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1. Realization Problems on Reachability Sequences

   Dippel, Matthew ; Sundaram, Ravi ; Varma, Akshar ( January 2020 , COCOON)

   The classical Erdös-Gallai theorem kicked off the study ofgraph realizability by characterizing degree sequences. We extend this line of research by investigating realizability of directed acyclic graphs (DAGs)given both a local constraint via degree sequences and a global constraint via a sequence of reachability values (number of nodes reachable from a given node). We show that, without degree constraints, DAG reachability realization is solvable in linear time, whereas it is strongly NP-complete given upper bounds on in-degree or out-degree. After defining a suitable notion of bicriteria approximation based on consistency, we give two approximation algorithms achieving O(logn)-reachability consistency and O(logn)-degreemore »consistency; the first, randomized, uses LP (Linear Program) rounding, while the second, deterministic, employs ak-setpacking heuristic. We end with two conjectures that we hope motivate further study of realizability with reachability constraints.« less
2. On the complexity of bidirected interleaved Dyck-reachability

   https://doi.org/10.1145/3434340  

   Li, Yuanbo ; Zhang, Qirun ; Reps, Thomas ( January 2021 , Proceedings of the ACM on Programming Languages)

   Many program analyses need to reason about pairs of matching actions, such as call/return, lock/unlock, or set-field/get-field. The family of Dyck languages { D k }, where D k has k kinds of parenthesis pairs, can be used to model matching actions as balanced parentheses. Consequently, many program-analysis problems can be formulated as Dyck-reachability problems on edge-labeled digraphs. Interleaved Dyck-reachability (InterDyck-reachability), denoted by D k ⊙ D k -reachability, is a natural extension of Dyck-reachability that allows one to formulate program-analysis problems that involve multiple kinds of matching-action pairs. Unfortunately, the general InterDyck-reachability problem is undecidable. In this paper, wemore »study variants of InterDyck-reachability on bidirected graphs , where for each edge ⟨ p , q ⟩ labeled by an open parenthesis ”( a ”, there is an edge ⟨ q , p ⟩ labeled by the corresponding close parenthesis ”) a ”, and vice versa . Language-reachability on a bidirected graph has proven to be useful both (1) in its own right, as a way to formalize many program-analysis problems, such as pointer analysis, and (2) as a relaxation method that uses a fast algorithm to over-approximate language-reachability on a directed graph. However, unlike its directed counterpart, the complexity of bidirected InterDyck-reachability still remains open. We establish the first decidable variant (i.e., D 1 ⊙ D 1 -reachability) of bidirected InterDyck-reachability. In D 1 ⊙ D 1 -reachability, each of the two Dyck languages is restricted to have only a single kind of parenthesis pair. In particular, we show that the bidirected D 1 ⊙ D 1 problem is in PTIME. We also show that when one extends each Dyck language to involve k different kinds of parentheses (i.e., D k ⊙ D k -reachability with k ≥ 2), the problem is NP-hard (and therefore much harder). We have implemented the polynomial-time algorithm for bidirected D 1 ⊙ D 1 -reachability. D k ⊙ D k -reachability provides a new over-approximation method for bidirected D k ⊙ D k -reachability in the sense that D k ⊙ D k -reachability can first be relaxed to bidirected D 1 ⊙ D 1 -reachability, and then the resulting bidirected D 1 ⊙ D 1 -reachability problem is solved precisely. We compare this D 1 ⊙ D 1 -reachability-based approach against another known over-approximating D k ⊙ D k -reachability algorithm. Surprisingly, we found that the over-approximation approach based on bidirected D 1 ⊙ D 1 -reachability computes more precise solutions, even though the D 1 ⊙ D 1 formalism is inherently less expressive than the D k ⊙ D k formalism.« less
3. A continuum limit for the PageRank algorithm

   https://doi.org/10.1017/S0956792521000097  

   YUAN, A. ; CALDER, J. ; OSTING, B. ( January 2021 , European Journal of Applied Mathematics)

   Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm and show how it can be interpreted as a numerical scheme on a directedmore »graph involving a type of normalised graph Laplacian . We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit partial differential equation. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.« less
4. ATP: Directed Graph Embedding with Asymmetric Transitivity Preservation

   https://doi.org/10.1609/aaai.v33i01.3301265  

   Sun, Jiankai ; Bandyopadhyay, Bortik ; Bashizade, Armin ; Liang, Jiongqian ; Sadayappan, P. ; Parthasarathy, Srinivasan ( July 2019 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Directed graphs have been widely used in Community Question Answering services (CQAs) to model asymmetric relationships among different types of nodes in CQA graphs, e.g., question, answer, user. Asymmetric transitivity is an essential property of directed graphs, since it can play an important role in downstream graph inference and analysis. Question difficulty and user expertise follow the characteristic of asymmetric transitivity. Maintaining such properties, while reducing the graph to a lower dimensional vector embedding space, has been the focus of much recent research. In this paper, we tackle the challenge of directed graph embedding with asymmetric transitivity preservation and thenmore »leverage the proposed embedding method to solve a fundamental task in CQAs: how to appropriately route and assign newly posted questions to users with the suitable expertise and interest in CQAs. The technique incorporates graph hierarchy and reachability information naturally by relying on a nonlinear transformation that operates on the core reachability and implicit hierarchy within such graphs. Subsequently, the methodology levers a factorization-based approach to generate two embedding vectors for each node within the graph, to capture the asymmetric transitivity. Extensive experiments show that our framework consistently and significantly outperforms the state-of-the-art baselines on three diverse realworld tasks: link prediction, and question difficulty estimation and expert finding in online forums like Stack Exchange. Particularly, our framework can support inductive embedding learning for newly posted questions (unseen nodes during training), and therefore can properly route and assign these kinds of questions to experts in CQAs.« less
5. AutoShrink: A Topology-aware NAS for Discovering Efficient Neural Architecture

   Zhang, Tunhou ; Cheng, Hsin-Pai ; Li, Zhenwen ; Yan, Feng ; Huang, Chengyu ; Li, Hai ; Chen, Yiran ( February 2020 , The Thirty-Fourth AAAI Conference on Artificial Intelligence)

   Resource is an important constraint when deploying Deep Neural Networks (DNNs) on mobile and edge devices. Existing works commonly adopt the cell-based search approach, which limits the flexibility of network patterns in learned cell structures. Moreover, due to the topology-agnostic nature of existing works, including both cell-based and node-based approaches, the search process is time consuming and the performance of found architecture may be sub-optimal. To address these problems, we propose AutoShrink, a topology-aware Neural Architecture Search (NAS) for searching efficient building blocks of neural architectures. Our method is node-based and thus can learn flexible network patterns in cell structuresmore »within a topological search space. Directed Acyclic Graphs (DAGs) are used to abstract DNN architectures and progressively optimize the cell structure through edge shrinking. As the search space intrinsically reduces as the edges are progressively shrunk, AutoShrink explores more flexible search space with even less search time. We evaluate AutoShrink on image classification and language tasks by crafting ShrinkCNN and ShrinkRNN models. ShrinkCNN is able to achieve up to 48% parameter reduction and save 34% Multiply-Accumulates (MACs) on ImageNet-1K with comparable accuracy of state-of-the-art (SOTA) models. Specifically, both ShrinkCNN and ShrinkRNN are crafted within 1.5 GPU hours, which is 7.2× and 6.7× faster than the crafting time of SOTA CNN and RNN models, respectively.« less