More effective randomized search heuristics for graph coloring through dynamic optimization

Dynamic optimization problems have gained significant attention in evolutionary computation as evolutionary algorithms (EAs) can easily adapt to changing environments. We show that EAs can solve the graph coloring problem for bipartite graphs more efficiently by using dynamic optimization. In our approach the graph instance is given incrementally such that the EA can reoptimize its coloring when a new edge introduces a conflict. We show that, when edges are inserted in a way that preserves graph connectivity, Randomized Local Search (RLS) efficiently finds a proper 2-coloring for all bipartite graphs. This includes graphs for which RLS and other EAs need exponential expected time in a static optimization scenario. We investigate different ways of building up the graph by popular graph traversals such as breadth-first-search and depth-first-search and analyse the resulting runtime behavior. We further show that offspring populations (e. g. a (1 + λ) RLS) lead to an exponential speedup in λ. Finally, an island model using 3 islands succeeds in an optimal time of Θ(m) on every m-edge bipartite graph, outperforming offspring populations. This is the first example where an island model guarantees a speedup that is not bounded in the number of islands

Time complexity analysis of RLS and (1 + 1) EA for the edge coloring problem

The edge coloring problem asks for an assignment of colors to edges of a graph such that no two incident edges share the same color and the number of colors is minimized. It is known that all graphs with maximum degree Δ can be colored with Δ or Δ + 1 colors, but it is NP-hard to determine whether Δ colors are sufficient. We present the first runtime analysis of evolutionary algorithms (EAs) for the edge coloring problem. Simple EAs such as RLS and (1+1) EA efficiently find (2Δ - 1)-colorings on arbitrary graphs and optimal colorings for even and odd cycles, paths, star graphs and arbitrary trees. A partial analysis for toroids also suggests efficient runtimes in bipartite graphs with many cycles. Experiments support these findings and investigate additional graph classes such as hypercubes, complete graphs and complete bipartite graphs. Theoretical and experimental results suggest that simple EAs find optimal colorings for all these graph classes in expected time O(Δℓ2m log m), where m is the number of edges and ℓ is the length of the longest simple path in the graph

Time complexity analysis of randomized search heuristics for the dynamic graph coloring problem

We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1) Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient

Exploring the feature space of TSP instances using quality diversity

Generating instances of different properties is key to algorithm selection methods that differentiate between the performance of different solvers for a given combinatorial optimization problem. A wide range of methods using evolutionary computation techniques has been introduced in recent years. With this paper, we contribute to this area of research by providing a new approach based on quality diversity (QD) that is able to explore the whole feature space. QD algorithms allow to create solutions of high quality within a given feature space by splitting it up into boxes and improving solution quality within each box. We use our QD approach for the generation of TSP instances to visualize and analyze the variety of instances differentiating various TSP solvers and compare it to instances generated by established approaches from the literature.Jakob Bossek, Frank Neuman

Evolutionary Diversity Optimization and the Minimum Spanning Tree Problem

In the area of evolutionary computation the calculation of diverse sets of high-quality solutions to a given optimization problem has gained momentum in recent years under the term evolutionary diversity optimization. Theoretical insights into the working principles of baseline evolutionary algorithms for diversity optimization are still rare. In this paper we study the well-known Minimum Spanning Tree problem (MST) in the context of diversity optimization where population diversity is measured by the sum of pairwise edge overlaps. Theoretical results provide insights into the fitness landscape of the MST diversity optimization problem pointing out that even for a population of µ = 2 fitness plateaus (of constant length) can be reached, but nevertheless diverse sets can be calculated in polynomial time. We supplement our theoretical results with a series of experiments for the unconstrained and constraint case where all solutions need to fulfill a minimal quality threshold. Our results show that a simple (µ + 1)-EA can effectively compute a diversified population of spanning trees of high quality.Jakob Bossek, Frank Neuman

Generating Instances with Performance Differences for More Than Just Two Algorithms

In recent years, Evolutionary Algorithms (EAs) have frequently been adopted to evolve instances for optimization problems that pose difficulties for one algorithm while being rather easy for a competitor and vice versa. Typically, this is achieved by either minimizing or maximizing the performance difference or ratio which serves as the fitness function. Repeating this process is useful to gain insights into strengths/weaknesses of certain algorithms or to build a set of instances with strong performance differences as a foundation for automatic per-instance algorithm selection or configuration. We contribute to this branch of research by proposing fitness-functions to evolve instances that show large performance differences for more than just two algorithms simultaneously. As a proof-of-principle, we evolve instances of the multi-component Traveling Thief Problem (TTP) for three incomplete TTP-solvers. Our results point out that our strategies are promising, but unsurprisingly their success strongly relies on the algorithms’ performance complementarity.Jakob Bossek, Markus Wagne

On the benefits of biased edge-exchange mutation for the multi-criteria spanning tree problem

Research has shown that for many single-objective graph problems where optimum solutions are composed of low weight sub-graphs, such as the minimum spanning tree problem (MST), mutation operators favoring low weight edges show superior performance. Intuitively, similar observations should hold for multi-criteria variants of such problems. In this work, we focus on the multi-criteria MST problem. A thorough experimental study is conducted where we estimate the probability of edges being part of non-dominated spanning trees as a function of the edges' non-domination level or domination count, respectively. Building on gained insights, we propose several biased one-edge-exchange mutation operators that differ in the used edge-selection probability distribution (biased towards edges of low rank). Our empirical analysis shows that among different graph types (dense and sparse) and edge weight types (both uniformly random and combinations of Euclidean and uniformly random) biased edge-selection strategies perform superior in contrast to the baseline uniform edge-selection. Our findings are in particular strong for dense graphs

Diversifying greedy sampling and evolutionary diversity optimisation for constrained monotone submodular functions

Submodular functions allow to model many real-world optimisation problems. This paper introduces approaches for computing diverse sets of high quality solutions for submodular optimisation problems with uniform and knapsack constraints. We first present diversifying greedy sampling approaches and analyse them with respect to the diversity measured by entropy and the approximation quality of the obtained solutions. Afterwards, we introduce an evolutionary diversity optimisation (EDO) approach to further improve diversity of the set of solutions.We carry out experimental investigations on popular submodular benchmark problems and analyse trade-offs in terms of solution quality and diversity of the resulting solution sets.Aneta Neumann, Jakob Bossek, Frank Neuman

Breeding Diverse Packings for the Knapsack Problem by Means of Diversity-Tailored Evolutionary Algorithms.

In practise, it is often desirable to provide the decision-maker with a rich set of diverse solutions of decent quality instead of just a single solution. In this paper we study evolutionary diversity optimization for the knapsack problem (KP). Our goal is to evolve a population of solutions that all have a profit of at least (1 - ϵ) · OPT, where OPT is the value of an optimal solution. Furthermore, they should differ in structure with respect to an entropy-based diversity measure. To this end we propose a simple (μ + 1)-EA with initial approximate solutions calculated by a well-known FPTAS for the KP. We investigate the effect of different standard mutation operators and introduce biased mutation and crossover which puts strong probability on flipping bits of low and/or high frequency within the population. An experimental study on different instances and settings shows that the proposed mutation operators in most cases perform slightly inferior in the long term, but show strong benefits if the number of function evaluations is severely limited.Jakob Bossek, Aneta Neumann, Frank Neuman