Approximation and Parameterized Runtime Analysis of Evolutionary Algorithms for the Maximum Cut Problem

The runtime analysis of evolutionary algorithms using crossover as search operator has recently produced remarkable results indicating benefits and drawbacks of crossover and illustrating its working principles. Virtually all these results are restricted to upper bounds on the running time of the crossover-based algorithms. This work addresses this lack of lower bounds and rigorously bounds the optimization time of simple algorithms using uniform crossover on the search space $$\{0,1\}^n$$ from below via two novel techniques called decoupling and family graphs. First, a simple steady-state crossover-based evolutionary algorithm without selection pressure is analyzed and shown that after $$O(\mu \log \mu )$$ generations, bit positions are sampled almost independently with marginal probabilities corresponding to the fraction of one-bits at the corresponding position in the initial population. In the presence of weak selective pressure induced by the probabilistic application of tournament selection, it is demonstrated that the inheritance probability at an arbitrary locus quickly approaches a uniform distribution over the initial population up to additive factors that depend on the effect of selection. Afterwards, the algorithm is analyzed by a novel generalization of the family tree technique originally introduced for mutation-only EAs. Using these so-called family graphs, almost tight lower bounds on the optimization time on the OneMax benchmark function are shown.

In this work we deal with a generalized variant of the multi-vehicle covering tour problem (m-CTP). The m-CTP consists of minimizing the total routing cost and satisfying the entire demand of all customers, without the restriction of visiting them all, so that each customer not included in any route is covered. In the m-CTP, only a subset of customers is visited to fulfil the total demand, but a restriction is put on the length of each route and the number of vertices that it contains. This paper tackles a generalized variant of the m-CTP, called the multi-vehicle multi-covering Tour Problem (mm-CTP), where a vertex must be covered several times instead of once. We study a particular case of the mm-CTP considering only the restriction on the number of vertices in each route and relaxing the constraint on the length (mm-CTP-p). A hybrid metaheuristic is developet by combining Genetic Algorithm (GA), Variable Neighborhood Descent method (VND), and a General Variable Neighborhood Search algorithm (GVNS) to solve the problem. Computational experiments show that our approaches are competitive with the Evolutionary Local Search (ELS) and Genetic Algorithm (GA), the methods proposed in the literature.

Binary optimization assumes a pervasive significance in the context of practical applications, such as knapsack problems, maximum cut problems, and critical node detection problems. Existing techniques including mathematical programming, heuristics, evolutionary computation, and neural networks have been employed to tackle binary optimization problems (BOPs), however, they grapple with the challenge of optimizing a large number of binary variables. In this paper, we propose a dimensionality reduction method to assist evolutionary algorithms in solving large-scale BOPs, which is achieved based on neural networks. The proposed method converts the optimization of a large number of binary variables into the optimization of a small number of network weights, resulting in a significant reduction in search space dimensionality. Crucially, the proposed method obviates the necessity for a training process, which eliminates the requirement for a priori knowledge and enhances the search efficiency. On six types of single-and multi-objective BOPs with up to 10 000 000 variables, the proposed method demonstrates superiority over top-tier evolutionary algorithms and neural network-based methods.

We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had perfc~rmance guarantees of ~for MAX CUT and ~for MAX 2SAT. Slight extensions of our analysis lead to a .79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximation algorithm for MAX SAT, where the best previously known approxim ation algorithms had performance guarantees of ~and ~, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of :semidefinite programming in the design of approximation algorithms.

In this paper, we study the Graph Isomorphism problem on graphs of bounded treewidth, bounded degree or bounded bandwidth. There are O(n c) algorithms which solve Graph Isomorphism on graphs which have a bound on the treewidth, degree or bandwidth, but the exponent c depends on this bound, where n is the number of vertices in input graphs. We introduce some new graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve Graph Isomorphism in O(n 2) time for graphs with bounded rooted path distance width, and in O(n 3) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is a NP-complete problem even if the input graphs are restricted to the class of trees. Moreover we show that the rooted path or tree distance width can be computed in O(ne) time and both path and tree distance width can be computed in O(n k+1) time, when they are bounded by a constant k, where e is the number of edges in input graphs. Finally, we study the relationships between the newly introduced parameters and other existing graph parameters.

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of an algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the \((1+1)\) evolutionary algorithm and the static, self-adjusting, and heavy-tailed \((1+(\lambda,\lambda))\) genetic algorithms on the OneMax benchmark. We are optimistic that the question of how to profit from good initial solutions is interesting beyond these first examples.

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of an algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the (1 + 1) evolutionary algorithm and the static, self-adjusting, and heavy-tailed (1 + (λ,λ)) genetic algorithms on the OneMax benchmark. We are optimistic that the question of how to profit from good initial solutions is interesting beyond these first examples.

Evolutionary algorithms are general-purpose randomized search heuristics. They have been successfully applied to many different kinds of search and optimization problems. They are popular in practice since they are easy to implement and easy to apply, and yet often find good solutions in reasonable time. This lead to an increased interest in gaining a thorough understanding of their strengths (and limitations). In the past decades, the theory of evolutionary algorithms has become an important and accepted field of research. This special issue Theory of Evolutionary Computation of Algorithmica contains expanded research from the theory track of the Genetic and Evolutionary Computation Conference and the International Conference on Parallel Problem Solving From Nature. All papers have been enhanced, extended, and undergone a critical journal reviewing process. The result is a collection of seven papers that all deal with theoretical aspects of evolutionary computation in a rigorous and accessible way. It contributes to the development of a better understanding of evolutionary algorithms, paving the way for a more informed application of these algorithms. Ten years after the first publication of a run-time analysis of the most simple evolutionary algorithm, the (1 + 1) evolutionary algorithm, Jens Jagerskupper reconsiders the result. By using the drift analysis method introduced by He and Yao in a sophisticated way, he obtains much more precise bounds on the expected optimization time. Drift analysis is also the topic of Pietro S. Oliveto’s and Carsten Witt’s article. They present a theorem that yields exponential lower bounds on the expected op-

In this paper, we evaluate stochastic-computing simulated annealing (SC-SA) for solving large-scale combinato-rial optimization problems. SC-SA is designed using stochastic computing, where the computation is realized using random bitstream, resulting in fast converging to the global minimum energy of the problems. The proposed SC-SA is compared with a typical SA and existing simulated-annealing (SA) processors on the maximum cut (MAX-CUT) problems, such as Gset that is a benchmark for SA. The simulation results show that SC-SA realizes hundreds of times faster than a typical SA. In addition, SC-SA achieves better MAX-CUT scores than other existing methods on K2000 that is a complete 2000-node optimization problem.

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich, FOCS'99, showed a lower bound of Ω(D + √n) for this problem, where D is the hop-diameter of G.

We revisit the problem of finding k paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the k paths. We provide a [k/2]-approximation algorithm for this problem, improving the best previous approximation factor of k − 1. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of O(n 3/4), where n is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be improved to \(O(\sqrt{n})\). While the problem is NP-hard, and even hard to approximate to within an O(logn) factor, we show that the problem is polynomially solvable when k is a constant. This settles an open problem posed by Omran et al. regarding the complexity of the problem for small values of k. We present most of our results in a more general form where each edge of the graph has a sharing cost and a sharing capacity, and there is vulnerability parameter r that determines the number of times an edge can be used among different paths before it is counted as a shared/vulnerable edge.KeywordsApproximation AlgorithmApproximation FactorNetwork Design ProblemSteiner Tree ProblemSparse GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Suppose that we are given two dominating sets \(D_s\) and \(D_t\) of a graph G whose cardinalities are at most a given threshold k. Then, we are asked whether there exists a sequence of dominating sets of G between \(D_s\) and \(D_t\) such that each dominating set in the sequence is of cardinality at most k and can be obtained from the previous one by either adding or deleting exactly one vertex. This decision problem is known to be PSPACE-complete in general. In this paper, we study the complexity of this problem from the viewpoint of graph classes. We first prove that the problem remains PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs. We then give a general scheme to construct linear-time algorithms and show that the problem can be solved in linear time for cographs, trees, and interval graphs. Furthermore, for these tractable cases, we can obtain a desired sequence if it exists such that the number of additions and deletions is bounded by O(n), where n is the number of vertices in the input graph.

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.