| Total: 21
In the area of multi-objective evolutionary algorithms (MOEAs), there is a trend of using an archive to store non-dominated solutions generated during the search. This is because 1) MOEAs may easily end up with the final population containing inferior solutions that are dominated by other solutions discarded during the search process and 2) the population that has a commensurable size of the problem's Pareto front is often not practical. In this paper, we theoretically show, for the first time, that using an archive can guarantee speed-ups for MOEAs. Specifically, we prove that for two well-established MOEAs (NSGA-II and SMS-EMOA) on two commonly studied problems (OneMinMax and LeadingOnesTrailingZeroes), using an archive brings a polynomial acceleration on the expected running time. The reason is that with an archive, the size of the population can reduce to a small constant; there is no need for the population to keep all the Pareto optimal solutions found. This contrasts existing theoretical studies for MOEAs where a population with a commensurable size of the problem's Pareto front is needed. The findings in this paper not only provide a theoretical confirmation for an increasingly popular practice in the design of MOEAs, but can also be beneficial to the theory community towards studying more practical MOEAs.
Machine learning has been adapted to help solve NP-hard combinatorial optimization problems. One prevalent way is learning to construct solutions by deep neural networks, which has been receiving more and more attention due to the high efficiency and less requirement for expert knowledge. However, many neural construction methods for Vehicle Routing Problems~(VRPs) focus on synthetic problem instances with specified node distributions and limited scales, leading to poor performance on real-world problems which usually involve complex and unknown node distributions together with large scales. To make neural VRP solvers more practical, we design an auxiliary policy that learns from the local transferable topological features, named local policy, and integrate it with a typical construction policy (which learns from the global information of VRP instances) to form an ensemble policy. With joint training, the aggregated policies perform cooperatively and complementarily to boost generalization. The experimental results on two well-known benchmarks, TSPLIB and CVRPLIB, of travelling salesman problem and capacitated VRP show that the ensemble policy significantly improves both cross-distribution and cross-scale generalization performance, and even performs well on real-world problems with several thousand nodes.
We propose a novel approach to improve multi-objective evolutionary algorithms by modifying crossover operations. Our approach uses a modifiable cross distribution and virtual point to rebalance the probability distribution of all crossover options. This design reduces runtime for typical pseudo-Boolean functions. Experiments and analysis show our approach effectively optimizes bi-objective problems COCZ and LOTZ in Θ(n) time during crossover, outperforming conventional crossover multi-objective evolutionary algorithms (C-MOEA) which require O(n log n) steps. For the tri-objective problem Hierarchical-COCZ, our approach guarantees an expected runtime of Θ(n2 log n), while C-MOEA needs at least Ω(n2 log n) and at most O(n2 log2 n) steps.
Neural Combinatorial Optimization (NCO) is an emerging domain where deep learning techniques are employed to address combinatorial optimization problems as a standalone solver. Despite their potential, existing NCO methods often suffer from inefficient search space exploration, frequently leading to local optima entrapment or redundant exploration of previously visited states. This paper introduces a versatile framework, referred to as Memory-Augmented Reinforcement for Combinatorial Optimization (MARCO), that can be used to enhance both constructive and improvement methods in NCO through an innovative memory module. MARCO stores data collected throughout the optimization trajectory and retrieves contextually relevant information at each state. This way, the search is guided by two competing criteria: making the best decision in terms of the quality of the solution and avoiding revisiting already explored solutions. This approach promotes a more efficient use of the available optimization budget. Moreover, thanks to the parallel nature of NCO models, several search threads can run simultaneously, all sharing the same memory module, enabling an efficient collaborative exploration. Empirical evaluations, carried out on the maximum cut, maximum independent set and travelling salesman problems, reveal that the memory module effectively increases the exploration, enabling the model to discover diverse, higher-quality solutions. MARCO achieves good performance in a low computational cost, establishing a promising new direction in the field of NCO.
Runtime analysis, as a branch of the theory of AI, studies how the number of iterations algorithms take before finding a solution (its runtime) depends on the design of the algorithm and the problem structure. Drift analysis is a state-of-the-art tool for estimating the runtime of randomised algorithms, such as bandit and evolutionary algorithms. Drift refers roughly to the expected progress towards the optimum per iteration. This paper considers the problem of deriving concentration tail-bounds on the runtime of algorithms. It provides a novel drift theorem that gives precise exponential tail-bounds given positive, weak, zero and even negative drift. Previously, such exponential tail bounds were missing in the case of weak, zero, or negative drift. Our drift theorem can be used to prove a strong concentration of the runtime/regret of algorithms in AI. For example, we prove that the regret of the RWAB bandit algorithm is highly concentrated, while previous analyses only considered the expected regret. This means that the algorithm obtains the optimum within a given time frame with high probability, i.e. a form of algorithm reliability. Moreover, our theorem implies that the time needed by the co-evolutionary algorithm RLS-PD to obtain a Nash equilibrium in a Bilinear max-min-benchmark problem is highly concentrated. However, we also prove that the algorithm forgets the Nash equilibrium, and the time until this occurs is highly concentrated. This highlights a weakness in the RLS-PD which should be addressed by future work.
The integer linear programming (ILP) problem is a fundamental research topic in operations research, and the local search method is an important class of algorithms for quickly solving many combinatorial optimization problems. With rapidly increasing computing power, parallelism turns out to be a promising approach to enhancing the efficiency of problem-solving. However, rare studies investigate parallel local search algorithms for solving the general ILP problem. We propose the first parallel local search framework (ParaILP) for solving the general ILP problem, based on two novel ideas: a new initialization method named polarity initialization to construct different initial solutions for local search threads and a cooperative evolution mechanism for managing and generating high-quality solutions using information shared by different threads. Extensive experiments demonstrate that ParaILP is significantly better than the state-of-the-art academic parallel solvers FiberSCIP and HiGHS, and is competitive with the state-of-the-art commercial parallel solver Gurobi. Experiments are also conducted to analyze the parallelization scalability and the effectiveness of our techniques.
Existing neural heuristics often train a deep architecture from scratch for each specific vehicle routing problem (VRP), ignoring the transferable knowledge across different VRP variants. This paper proposes the cross-problem learning to assist heuristics training for different downstream VRP variants. Particularly, we modularize neural architectures for complex VRPs into 1) the backbone Transformer for tackling the travelling salesman problem (TSP), and 2) the additional lightweight modules for processing problem-specific features in complex VRPs. Accordingly, we propose to pre-train the backbone Transformer for TSP, and then apply it in the process of fine-tuning the Transformer models for each target VRP variant. On the one hand, we fully fine-tune the trained backbone Transformer and problem-specific modules simultaneously. On the other hand, we only fine-tune small adapter networks along with the modules, keeping the backbone Transformer still. Extensive experiments on typical VRPs substantiate that 1) the full fine-tuning achieves significantly better performance than the one trained from scratch, and 2) the adapter-based fine-tuning also delivers comparable performance while being notably parameter-efficient. Furthermore, we empirically demonstrate the favorable effect of our method in terms of cross-distribution application and versatility.
Peptide vaccines are growing in significance for fighting diverse diseases. Machine learning has improved the identification of peptides that can trigger immune responses, and the main challenge of peptide vaccine design now lies in selecting an effective subset of peptides due to the allelic diversity among individuals. Previous works mainly formulated this task as a constrained optimization problem, aiming to maximize the expected number of peptide-Major Histocompatibility Complex (peptide-MHC) bindings across a broad range of populations by selecting a subset of diverse peptides with limited size; and employed a greedy algorithm, whose performance, however, may be limited due to the greedy nature. In this paper, we propose a new framework PVD-EMO based on Evolutionary Multi-objective Optimization, which reformulates Peptide Vaccine Design as a bi-objective optimization problem that maximizes the expected number of peptide-MHC bindings and minimizes the number of selected peptides simultaneously, and employs a Multi-Objective Evolutionary Algorithm (MOEA) to solve it. We also incorporate warm-start and repair strategies into MOEAs to improve efficiency and performance. We prove that the warm-start strategy ensures that PVD-EMO maintains the same worst-case approximation guarantee as the previous greedy algorithm, and meanwhile, the EMO framework can help avoid local optima. Experiments on a peptide vaccine design for COVID-19, caused by the SARS-CoV-2 virus, demonstrate the superiority of PVD-EMO.
Neural combinatorial optimization (NCO) is a promising learning-based approach to solving various vehicle routing problems without much manual algorithm design. However, the current NCO methods mainly focus on the in-distribution performance, while the real-world problem instances usually come from different distributions. A costly fine-tuning approach or generalized model retraining from scratch could be needed to tackle the out-of-distribution instances. Unlike the existing methods, this work investigates an efficient prompt learning approach in NCO for cross-distribution adaptation. To be concrete, we propose a novel prompt learning method to facilitate fast zero-shot adaptation of a pre-trained model to solve routing problem instances from different distributions. The proposed model learns a set of prompts among various distributions and then selects the best-matched one to prompt a pre-trained attention model for each problem instance. Extensive experiments show that the proposed prompt learning approach facilitates the fast adaptation of pre-trained routing models. It also outperforms existing generalized models on both in-distribution prediction and zero-shot generalization to a diverse set of new tasks. Our code implementation is available online at https://github.com/FeiLiu36/PromptVRP.
This work proposes a new learning-to-search benchmark and uses AI to discover new mathematical knowledge related to an open conjecture of Erdos (1975) in extremal graph theory. The problem is to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum---jump-starting the search for larger graphs using good graphs found at smaller sizes---we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.
Quality-Diversity (QD) algorithms are a new type of Evolutionary Algorithms (EAs), aiming to find a set of high-performing, yet diverse solutions. They have found many successful applications in reinforcement learning and robotics, helping improve the robustness in complex environments. Furthermore, they often empirically find a better overall solution than traditional search algorithms which explicitly search for a single highest-performing solution. However, their theoretical analysis is far behind, leaving many fundamental questions unexplored. In this paper, we try to shed some light on the optimization ability of QD algorithms via rigorous runtime analysis. By comparing the popular QD algorithm MAP-Elites with (\mu+1)-EA (a typical EA focusing on finding better objective values only), we prove that on two NP-hard problem classes with wide applications, i.e., monotone approximately submodular maximization with a size constraint, and set cover, MAP-Elites can achieve the (asymptotically) optimal polynomial-time approximation ratio, while (\mu+1)-EA requires exponential expected time on some instances. This provides theoretical justification for that QD algorithms can be helpful for optimization, and discloses that the simultaneous search for high-performing solutions with diverse behaviors can provide stepping stones to good overall solutions and help avoid local optima.
We propose Expected Work Search (EWS), a new game solving algorithm. EWS combines win rate estimation, as used in Monte Carlo Tree Search, with proof size estimation, as used in Proof Number Search. The search efficiency of EWS stems from minimizing a novel notion of Expected Work, which predicts the expected computation required to solve a position. EWS outperforms traditional solving algorithms on the games of Go and Hex. For Go, we present the first solution to the empty 5x5 board with the commonly used positional superko ruleset. For Hex, our algorithm solves the empty 8x8 board in under 4 minutes. Experiments show that EWS succeeds both with and without extensive domain-specific knowledge.
In the real world, there exist a class of optimization problems that multiple (local) optimal solutions in the solution space correspond to a single point in the objective space. In this paper, we theoretically show that for such multimodal problems, a simple method that considers the diversity of solutions in the solution space can benefit the search in evolutionary algorithms (EAs). Specifically, we prove that the proposed method, working with crossover, can help enhance the exploration, leading to polynomial or even exponential acceleration on the expected running time. This result is derived by rigorous running time analysis in both single-objective and multi-objective scenarios, including (mu+1)-GA solving the widely studied single-objective problem, Jump, and NSGA-II and SMS-EMOA (two well-established multi-objective EAs) solving the widely studied bi-objective problem, OneJumpZeroJump. Experiments are also conducted to validate the theoretical results. We hope that our results may encourage the exploration of diversity maintenance in the solution space for multi-objective optimization, where existing EAs usually only consider the diversity in the objective space and can easily be trapped in local optima.
This paper provides a theoretical study on Multi-Objective Heuristic Search. We first classify states in the state space into must-expand, maybe-expand, and never-expand states and then transfer these definitions to nodes in the search tree. We then formalize a framework that generalizes A* to Multi-Objective Search. We study different ways to order nodes under this framework and their relation to traditional tie-breaking policies and provide theoretical findings. Finally, we study and empirically compare different ordering functions.
The maximum k-plex problem (MKPP) is an significant relaxation version of the maximum clique problem with extensive applications. Recently, lots of researchers have proposed many heuristic algorithms based on various methods to solve the MKPP. In this work, to further improve the performance of solving the MKPP, we propose an efficient local search algorithm based on three main ideas. First, we propose a relaxed bounded configuration checking strategy that considers two kinds of historical searching information to relax the restricted strength of configuration checking and the forbidden condition of candidate vertices for the operation Add, respectively. Second, we present a novel solution information-based vertex selection strategy based on two kinds of solution information to select high-quality candidate vertices. Third, we define the solution core and then introduce a core-based perturbation strategy to help the algorithm jump out of local optima. The experimental results show that the proposed algorithm significantly outperforms the state-of-the-art MKPP algorithms in almost all the instances.
The Traveling Thief Problem (TTP) is a challenging combinatorial optimization problem with broad practical applications. TTP combines two NP-hard problems: the Traveling Salesman Problem (TSP) and Knapsack Problem (KP). While a number of machine learning and deep learning based algorithms have been developed for TSP and KP, there is limited research dedicated to TTP. In this paper, we present the first reinforcement learning based multi-start neighborhood search algorithm, denoted by ReinforceNS, for solving TTP. To accelerate the search, we employ a pre-processing procedure for neighborhood reduction. A TSP routing and an iterated greedy packing are independently utilized to construct a high-quality initial solution, further improved by a reinforcement learning based neighborhood search. Additionally, a post-optimization procedure is devised for continued solution improvement. We conduct extensive experiments on 60 commonly used benchmark instances with 76 to 33810 cities in the literature. The experimental results demonstrate that our proposed ReinforceNS algorithm outperforms three state-of-the-art algorithms in terms of solution quality with the same time limit. In particular, ReinforceNS achieves 12 new results for 18 instances publicly reported in a recent TTP competition. We also perform an additional experiment to validate the effectiveness of the reinforcement learning strategy.
The Latin square completion (LSC) problem aims to assign n symbols to the empty cells of a partially filled Latin square such that in each row and each column, each symbol appears exactly once. In this paper, we propose a swap relaxation-based fast local search algorithm called SRLS for solving the LSC problem. First, it introduces a novel search space definition, which forbids row conflicts based on which a swap-based neighborhood is defined. Second, a color domain relaxation technique is employed in the swap-based neighborhood by temporarily accepting the violation of some constraints to connect high-quality solutions. Third, two effective scoring functions are adopted to select neighborhood moves minimizing the number of conflicting edges as well as the number of color domain violations. Finally, SRLS employs an adaptive restart mechanism to balance the exploitation and exploration of the search. Extensive experiments on 1819 public benchmark instances demonstrate that SRLS outperforms the state-of-the-art algorithms in the literature in terms of both success rate and computational efficiency.
The Minimum Dominating Set Problem (MDSP) is an important NP-Hard optimization problem with many applications in various domains. This paper designs two exact algorithms for MDSP that use the same Branch-and-Bound framework. However, one uses LP relaxations as lower bounds for pruning the search space, and the other one is a pure combinatorial algorithm. The two algorithms possess a distinct advantage. Performance experiments on standard test datasets reveal that our combinatorial algorithm is over 1000 times faster than the previous state-of-the-art exact algorithm presented in IJCAI 2023, and our LP Relaxation algorithm can even enhance the speed of our combinatorial algorithm by over 100 times. However, our combinatorial algorithm still outperform the LP Relaxation algorithm on very dense graphs.
Submodular function maximization has been studied extensively in recent years due to its numerous applications in machine learning and artificial intelligence. We study a natural online variant of this problem on massive streaming data in which elements arrive one-by-one and the algorithm has to maintain a solution under cardinality constraint, i.e., k. Upon arrival of an element, the algorithm to maximize a monotone submodular function has to decide whether to accept the element and may replace a previously chosen element. Existing algorithms cannot simultaneously achieve optimal performance in terms of competitive ratio, memory complexity and running time. Also, the algorithm with best competitive ratio performs poorly in practice. In this paper, we propose a new algorithm OnlineAdaptive with optimal performance by exploiting adaptive thresholds to decide the acceptance of arriving elements by replacement. We prove that the competitive ratio of OnlineAdaptive is at least 1/4, and the ratio is about 0.2959 when k>=4 and approaches 0.3178 when k tends to infinity. In addition, OnlineAdaptive only needs O(k) memory and just performs one oracle per element. Experiments on diverse datasets confirm that OnlineAdaptive outperforms existing algorithms in both quality and efficiency.
Given a directed graph, a set of source nodes, a target node and a budget, we study the problem of maximizing the number of source nodes disconnected from the target node by removing edges not exceeding the budget. Our model is mainly motivated by a cyber security use case where we need to minimize the attack surface of a Windows Active Directory system. In these high-profile attacks, the attackers first compromise a source (i.e., a compromised user node) and then laterally move to a destination (i.e., a high-privileged admin node). Our aim is to minimize the number of users with a path to the admin. We first prove that the problem is NP-hard. Algorithms for exact optimality usually struggle to converge on graphs that approach real-world network scales and therefore are not practical for usage. In light of this, we study anytime algorithms that return an acceptable result whenever the algorithm is terminated, and can improve optimality by allowing longer computational time. We observe the source connectivity of directed graphs, based on which we propose a novel anytime algorithm---the spiral algorithm. We also develop two Monte Carlo Tree Search (MCTS) algorithms as a baseline to study the performance of typical anytime algorithms for our problem, and show that the spiral algorithm improves the optimality at a significantly faster speed and therefore exhibits better anytime behavior compared with MCTS.
We present an evolutionary algorithm evo-SMC for the problem of Submodular Maximization under Cost constraints (SMC). Our algorithm achieves 1/2-approximation with a high probability 1-1/n within O(n2 K) iterations, where K denotes the maximum size of a feasible solution set with cost constraint B. To the best of our knowledge, this is the best approximation guarantee offered by evolutionary algorithms for this problem. We further refine evo-SMC and develop st-evo-SMC. This stochastic version yields a significantly faster algorithm while maintaining the approximation ratio of 1/2, with probability 1-epsilon. The required number of iterations reduces to O(nK log(1/epsilon)/p), where the user defined parameters p represents the stochasticity probability, and epsilon denotes the error threshold. Finally, the empirical evaluations carried out through extensive experimentation substantiate the efficiency and effectiveness of our proposed algorithms. Our algorithms consistently outperform existing methods, producing higher-quality solutions.