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Evolutionary algorithms (EAs) have been widely and successfully applied to solve multi-objective optimization problems, due to their nature of population-based search. Population update is a key component in multi-objective EAs (MOEAs), and it is performed in a greedy, deterministic manner. That is, the next-generation population is formed by selecting the first population-size ranked solutions (based on some selection criteria, e.g., non-dominated sorting, crowdedness and indicators) from the collections of the current population and newly-generated solutions. In this paper, we question this practice. We analytically present that introducing randomness into the population update procedure in MOEAs can be beneficial for the search. More specifically, we prove that the expected running time of a well-established MOEA (SMS-EMOA) for solving a commonly studied bi-objective problem, OneJumpZeroJump, can be exponentially decreased if replacing its deterministic population update mechanism by a stochastic one. Empirical studies also verify the effectiveness of the proposed stochastic population update method. This work is an attempt to challenge a common practice for the population update in MOEAs. Its positive results, which might hold more generally, should encourage the exploration of developing new MOEAs in the area.
The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size N >= 4((n-1) wmax + 1) computes all extremal points of the Pareto front in an expected number of O(m^2 n wmax log(n wmax)) iterations, where n is the number of vertices, m the number of edges, and wmax is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of |F|, the number of extremal points of the Pareto front, a set that can be as large as n wmax. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.
In influence maximization (IM), the goal is to find a set of seed nodes in a social network that maximizes the influence spread. While most IM problems focus on classical influence cascades (e.g., Independent Cascade and Linear Threshold) which assume individual influence cascade probability is independent of the number of neighbors, recent studies by sociologists show that many influence cascades follow a pattern called complex contagion (CC), where influence cascade probability is much higher when more neighbors are influenced. Nonetheless, there are very limited studies for complex contagion influence maximization (CCIM) problems. This is partly because CC is non-submodular, the solution of which has been an open challenge. In this study, we propose the first reinforcement learning (RL) approach to CCIM. We find that a key obstacle in applying existing RL approaches to CCIM is the reward sparseness issue, which comes from two distinct sources. We then design a new RL algorithm that uses the CCIM problem structure to address the issue. Empirical results show that our approach achieves the state-of-the-art performance on 9 real-world networks.
The recent popularity of Wordle has revived interest in guessing games. We develop a general method for finding optimal strategies for guessing games while avoiding an exhaustive search. Our main contribution are several theorems that build towards a general theory to prove optimality of a strategy for a guessing game. This work is developed to apply to any guessing game, but we use Wordle as an example to present concrete results.
In single-objective optimization, it is well known that evolutionary algorithms also without further adjustments can stand a certain amount of noise in the evaluation of the objective function. In contrast, this question is not at all understood for multi-objective optimization. In this work, we conduct the first mathematical runtime analysis of a simple multi-objective evolutionary algorithm (MOEA) on a classic benchmark in the presence of noise in the objective function. We prove that when bit-wise prior noise with rate p <= alpha/n, alpha a suitable constant, is present, the simple evolutionary multi-objective optimizer (SEMO) without any adjustments to cope with noise finds the Pareto front of the OneMinMax benchmark in time O(n^2 log n), just as in the case without noise. Given that the problem here is to arrive at a population consisting of n+1 individuals witnessing the Pareto front, this is a surprisingly strong robustness to noise (comparably simple evolutionary algorithms cannot optimize the single-objective OneMax problem in polynomial time when p = omega(log(n)/n)). Our proofs suggest that the strong robustness of the MOEA stems from its implicit diversity mechanism designed to enable it to compute a population covering the whole Pareto front. Interestingly this result only holds when the objective value of a solution is determined only once and the algorithm from that point on works with this, possibly noisy, objective value. We prove that when all solutions are reevaluated in each iteration, then any noise rate p = omega(log(n)/n^2) leads to a super-polynomial runtime. This is very different from single-objective optimization, where it is generally preferred to reevaluate solutions whenever their fitness is important and where examples are known such that not reevaluating solutions can lead to catastrophic performance losses.
Finding diverse solutions to optimization problems has been of practical interest for several decades, and recently enjoyed increasing attention in research. While submodular optimization has been rigorously studied in many fields, its diverse solutions extension has not. In this study, we consider the most basic variants of submodular optimization, and propose two simple greedy algorithms, which are known to be effective at maximizing monotone submodular functions. These are equipped with parameters that control the trade-off between objective and diversity. Our theoretical contribution shows their approximation guarantees in both objective value and diversity, as functions of their respective parameters. Our experimental investigation with maximum vertex coverage instances demonstrates their empirical differences in terms of objective-diversity trade-offs.
Video games feature a dynamic environment where locations of objects (e.g., characters, equipment, weapons, vehicles etc.) frequently change within the game world. Although searching for relevant nearby objects in such a dynamic setting is a fundamental operation, this problem has received little research attention. In this paper, we propose a simple lightweight index, called Grid Tree, to store objects and their associated textual data. Our index can be efficiently updated with the underlying updates such as object movements, and supports a variety of object search queries, including k nearest neighbors (returning the k closest objects), keyword k nearest neighbors (returning the k closest objects that satisfy query keywords), and several other variants. Our extensive experimental study, conducted on standard game maps benchmarks and real-world keywords, demonstrates that our approach has up to 2 orders of magnitude faster update times for moving objects compared to state-of-the-art approaches such as navigation mesh and IR-tree. At the same time, query performance of our approach is similar to or better than that of IR-tree and up to two orders of magnitude faster than the other competitor.
Learning-augmented algorithms have been attracting increasing interest, but have only recently been considered in the setting of explorable uncertainty where precise values of uncertain input elements can be obtained by a query and the goal is to minimize the number of queries needed to solve a problem. We study learning-augmented algorithms for sorting and hypergraph orientation under uncertainty, assuming access to untrusted predictions for the uncertain values. Our algorithms provide improved performance guarantees for accurate predictions while maintaining worst-case guarantees that are best possible without predictions. For sorting, our algorithm uses the optimal number of queries for accurate predictions and at most twice the optimal number for arbitrarily wrong predictions. For hypergraph orientation, for any γ≥2, we give an algorithm that uses at most 1+1/γ times the optimal number of queries for accurate predictions and at most γ times the optimal number for arbitrarily wrong predictions. These tradeoffs are the best possible. We also consider different error metrics and show that the performance of our algorithms degrades smoothly with the prediction error in all the cases where this is possible.
In the NP-hard Max c-Cut problem, one is given an undirected edge-weighted graph G and wants to color the vertices of G with c colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with c=2 is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS-Max c-Cut where we are additionally given a vertex coloring f and an integer k and the task is to find a better coloring f' that differs from f in at most k entries, if such a coloring exists; otherwise, f is k-optimal. We show that LS-Max c-Cut presumably cannot be solved in g(k) · nᴼ⁽¹⁾ time even on bipartite graphs, for all c ≥ 2. We then show an algorithm for LS-Max c-Cut with running time O((3eΔ)ᵏ · c · k³ · Δ · n), where Δ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for state-of-the-art heuristics for Max c-Cut. We show that using parameterized local search, the results of this heuristic can be further improved on a set of standard benchmark instances.
One of the most common problem-solving heuristics is by analogy. For a given problem, a solver can be viewed as a strategic walk on its fitness landscape. Thus if a solver works for one problem instance, we expect it will also be effective for other instances whose fitness landscapes essentially share structural similarities with each other. However, due to the black-box nature of combinatorial optimization, it is far from trivial to infer such similarity in real-world scenarios. To bridge this gap, by using local optima network as a proxy of fitness landscapes, this paper proposed to leverage graph data mining techniques to conduct qualitative and quantitative analyses to explore the latent topological structural information embedded in those landscapes. In our experiments, we use the number partitioning problem as the case and our empirical results are inspiring to support the overall assumption of the existence of structural similarity between landscapes within neighboring dimensions. Besides, experiments on simulated annealing demonstrate that the performance of a meta-heuristic solver is similar on structurally similar landscapes.
The Minimum Dominating Set (MDS) problem is a classic NP-hard combinatorial optimization problem with many practical applications. Solving MDS is extremely challenging in computation. Previous work on exact algorithms mainly focuses on improving the theoretical time complexity and existing practical algorithms for MDS are almost based on heuristic search. In this paper, we propose a novel lower bound and an exact algorithm for MDS. The algorithm implements a branch-and-bound (BnB) approach and employs the new lower bound to reduce search space. Extensive empirical results show that the new lower bound is efficient in reduction of the search space and the new algorithm is effective for the standard instances and real-world instances. To the best of our knowledge, this is the first effective BnB algorithm for MDS.
A k-plex of a graph G is an induced subgraph in which every vertex has at most k-1 nonadjacent vertices. The Maximum k-plex Problem (MKP) consists in finding a k-plex of the largest size, which is NP-hard and finds many applications. Existing exact algorithms mainly implement a branch-and-bound approach and improve performance by integrating effective upper bounds and graph reduction rules. In this paper, we propose a refined upper bound, which can derive a tighter upper bound than existing methods, and an inprocessing strategy, which performs graph reduction incrementally. We implement a new BnB algorithm for MKP that employs the two components to reduce the search space. Extensive experiments show that both the refined upper bound and the inprocessing strategy are very efficient in the reduction of search space. The new algorithm outperforms the state-of-the-art algorithms on the tested benchmarks significantly.
Levin Tree Search (LTS) is a search algorithm that makes use of a policy (a probability distribution over actions) and comes with a theoretical guarantee on the number of expansions before reaching a goal node, depending on the quality of the policy. This guarantee can be used as a loss function, which we call the LTS loss, to optimize neural networks representing the policy (LTS+NN). In this work we show that the neural network can be substituted with parameterized context models originating from the online compression literature (LTS+CM). We show that the LTS loss is convex under this new model, which allows for using standard convex optimization tools, and obtain convergence guarantees to the optimal parameters in an online setting for a given set of solution trajectories --- guarantees that cannot be provided for neural networks. The new LTS+CM algorithm compares favorably against LTS+NN on several benchmarks: Sokoban (Boxoban), The Witness, and the 24-Sliding Tile puzzle (STP). The difference is particularly large on STP, where LTS+NN fails to solve most of the test instances while LTS+CM solves each test instance in a fraction of a second. Furthermore, we show that LTS+CM is able to learn a policy that solves the Rubik's cube in only a few hundred expansions, which considerably improves upon previous machine learning techniques.
Recent research on bidirectional heuristic search (BiHS) is based on the must-expand pairs theory (MEP theory), which describes which pairs of nodes must be expanded during the search to guarantee the optimality of solutions. A separate line of research in BiHS has proposed algorithms that use lower bounds that are derived from consistent heuristics during search. This paper links these two directions, providing a comprehensive unifying view and showing that both existing and novel algorithms can be derived from the MEP theory. An extended set of bounds is formulated, encompassing both previously discovered bounds and new ones. Finally, the bounds are empirically evaluated by their contribution to the efficiency of the search
The subgraph isomorphism problem (SIP) is a challenging problem with wide practical applications. In the last decade, despite being a theoretical hard problem, researchers design various algorithms for solving SIP. In this work, we propose three main heuristics and develop an improved exact algorithm for SIP. First, we design a probing search procedure to try whether the search procedure can successfully obtain a solution at first sight. Second, we design a novel matching ordering as a value-ordering heuristic, which uses some useful information obtained from the probing search procedure to preferentially select some promising target vertices. Third, we discuss the characteristics of different propagation methods in the context of SIP and present an adaptive propagation method to make a good balance between these methods. Experimental results on a broad range of real-world benchmarks show that our proposed algorithm performs better than state-of-the-art algorithms for the SIP.
Given a graph, the k-plex is a vertex set in which each vertex is not adjacent to at most k-1 other vertices in the set. The maximum k-plex problem, which asks for the largest k-plex from a given graph, is an important but computationally challenging problem in applications like graph search and community detection. So far, there is a number of empirical algorithms without sufficient theoretical explanations on the efficiency. We try to bridge this gap by defining a novel parameter of the input instance, g_k(G), the gap between the degeneracy bound and the size of maximum k-plex in the given graph, and presenting an exact algorithm parameterized by g_k(G). In other words, we design an algorithm with running time polynomial in the size of input graph and exponential in g_k(G) where k is a constant. Usually, g_k(G) is small and bounded by O(log(|V|)) in real-world graphs, indicating that the algorithm runs in polynomial time. We also carry out massive experiments and show that the algorithm is competitive with the state-of-the-art solvers. Additionally, for large k values such as 15 and 20, our algorithm has superior performance over existing algorithms.
The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective OneMinMax problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points - a small constant factor more than the size of the Pareto front, as suggested for this algorithm - computes the complete Pareto front of the 3-objective OneMinMax benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in the previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives.