sparsifiers of planar graphs.Our first contribution identifies a subset of the minimum terminal cuts, which we call elementary, that generates all the others. Consequently, H is a cut sparsifier if and only if it preserves all the elementary terminal cuts (up to this factor q). This structural characterization can reduce the number of requirements, and thus lead to simpler proofs and to improved bounds on the size of H. For example, it leads to a small improvement in the mimicking-network size for planar graphs, which is actually near-optimal by a very recent lower bound [Karpov, Pilipczuk, and Zych-Pawlewicz, arXiv:1706.06086].Our second and main contribution is to refine the known bounds in terms of γ = γ(G), which is defined as the minimum number of faces that are incident to all the terminals in a planar graph G. We prove that the number of elementary terminal cuts is O((2k/γ) 2γ ) (compared to O(2 k ) terminal cuts), and furthermore obtain a mimicking-network of size O(γ2 2γ k 4 ), which is near-optimal as a function of γ by the aforementioned recent lower bound. The main challenge here is that the mimicking-network size is smaller than the number of elementary terminal cuts (requirements), and indeed our analysis breaks the elementary terminal cuts even further, and carefully counting these fragments yields a smaller number of requirements.Our third contribution is a duality between cut sparsification and distance sparsification for certain planar graphs, when the sparsifier H is required to be a minor of G. This duality connects problems that were previously studied separately, implying new results, new proofs of known results, and equivalences between open gaps.H instead of on G, using less resources like runtime and memory, or achieving better accuracy when the solution is approximate. This paradigm has lead to remarkable successes, such as faster runtimes for fundamental problems, and the introduction of important concepts, from spanners [PU89] to cut and spectral sparsifiers [BK15, ST11]. In these examples, H is a subgraph of G with the same vertex set but sparse, and is sometimes called an edge sparsifier. In contrast, we aim to reduce the number of vertices in G, using so-called vertex sparsifiers.In the vertex-sparsification scenario, G has k designated vertices called terminals, and the goal is to construct a small graph H that contains these terminals, and maintains some of their features inside G, like distances or cuts. Throughout, a k-terminal netw...

The relationship between the sparsest cut and the maximum concurrent multi-flow in graphs has been studied extensively. For general graphs, the worst-case gap between these two quantities is now settled: When there are k terminal pairs, the flow-cut gap is O(log k), and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flow-cut gap in planar networks is O(1), while the known bounds place the gap somewhere between 2 (Lee and Raghavendra, 2003) and O( √ log k) (Rao, 1999).A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flow-cut gap is exactly 1. This setting can be generalized by considering planar networks where the terminals lie on one of γ > 1 faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flow-cut gap is bounded by a function of γ, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most 3γ. We significantly improve these asymptotics by establishing that the flow-cut gap is O(log γ). This is achieved by showing that the edge-weighted shortest-path metric induced on the terminals admits a stochastic embedding into trees with distortion O(log γ). The latter result is tight, e.g., for a square planar lattice on Θ(γ) vertices.The preceding results refer to the setting of edge-capacitated networks. For vertex-capacitated networks, it can be significantly more challenging to control flow-cut gaps. While there is no exact vertex-capacitated version of the Okamura-Seymour Theorem, an approximate version holds; Lee, Mendel, and Moharrami (2015) showed that the vertex-capacitated flow-cut gap is O(1) on planar networks whose terminals lie on a single face. We prove that the flow-cut gap is O(γ) for vertex-capacitated instances when the terminals lie on at most γ faces. In fact, this result holds in the more general setting of submodular vertex capacities.

In the field of People Intelligence, Emotion Analytics is one of the emerging and growing challenges. A prominent study field is analyzing an individual's emotional state from textual data, as well as recognizing emotions from audio and video recordings. Current Artificial Intelligence approaches for Emotion Analytics based on machine and deep learning and neural networks based on classic data science approaches assume rational people's decision-making process. People's decision-making is irrational. As a result of recent quantum cognition advancements, we show that emotional judgments from one modality may be incompatible with judgments from another, and they cannot be assessed together to produce a final judgment. As a result, the cognitive process exhibits "quantum-like" biases that classical AI approaches based on probability models challenged to analyze. As a result, we offer an emotion analytics approach based on the quantum data science method for predicting people's emotions by a fundamentally novel assessment method.

No abstract

No abstract