Decompositions and Algorithms for the Disjoint Paths Problem in Planar Graphs

Στο πρόβλημα των Διακεκριμενων Μονοπατιων μας ζητείται να εξετάσουμε, δεδομένου ενός γραφήματος G και ενος συνόλου k ζευγών τερματικών,αν τα ζεύγη των τερματικών μπορούν να συνδεθούν με διακεκριμένα μονοπάτια. Στα "Graph Minors", μια σειρά 23 εργασιών μεταξύ 1984 και 2011, οι Neil Robertson και Paul D. Seymour, ανάμεσα σε άλλα σπουδαία αποτελέσματα που επηρέασαν βαθιά την Θεωρία Γραφημάτων, παρουσίασαν έναν f(k)*n^3 αλγόριθμο για το πρόβλημα των Διακεκριμενων Μονοπατιων. Για να το καταφέρουν αυτό, εισήγαγαν την "τεχνκή της άσχετης κορυφής" σύμφωνα με την οποία σε κάθε στιγμιότυπο δεντροπλάτους μεγαλύτερου του g(k) υπάρχει μια "άσχετη" κορυφή της οποίας η αφαίρεση δημιουργεί ένα ισοδύναμο στιγμιότυπο του προβλήματος. Εδώ μελετάμε το πρόβλημα σε επίπεδα γραφήματα και αποδεικνύουμε ότι για κάθε σταθερό k κάθε στιγμιότυπο του προβλήματος των Διακεκριμενων Μονοπατιων σε επιπεδα γραφηματα μπορεί να μετασχηματιστεί σε ένα ισοδύναμο που έχει φραγμένο δενδροπλάτος, αφαιρώντας ταυτόχρονα ένα σύνολο κορυφών από το δεδομένο επίπεδο γράφημα. Ως συνέπεια αυτού, το πρόβλημα των Διακεκριμένων Μονοπατιών σε επίπεδα γραφήματα μπορεί να λυθεί σε γραμμικό χρόνο για κάθε σταθερό πλήθος τερματικών.> In the Disjoint Paths Problem, given a graph G and a set of k pairs of terminals, we ask whether the pairs of terminals can be linked by pairwise disjoint paths. > In the Graph Minors series of 23 papers between 1984 and 2011, Neil Robertson and Paul D. Seymour, among other great results that heavily influenced Graph Theory, provided an f(k)\cdot n^{3} algorithm for the Disjoint Paths Problem. To achieve this, they introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. > > We study the problem in planar graphs and we prove that for every fixed k every instance of the Planar Disjoint Paths Problem can be transformed to an equivalent one that has bounded treewidth, by simultaneously discarding a set of vertices of the given planar graph. As a consequence the Planar Disjoint Paths Problem can be solved in linear time for every fixed number of terminals

Influence Maximization in Social Networks: A Survey

Online social networks have become an important platform for people to communicate, share knowledge and disseminate information. Given the widespread usage of social media, individuals' ideas, preferences and behavior are often influenced by their peers or friends in the social networks that they participate in. Since the last decade, influence maximization (IM) problem has been extensively adopted to model the diffusion of innovations and ideas. The purpose of IM is to select a set of k seed nodes who can influence the most individuals in the network. In this survey, we present a systematical study over the researches and future directions with respect to IM problem. We review the information diffusion models and analyze a variety of algorithms for the classic IM algorithms. We propose a taxonomy for potential readers to understand the key techniques and challenges. We also organize the milestone works in time order such that the readers of this survey can experience the research roadmap in this field. Moreover, we also categorize other application-oriented IM studies and correspondingly study each of them. What's more, we list a series of open questions as the future directions for IM-related researches, where a potential reader of this survey can easily observe what should be done next in this field

Discovering Dynamic Causal Space for DAG Structure Learning

Discovering causal structure from purely observational data (i.e., causal discovery), aiming to identify causal relationships among variables, is a fundamental task in machine learning. The recent invention of differentiable score-based DAG learners is a crucial enabler, which reframes the combinatorial optimization problem into a differentiable optimization with a DAG constraint over directed graph space. Despite their great success, these cutting-edge DAG learners incorporate DAG-ness independent score functions to evaluate the directed graph candidates, lacking in considering graph structure. As a result, measuring the data fitness alone regardless of DAG-ness inevitably leads to discovering suboptimal DAGs and model vulnerabilities. Towards this end, we propose a dynamic causal space for DAG structure learning, coined CASPER, that integrates the graph structure into the score function as a new measure in the causal space to faithfully reflect the causal distance between estimated and ground truth DAG. CASPER revises the learning process as well as enhances the DAG structure learning via adaptive attention to DAG-ness. Grounded by empirical visualization, CASPER, as a space, satisfies a series of desired properties, such as structure awareness and noise robustness. Extensive experiments on both synthetic and real-world datasets clearly validate the superiority of our CASPER over the state-of-the-art causal discovery methods in terms of accuracy and robustness.Comment: Accepted by KDD 2023. Our codes are available at https://github.com/liuff19/CASPE

FJMP: Factorized Joint Multi-Agent Motion Prediction

Multi-agent motion prediction is an important problem in an autonomous driving pipeline, and it involves forecasting the future behaviour of multiple agents in complex driving environments. Autonomous vehicles (AVs) should produce accurate predictions of future agent behaviour in order to make safe and informed plans in safety-critical scenarios. Importantly, AVs should generate scene-consistent future predictions that predict the joint future behaviour of multiple agents, as this enables reasoning about potential future multi-agent interactions, which are critical for downstream planning. In this thesis, we address the problem of generating a set of scene-level, or joint, future trajectory predictions in multi-agent driving scenarios. To this end, we propose FJMP, a Factorized Joint Motion Prediction framework for multi-agent interactive driving scenarios. FJMP models the future scene interaction dynamics as a sparse directed interaction graph, where nodes represent agents and edges denote explicit interactions between agents. We then prune the graph into a directed acyclic graph (DAG) and decompose the joint prediction task into a sequence of marginal and conditional predictions according to the partial ordering of the DAG, where joint future trajectories are decoded using a directed acyclic graph neural network (DAGNN). We conduct experiments on two autonomous driving datasets and demonstrate that FJMP produces more accurate and scene-consistent joint trajectory predictions than existing approaches. Importantly, we show that FJMP produces superior joint forecasts compared to non-factorized approaches on the most interactive and kinematically interesting agents, which highlights the benefit of our proposed factorization

Markov Tensor Theory and Cascade, Reachability, and Routing in Complex Networks

University of Minnesota Ph.D. dissertation. 2017. Major: Electrical Engineering. Advisor: Zhi-Li Zhang. 1 computer file (PDF); 180 pages.In this dissertation, we study and characterize the networks as the medium and substrate for communications, interactions, and flows by addressing various crucial problems under the general topics of cascade, reachability, and routing. These are general problem domains common in several applications and from a variety of networks. We address these problems in a unified way by a theoretical platform that we have developed in this research, which we call Markov Tensor Theory. How does a phenomena, influence, or a failure cascade in a network and what are the key factors in this cascade? We study the influence cascade in social networks and introduce the Heat Conduction (HC) Model which captures both social influence and non-social influence, and extends many of the existing non-progressive models. We then prove that selecting the optimal seed set of influential nodes for maximizing the influence spread is NP-hard for HC, however, by establishing the submodularity of influence spread, we tackle the influence maximization problem with a scalable and provably near-optimal greedy algorithm. We also study failure cascade in inter-dependent networks where we considered the effects of cascading failures both within and across different layers. In this study, we investigate how different couplings (i.e., inter-dependencies) between network elements across layers affect the cascading failure dynamics. How failures or disruptions affect the network in terms of reachability of entities from each other, how to identify the reachabilities efficiently after failures, and who are the pivotal players in the reachabilities? We develop an oracle to answer dynamic reachabilities efficiently for failure-prone networks with frequent reachability query requirement. Founded on the concept of reachability, we also introduce and provide a formulation for finding articulation points, measuring network load balancing, and computing pivotality ranking of nodes. Once the reachabilities are determined, how to quickly and robustly route a flow from a part of the network to the other part of a network under the failures? To avoid solely relying on the shortest path and generate alternative paths on one hand, and to correct the degeneracy of hitting time distance, on the other hand, we develop a novel routing continuum method from shortest-path routing to all-path routing which provides both a closed form formulation for computing the continuum distances and an efficient routing strategy. We also devise an oracle for efficiently answering to single-source shortest path queries as well as finding the replacement paths in the case of multiple failures. For these studies, we develop Markov Tensor Theory as a platform of powerful theories and tools founded on Markov chain theory and random walk methods which supports the general weighted and directed networks

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well-known consensus and adjacency dynamics.Comment: 8 page