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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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.
Domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming, has shown promising performance on multiple combinatorial optimization problems compared with mixed integer programming (MIP) and constraint programming (CP). The current DIDP solvers are based on heuristic search, and the state-of-the-art solver, complete anytime beam search (CABS), uses beam search. However, the current DIDP solvers cannot utilize multiple threads, unlike state-of-the-art MIP and CP solvers. In this paper, we propose three parallel beam search algorithms and develop multi-thread implementations of CABS. With 32 threads, our multi-thread DIDP solvers achieve 9 to 39 times speedup on average and significant performance improvement over the sequential solver, finding the new best solutions for two instances of the traveling salesperson problem with time windows. In addition, our solvers outperform multi-thread MIP and CP solvers in four of the six combinatorial optimization problems evaluated.
Anytime heuristic search algorithms try to find a (potentially suboptimal) solution as quickly as possible and then work to find better and better solutions until an optimal solution is obtained or time is exhausted. The most widely-known anytime search algorithms are based on best-first search. In this paper, we propose a new algorithm, rectangle search, that is instead based on beam search, a variant of breadth-first search. It repeatedly explores alternatives at all depth levels and is thus best-suited to problems featuring deep local minima. Experiments using a variety of popular search benchmarks suggest that rectangle search is competitive with fixed-width beam search and often performs better than the previous best anytime search algorithms.
Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), as they significantly tighten the dual bounds and improve the solving performance. A key problem for cuts is when to stop cuts generation, which is important for the efficiency of solving MILPs. However, many modern MILP solvers employ hard-coded heuristics to tackle this problem, which tends to neglect underlying patterns among MILPs from certain applications. To address this challenge, we formulate the cuts generation stopping problem as a reinforcement learning problem and propose a novel hybrid graph representation model (HYGRO) to learn effective stopping strategies. An appealing feature of HYGRO is that it can effectively capture both the dynamic and static features of MILPs, enabling dynamic decision-making for the stopping strategies. To the best of our knowledge, HYGRO is the first data-driven method to tackle the cuts generation stopping problem. By integrating our approach with modern solvers, experiments demonstrate that HYGRO significantly improves the efficiency of solving MILPs compared to competitive baselines, achieving up to 31% improvement.
The maximum vertex-weighted clique problem (MVWCP) and the maximum edge-weighted clique problem (MEWCP) are two natural extensions of the fundamental maximum clique problem. In this paper, we systematically study MEWCP and make the following major contributions: (1) We show that MEWCP is NP-hard even when the minimum degree of the graph is n-2, in contrast to MVWCP which is polynomial-time solvable when the minimum degree of the graph is at least n-3. This result distinguishes the complexity of the two problems for the first time. (2) To address MEWCP, we develop an efficient branch-and-bound algorithm called MEWCat with both practical and theoretical performance guarantees. In practice, MEWCat utilizes a new upper bound tighter than existing ones, which allows for more efficient pruning of branches. In theory, we prove a running-time bound of O*(1.4423^n) for MEWCat, which breaks the trivial bound of O*(2^n) in the research line of practical exact MEWCP solvers for the first time. (3) Empirically, we evaluate the performance of MEWCat on various benchmark instances. The experiments demonstrate that MEWCat outperforms state-of-the-art exact solvers significantly. For instance, on 16 DIMACS graphs that the state-of-the-art solver BBEWC fails to solve within 7200 seconds, MEWCat solves all of them with an average time of less than 1000 seconds. On real-world graphs, MEWCat achieves an average speedup of over 36x.
Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the DM has subjective preferences. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax, OneJumpZeroJump problems are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.
Discrete optimization belongs to the set of N P-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms, which reach optimal solutions by iteratively adding inequalities known as cuts to refine a feasible set. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. In this work, we propose a method for accelerating cutting-plane algorithms via reinforcement learning. Our approach uses learned policies as surrogates for N P-hard elements of the cut generating procedure in a way that (i) accelerates convergence, and (ii) retains guarantees of optimality. We apply our method on two types of problems where cutting-plane algorithms are commonly used: stochastic optimization, and mixed-integer quadratic programming. We observe the benefits of our method when applied to Benders decomposition (stochastic optimization) and iterative loss approximation (quadratic programming), achieving up to 45% faster average convergence when compared to modern alternative algorithms.
People want to rely on optimization algorithms for complex decisions but verifying the optimality of the solutions can then become a valid concern, particularly for critical decisions taken by non-experts in optimization. One example is the shortest-path problem on a network, occurring in many contexts from transportation to logistics to telecommunications. While the standard shortest-path problem is both solvable in polynomial time and certifiable by duality, introducing side constraints makes solving and certifying the solutions much harder. We propose a proof system for constrained shortest-path problems, which gives a set of logical rules to derive new facts about feasible solutions. The key trait of the proposed proof system is that it specifically includes high-level graph concepts within its reasoning steps (such as connectivity or path structure), in contrast to, e.g., using linear combinations of model constraints. Thus, using our proof system, we can provide a step-by-step, human-auditable explanation showing that the path given by an external solver cannot be improved. Additionally, to maximize the advantages of this setup, we propose a proof search procedure that specifically aims to find small proofs of this form using a procedure similar to A* search. We evaluate our proof system on constrained shortest path instances generated from real-world road networks and experimentally show that we may indeed derive more interpretable proofs compared to an integer programming approach, in some cases leading to much smaller proofs.
Deep-reinforcement-learning (DRL) based neural combinatorial optimization (NCO) methods have demonstrated efficiency without relying on the guidance of optimal solutions. As the most mainstream among them, the learning constructive heuristic (LCH) achieves high-quality solutions through a rapid autoregressive solution construction process. However, these LCH-based methods are deficient in convergency, and there is still a performance gap compared to the optimal. Intuitively, learning to regret some steps in the solution construction process is helpful to the training efficiency and network representations. This article proposes a novel regret-based mechanism for an advanced solution construction process. Our method can be applied as a plug-in to any existing LCH-based DRL-NCO method. Experimental results demonstrate the capability of our work to enhance the performance of various NCO models. Results also show that the proposed LCH-Regret outperforms the previous modification methods on several typical combinatorial optimization problems. The code and Supplementary File are available at https://github.com/SunnyR7/LCH-Regret.
Combinatorial Optimization (CO) problems over graphs appear routinely in many applications such as in optimizing traffic, viral marketing in social networks, and matching for job allocation. Due to their combinatorial nature, these problems are often NP-hard. Existing approximation algorithms and heuristics rely on the search space to find the solutions and become time-consuming when this space is large. In this paper, we design a neural method called COMBHelper to reduce this space and thus improve the efficiency of the traditional CO algorithms based on node selection. Specifically, it employs a Graph Neural Network (GNN) to identify promising nodes for the solution set. This pruned search space is then fed to the traditional CO algorithms. COMBHelper also uses a Knowledge Distillation (KD) module and a problem-specific boosting module to bring further efficiency and efficacy. Our extensive experiments show that the traditional CO algorithms with COMBHelper are at least 2 times faster than their original versions.
Enhancing the generalization performance of neural networks given limited data availability remains a formidable challenge, due to the model selection trade-off between training error and generalization gap. To handle this challenge, we present a posterior optimization issue, specifically designed to reduce the generalization error of trained neural networks. To operationalize this concept, we propose a Doubly-Robust Boosting machine (DRBoost) which consists of a statistical learner and a zero-order optimizer. The statistical learner reduces the model capacity and thus the generalization gap; the zero-order optimizer minimizes the training error in a gradient-free manner. The two components cooperate to reduce the generalization error of a fully trained neural network in a doubly robust manner. Furthermore, the statistical learner alleviates the multicollinearity in the discriminative layer and enhances the generalization performance. The zero-order optimizer eliminates the reliance on gradient calculation and offers more flexibility in learning objective selection. Experiments demonstrate that DRBoost improves the generalization performance of various prevalent neural network backbones effectively.
Light field microscopy is a high-speed 3D imaging technique that records the light field from multiple angles by the microlens array(MLA), thus allowing us to obtain information about the light source from a single image only. For the fundamental problem of neuron localization, we improve the method of combining depth-dependent dictionary with sparse coding in this paper. In order to obtain higher localization accuracy and good noise immunity, we propose an inertial proximal gradient acceleration algorithm with dry friction, Fast-IPGDF. By preventing falling into a local minimum, our algorithm achieves better convergence and converges quite fast, which improves the speed and accuracy of obtaining the locolization of the light source based on the matching depth of epipolar plane images (EPI). We demonstrate the effectiveness of the algorithm for localizing non-scattered fluorescent beads in both noisy and non-noisy environments. The experimental results show that our method can achieve simultaneous localization of multiple point sources and effective localization in noisy environments. Compared to existing studies, our method shows significant improvements in both localization accuracy and speed.