We initiate a systematic study on $\mathit{dynamic}$ $\mathit{influence}$ $\mathit{maximization}$ (DIM). In the DIM problem, one maintains a seed set $S$ of at most $k$ nodes in a dynamically involving social network, with the goal of maximizing the expected influence spread while minimizing the amortized updating cost. We consider two evolution models. In the $\mathit{incremental}$ model, the social network gets enlarged over time and one only introduces new users and establishes new social links, we design an algorithm that achieves $(1-1/e-\epsilon)$-approximation to the optimal solution and has $k \cdot\mathsf{poly}(\log n, \epsilon^{-1})$ amortized running time, which matches the state-of-art offline algorithm with only poly-logarithmic overhead. In the $\mathit{fully}$ $\mathit{dynamic}$ model, users join in and leave, influence propagation gets strengthened or weakened in real time, we prove that under the Strong Exponential Time Hypothesis (SETH), no algorithm can achieve $2^{-(\log n)^{1-o(1)}}$-approximation unless the amortized running time is $n^{1-o(1)}$. On the technical side, we exploit novel adaptive sampling approaches that reduce DIM to the dynamic MAX-k coverage problem, and design an efficient $(1-1/e-\epsilon)$-approximation algorithm for it. Our lower bound leverages the recent developed distributed PCP framework.
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Safety is critical in autonomous robotic systems. A safe control law ensures forward invariance of a safe set (a subset in the state space). It has been extensively studied regarding how to derive a safe control law with a control-affine analytical dynamic model. However, in complex environments and tasks, it is challenging and time-consuming to obtain a principled analytical model of the system. In these situations, data-driven learning is extensively used and the learned models are encoded in neural networks. How to formally derive a safe control law with Neural Network Dynamic Models (NNDM) remains unclear due to the lack of computationally tractable methods to deal with these black-box functions. In fact, even finding the control that minimizes an objective for NNDM without any safety constraint is still challenging. In this work, we propose MIND-SIS (Mixed Integer for Neural network Dynamic model with Safety Index Synthesis), the first method to derive safe control laws for NNDM. The method includes two parts: 1) SIS: an algorithm for the offline synthesis of the safety index (also called as barrier function), which uses evolutionary methods and 2) MIND: an algorithm for online computation of the optimal and safe control signal, which solves a constrained optimization using a computationally efficient encoding of neural networks. It has been theoretically proved that MIND-SIS guarantees forward invariance and finite convergence. And it has been numerically validated that MIND-SIS achieves safe and optimal control of NNDM. From our experiments, the optimality gap is less than $10^{-8}$, and the safety constraint violation is $0$.
We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. The expected running time of our algorithm is near-linear in $n$ and a fixed polynomial in $\Delta$, where $n$ is the number of variables and $\Delta$ is the max degree of constraints. Previously, up to similar conditions, sampling algorithms with running time polynomial in both $n$ and $\Delta$, only existed for the almost atomic case, where each constraint is violated by a small number of forbidden local configurations. Our sampling approach departs from all previous fast algorithms for sampling LLL, which were based on Markov chains. A crucial step of our algorithm is a recursive marginal sampler that is of independent interests. Within a local lemma regime, this marginal sampler can draw a random value for a variable according to its marginal distribution, at a local cost independent of the size of the CSP.
On-demand delivery has become increasingly popular around the world. Motivated by a large grocery chain store who offers fast on-demand delivery services, we model and solve a stochastic dynamic driver dispatching and routing problem for last-mile delivery systems where on-time performance is the main target. The system operator needs to dispatch a set of drivers and specify their delivery routes facing random demand that arrives over a fixed number of periods. The resulting stochastic dynamic program is challenging to solve due to the curse of dimensionality. We propose a novel structured approximation framework to approximate the value function via a parametrized dispatching and routing policy. We analyze the structural properties of the approximation framework and establish its performance guarantee under large-demand scenarios. We then develop efficient exact algorithms for the approximation problem based on Benders decomposition and column generation, which deliver verifiably optimal solutions within minutes. The evaluation results on a real-world data set show that our framework outperforms the current policy of the company by 36.53% on average in terms of delivery time. We also perform several policy experiments to understand the value of dynamic dispatching and routing with varying fleet sizes and dispatch frequencies.
Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\frac{\Delta}{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.
Network-aware cascade size prediction aims to predict the final reposted number of user-generated information via modeling the propagation process in social networks. Estimating the user's reposting probability by social influence, namely state activation plays an important role in the information diffusion process. Therefore, Graph Neural Networks (GNN), which can simulate the information interaction between nodes, has been proved as an effective scheme to handle this prediction task. However, existing studies including GNN-based models usually neglect a vital factor of user's preference which influences the state activation deeply. To that end, we propose a novel framework to promote cascade size prediction by enhancing the user preference modeling according to three stages, i.e., preference topics generation, preference shift modeling, and social influence activation. Our end-to-end method makes the user activating process of information diffusion more adaptive and accurate. Extensive experiments on two large-scale real-world datasets have clearly demonstrated the effectiveness of our proposed model compared to state-of-the-art baselines.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
Influence Maximization (IM), which aims to select a set of users from a social network to maximize the expected number of influenced users, is an evergreen hot research topic. Its research outcomes significantly impact real-world applications such as business marketing. The booming location-based network platforms of the last decade appeal to the researchers embedding the location information into traditional IM research. In this survey, we provide a comprehensive review of the existing location-driven IM studies from the perspective of the following key aspects: (1) a review of the application scenarios of these works, (2) the diffusion models to evaluate the influence propagation, and (3) a comprehensive study of the approaches to deal with the location-driven IM problems together with a particular focus on the accelerating techniques. In the end, we draw prospects into the research directions in future IM research.
We study dynamic algorithms for the problem of maximizing a monotone submodular function over a stream of $n$ insertions and deletions. We show that any algorithm that maintains a $(0.5+\epsilon)$-approximate solution under a cardinality constraint, for any constant $\epsilon>0$, must have an amortized query complexity that is $\mathit{polynomial}$ in $n$. Moreover, a linear amortized query complexity is needed in order to maintain a $0.584$-approximate solution. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve $(0.5-\epsilon)$-approximation with a $\mathsf{poly}\log(n)$ amortized query complexity. On the positive side, when the stream is insertion-only, we present efficient algorithms for the problem under a cardinality constraint and under a matroid constraint with approximation guarantee $1-1/e-\epsilon$ and amortized query complexities $\smash{O(\log (k/\epsilon)/\epsilon^2)}$ and $\smash{k^{\tilde{O}(1/\epsilon^2)}\log n}$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
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