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Takashi Horiyama, Yasuaki Kobayashi, Hirotaka Ono, Kazuhisa Seto, Ryu Suzuki

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

Mingyu Guo, Jialiang Li, Aneta Neumann, Frank Neumann, Hung Nguyen

We propose a combinatorial optimisation model called Limited Query Graph Connectivity Test. We consider a graph whose edges have two possible states (On/Off). The edges' states are hidden initially. We could query an edge to reveal its state. Given a source s and a destination t, we aim to test s−t connectivity by identifying either a path (consisting of only On edges) or a cut (consisting of only Off edges). We are limited to B queries, after which we stop regardless of whether graph connectivity is established. We aim to design a query policy that minimizes the expected number of queries. Our model is mainly motivated by a cyber security use case where we need to establish whether attack paths exist in a given network, between a source (i.e., a compromised user node) and a destination (i.e., a high-privilege admin node). Edge query is resolved by manual effort from the IT admin, which is the motivation behind query minimization. Our model is highly related to Stochastic Boolean Function Evaluation (SBFE). There are two existing exact algorithms for SBFE that are prohibitively expensive. We propose a signifcantly more scalable exact algorithm. While previous exact algorithms only scale for trivial graphs (i.e., past works experimented on at most 20 edges), we empirically demonstrate that our algorithm is scalable for a wide range of much larger practical graphs (i.e., graphs representing Windows domain networks with tens of thousands of edges). We also propose three heuristics. Our best-performing heuristic is via limiting the planning horizon of the exact algorithm. The other two are via reinforcement learning (RL) and Monte Carlo tree search (MCTS). We also derive an algorithm for computing the performance lower bound. Experimentally, we show that all our heuristics are near optimal. The heuristic building on the exact algorithm outperforms all other heuristics, surpassing RL, MCTS and eight existing heuristics ported from SBFE and related literature.

Longkun Guo, Chaoqi Jia, Kewen Liao, Zhigang Lu, Minhui Xue

Center-based clustering has attracted significant research interest from both theory and practice. In many practical applications, input data often contain background knowledge that can be used to improve clustering results. In this work, we build on widely adopted k-center clustering and model its input background knowledge as must-link (ML) and cannot-link (CL) constraint sets. However, most clustering problems including k-center are inherently NP-hard, while the more complex constrained variants are known to suffer severer approximation and computation barriers that significantly limit their applicability. By employing a suite of techniques including reverse dominating sets, linear programming (LP) integral polyhedron, and LP duality, we arrive at the first efficient approximation algorithm for constrained k-center with the best possible ratio of 2. We also construct competitive baseline algorithms and empirically evaluate our approximation algorithm against them on a variety of real datasets. The results validate our theoretical findings and demonstrate the great advantages of our algorithm in terms of clustering cost, clustering quality, and running time.

Kaan Gokcesu, Hakan Gökcesu

We study the problem of global optimization, where we analyze the performance of the Piyavskii--Shubert algorithm and its variants. For any given time duration T, instead of the extensively studied simple regret (which is the difference of the losses between the best estimate up to T and the global minimum), we study the cumulative regret up to time T. For L-Lipschitz continuous functions, we show that the cumulative regret is O(L logT). For H-Lipschitz smooth functions, we show that the cumulative regret is O(H). We analytically extend our results for functions with Hölder continuous derivatives, which cover both the Lipschitz continuous and the Lipschitz smooth functions, individually. We further show that a simpler variant of the Piyavskii-Shubert algorithm performs just as well as the traditional variants for the Lipschitz continuous or the Lipschitz smooth functions. We further extend our results to broader classes of functions, and show that, our algorithm efficiently determines its queries; and achieves nearly minimax optimal (up to log factors) cumulative regret, for general convex or even concave regularity conditions on the extrema of the objective (which encompasses many preceding regularities). We consider further extensions by investigating the performance of the Piyavskii-Shubert variants in the scenarios with unknown regularity, noisy evaluation and multivariate domain.

Simon Dold, Malte Helmert

Classical planning considers a given task and searches for a plan to solve it. Some tasks are harder to solve than others. We can measure the 'hardness' of a task with the novelty width and the correlation complexity. In this work, we compare these measures. Additionally, we introduce the river measure, a new measure that is based on potential heuristics and therefore similar to the correlation complexity but also comparable to the novelty width. We show that the river measure is upper bounded by the correlation complexity and by the novelty width +1. Furthermore, we show that we can convert a planning task with a polynomial blowup of the task size to ensure that a heuristic of dimension 2 exists that gives rise to backtrack-free search.

Benjamin Doerr, Aymen Echarghaoui, Mohammed Jamal, Martin S. Krejca

Most evolutionary algorithms used in practice heavily employ crossover. In contrast, the rigorous understanding of how crossover is beneficial is largely lagging behind. In this work, we make a considerable step forward by analyzing the population dynamics of the (µ+1) genetic algorithm when optimizing the Jump benchmark. We observe (and prove via mathematical means) that once the population contains two different individuals on the local optimum, the diversity in the population increases in expectation. From this drift towards more diverse states, we show that a diversity suitable for crossover to be effective is reached quickly and, more importantly, then persists for a time that is at least exponential in the population size µ. This drastically improves over the previously best known guarantee, which is only quadratic in µ. Our new understanding of the population dynamics easily gives stronger performance guarantees. In particular, we derive that population sizes logarithmic in the problem size n suffice to gain an Ω(n)-factor runtime improvement from crossover (previous works achieved comparable bounds only with µ = Θ(n) or a non-standard mutation rate).

Zhe Chen, Daniel Harabor, Jiaoyang Li, Peter J. Stuckey

Multi-Agent Path Finding (MAPF) is a fundamental problem in robotics that asks us to compute collision-free paths for a team of agents, all moving across a shared map. Although many works appear on this topic, all current algorithms struggle as the number of agents grows. The principal reason is that existing approaches typically plan free-flow optimal paths, which creates congestion. To tackle this issue, we propose a new approach for MAPF where agents are guided to their destination by following congestion-avoiding paths. We evaluate the idea in two large-scale settings: one-shot MAPF, where each agent has a single destination, and lifelong MAPF, where agents are continuously assigned new destinations. Empirically, we report large improvements in solution quality for one-short MAPF and in overall throughput for lifelong MAPF.

Xianrun Chen, Dachuan Xu, Yicheng Xu, Yong Zhang

Clustering is one of the most fundamental tools in artificial intelligence, machine learning, and data mining. In this paper, we follow one of the recent mainstream topics of clustering, Sum of Radii (SoR), which naturally arises as a balance between the folklore k-center and k-median. SoR aims to determine a set of k balls, each centered at a point in a given dataset, such that their union covers the entire dataset while minimizing the sum of radii of the k balls. We propose a general technical framework to overcome the challenge posed by varying radii in SoR, which yields fixed-parameter tractable (fpt) algorithms with respect to k (i.e., whose running time is f(k) ploy(n) for some f). Our framework is versatile and obtains fpt approximation algorithms with constant approximation ratios for SoR as well as its variants in general metrics, such as Fair SoR and Matroid SoR, which significantly improve the previous results.

Samantha Chen, Puoya Tabaghi, Yusu Wang

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the space. However, measures supported on trees admit a closed-form optimal transport that can be computed in linear time. In this paper, we aim to find an optimal tree structure for a given discrete metric space so that the tree-Wasserstein distance approximates the optimal transport distance in the original space. One of our key ideas is to cast the problem in ultrametric spaces. This helps us optimize over the space of ultrametric trees --- a mixed-discrete and continuous optimization problem --- via projected gradient decent over the space of ultrametric matrices. During optimization, we project the parameters to the ultrametric space via a hierarchical minimum spanning tree algorithm, equivalent to the closest projection to ultrametrics under the supremum norm. Experimental results on real datasets show that our approach outperforms previous approaches (e.g. Flowtree, Quadtree) in approximating optimal transport distances. Finally, experiments on synthetic data generated on ground truth trees show that our algorithm can accurately uncover the underlying trees.

Qingyun Chen, Sungjin Im, Benjamin Moseley, Chenyang Xu, Ruilong Zhang

The feedback arc set problem is one of the most fundamental and well-studied ranking problems where n objects are to be ordered based on their pairwise comparison. The problem enjoys several efficient approximation algorithms in the offline setting. Unfortunately, online there are strong lower bounds on the competitive ratio establishing that no algorithm can perform well in the worst case. This paper introduces a new beyond-worst-case model for online feedback arc set. In the model, a sample of the input is given to the algorithm offline before the remaining instance is revealed online. This models the case in practice where yesterday's data is available and is similar to today's online instance. This sample is drawn from a known distribution which may not be uniform. We design an online algorithm with strong theoretical guarantees. The algorithm has a small constant competitive ratio when the sample is uniform---if not, we show we can recover the same result by adding a provably minimal sample. Empirical results validate the theory and show that such algorithms can be used on temporal data to obtain strong results.

Lulu Cao, Yufei Liu, Zhenzhong Wang, Dejun Xu, Kai Ye, Kay Chen Tan, Min Jiang

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

Florent Avellaneda, Roger Villemaire

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.

Ahmed Abbas, Paul Swoboda

We present a fast, scalable, data-driven approach for solving relaxations of 0-1 integer linear programs. We use a combination of graph neural networks (GNN) and a Lagrange decomposition based algorithm. We make the latter differentiable for end-to-end training and use GNNs to predict its algorithmic parameters. This allows to retain the algorithm's theoretical properties including dual feasibility and guaranteed non-decrease in the lower bound while improving it via training. We overcome suboptimal fixed points of the basic solver by additional non-parametric GNN update steps maintaining dual feasibility. For training we use an unsupervised loss. We train on smaller problems and test on larger ones showing strong generalization performance with a GNN comprising only around 10k parameters. Our solver achieves significantly faster performance and better dual objectives than its non-learned version, achieving close to optimal objective values of LP relaxations of very large structured prediction problems and on selected combinatorial ones. In particular, we achieve better objective values than specialized approximate solvers for specific problem classes while retaining their efficiency. Our solver has better any-time performance over a large time period compared to a commercial solver.

Weijia Zhang, Chun Kai Ling, Xuanhui Zhang

Censoring is the central problem in survival analysis where either the time-to-event (for instance, death), or the time-to censoring (such as loss of follow-up) is observed for each sample. The majority of existing machine learning-based survival analysis methods assume that survival is conditionally independent of censoring given a set of covariates; an assumption that cannot be verified since only marginal distributions is available from the data. The existence of dependent censoring, along with the inherent bias in current estimators has been demonstrated in a variety of applications, accentuating the need for a more nuanced approach. However, existing methods that adjust for dependent censoring require practitioners to specify the ground truth copula. This requirement poses a significant challenge for practical applications, as model misspecification can lead to substantial bias. In this work, we propose a flexible deep learning-based survival analysis method that simultaneously accommodate for dependent censoring and eliminates the requirement for specifying the ground truth copula. We theoretically prove the identifiability of our model under a broad family of copulas and survival distributions. Experiments results from a wide range of datasets demonstrate that our approach successfully discerns the underlying dependency structure and significantly reduces survival estimation bias when compared to existing methods.

Shenbao Yu, Yifeng Zeng, Fan Yang, Yinghui Pan

Knowing a prerequisite structure among skills in a subject domain effectively enables several educational applications, including intelligent tutoring systems and curriculum planning. Traditionally, educators or domain experts use intuition to determine the skills' prerequisite relationships, which is time-consuming and prone to fall into the trap of blind spots. In this paper, we focus on inferring the prerequisite structure given access to students' performance on exercises in a subject. Nevertheless, it is challenging since students' mastery of skills can not be directly observed, but can only be estimated, i.e., its latency in nature. To tackle this problem, we propose a causal-driven skill prerequisite structure discovery (CSPS) method in a two-stage learning framework. In the first stage, we learn the skills' correlation relationships presented in the covariance matrix from the student performance data while, through the predicted covariance matrix in the second stage, we consider a heuristic method based on conditional independence tests and standardized partial variance to discover the prerequisite structure. We demonstrate the performance of the new approach with both simulated and real-world data. The experimental results show the effectiveness of the proposed model for identifying the skills' prerequisite structure.

Hantao Yang, Xutong Liu, Zhiyong Wang, Hong Xie, John C. S. Lui, Defu Lian, Enhong Chen

We study the problem of federated contextual combinatorial cascading bandits, where agents collaborate under the coordination of a central server to provide tailored recommendations to users. Existing works consider either a synchronous framework, necessitating full agent participation and global synchronization, or assume user homogeneity with identical behaviors. We overcome these limitations by considering (1) federated agents operating in an asynchronous communication paradigm, where no mandatory synchronization is required and all agents communicate independently with the server, (2) heterogeneous user behaviors, where users can be stratified into latent user clusters, each exhibiting distinct preferences. For this setting, we propose a UCB-type algorithm with delicate communication protocols. Through theoretical analysis, we give sub-linear regret bounds on par with those achieved in the synchronous framework, while incurring only logarithmic communication costs. Empirical evaluation on synthetic and real-world datasets validates our algorithm's superior performance in terms of regrets and communication costs.

Kevin Xia, Elias Bareinboim

The ability of humans to understand the world in terms of cause and effect relationships, as well as their ability to compress information into abstract concepts, are two hallmark features of human intelligence. These two topics have been studied in tandem under the theory of causal abstractions, but it is an open problem how to best leverage abstraction theory in real-world causal inference tasks, where the true model is not known, and limited data is available in most practical settings. In this paper, we focus on a family of causal abstractions constructed by clustering variables and their domains, redefining abstractions to be amenable to individual causal distributions. We show that such abstractions can be learned in practice using Neural Causal Models, allowing us to utilize the deep learning toolkit to solve causal tasks (identification, estimation, sampling) at different levels of abstraction granularity. Finally, we show how representation learning can be used to learn abstractions, which we apply in our experiments to scale causal inferences to high dimensional settings such as with image data.

Marcel Wienöbst, Benito Van der Zander, Maciej Liśkiewicz

Causal effect estimation from observational data is a fundamental task in empirical sciences. It becomes particularly challenging when unobserved confounders are involved in a system. This paper focuses on front-door adjustment – a classic technique which, using observed mediators allows to identify causal effects even in the presence of unobserved confounding. While the statistical properties of the front-door estimation are quite well understood, its algorithmic aspects remained unexplored for a long time. In 2022, Jeong, Tian, and Bareinboim presented the first polynomial-time algorithm for finding sets satisfying the front-door criterion in a given directed acyclic graph (DAG), with an O(n³(n+m)) run time, where n denotes the number of variables and m the number of edges of the causal graph. In our work, we give the first linear-time, i.e., O(n+m), algorithm for this task, which thus reaches the asymptotically optimal time complexity. This result implies an O(n(n+m)) delay enumeration algorithm of all front-door adjustment sets, again improving previous work by a factor of n³. Moreover, we provide the first linear-time algorithm for finding a minimal front-door adjustment set. We offer implementations of our algorithms in multiple programming languages to facilitate practical usage and empirically validate their feasibility, even for large graphs.

Gabriele Venturato, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt

Decision making under uncertainty in dynamic environments is a fundamental AI problem in which agents need to determine which decisions (or actions) to make at each time step to maximise their expected utility. Dynamic decision networks (DDNs) are an extension of dynamic Bayesian networks with decisions and utilities. DDNs can be used to compactly represent Markov decision processes (MDPs). We propose a novel algorithm called mapl-cirup that leverages knowledge compilation techniques developed for (dynamic) Bayesian networks to perform inference and gradient-based learning in DDNs. Specifically, we knowledge-compile the Bellman update present in DDNs into dynamic decision circuits and evaluate them within an (algebraic) model counting framework. In contrast to other exact symbolic MDP approaches, we obtain differentiable circuits that enable gradient-based parameter learning.

Russell Tsuchida, Cheng Soon Ong, Dino Sejdinovic

We introduce squared neural Poisson point processes (SNEPPPs) by parameterising the intensity function by the squared norm of a two layer neural network. When the hidden layer is fixed and the second layer has a single neuron, our approach resembles previous uses of squared Gaussian process or kernel methods, but allowing the hidden layer to be learnt allows for additional flexibility. In many cases of interest, the integrated intensity function admits a closed form and can be computed in quadratic time in the number of hidden neurons. We enumerate a far more extensive number of such cases than has previously been discussed. Our approach is more memory and time efficient than naive implementations of squared or exponentiated kernel methods or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in a reparameterisation of the final layer of the intensity function can be obtained by solving a (strongly) convex optimisation problem using projected gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and provide a software implementation.