Additional Dimensions to the Study of Funnels in Combinatorial Landscapes

The global structure of travelling salesman's fitness landscapes has recently revealed the presence of multiple `funnels'. This implies that local optima are organised into several clusters, so that a particular local optimum largely belongs to a particular funnel. Such a global structure can increase search difficulty, especially, when the global optimum is located in a deep, narrow funnel. Our study brings more precision (and dimensions) to the notion of funnels with a data-driven approach using Local Optima Networks and the Chained Lin-Kernighan heuristic. We start by exploring the funnel 'floors', characterising them using the notion of communities from complex networks. We then analyse the more complex funnel 'basins'. Since their depth is relevant to search, we visualise them in 3D. Our study, across a set of TSP instances, reveals a multi-funnel structure in most of them. However, the specific topology varies across instances and relates to search difficulty. Finally, including a stronger perturbation into Chained Lin-Kernighan proved to smooth the funnel structure, reducing the number of funnels and enlarging the valley leading to global optima

The Effect of Landscape Funnels in QAPLIB Instances

The effectiveness of common metaheuristics on combinatorial optimisation problems can be limited by certain characteristics of the fitness landscape. We use the local optima network model to compress the ‘inherent structure’ of a problem space into a network whose structure relates to the empirical hardness of the underlying landscape. Monotonic sequences are used on the local optima networks of a benchmark set of QAP instances (QAPLIB) to expose landscape funnels. The results suggest links between features of these structures and lowered metaheuristic performance

Comparing Communities of Optima with Funnels in Combinatorial Fitness Landscapes

The existence of sub-optimal funnels in combinatorial fitness landscapes has been linked to search difficulty. The exact nature of these structures — and how commonly they appear — is not yet fully understood. Improving our understanding of funnels could help with designing effective diversification mechanisms for a ‘smoothing’ effect, making optimisation easier. We model fitness landscapes as local optima networks. The relationship between communities of local optima found by network clustering algorithms and funnels is explored. Funnels are identified using the notion of monotonic sequences from the study of energy landscapes in theoretical chemistry. NK Landscapes and the Quadratic Assignment Problem are used as case studies. Our results show that communities are linked to funnels. The analysis exhibits relationships between these landscape structures and the performance of trajectory-based metaheuristics such as Simulated Annealing (SA) and Iterated Local Search (ILS). In particular, ILS gets trapped in funnels, and modular communities of optima slow it down. The funnels contribute to lower success for SA. We show that increasing the strength of ILS perturbation helps to ‘smooth’ the funnels and improves performance in multi-funnel landscapes.Authors listed as ECOM Trac

Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study

Phase transitions play an important role in understanding search difficulty in combinatorial optimisation. However, previous attempts have not revealed a clear link between fitness landscape properties and the phase transition. We explore whether the global landscape structure of the number partitioning problem changes with the phase transition. Using the local optima network model, we analyse a number of instances before, during, and after the phase transition. We compute relevant network and neutrality metrics; and importantly, identify and visualise the funnel structure with an approach (monotonic sequences) inspired by theoretical chemistry. While most metrics remain oblivious to the phase transition, our results reveal that the funnel structure clearly changes. Easy instances feature a single or a small number of dominant funnels leading to global optima; hard instances have a large number of suboptimal funnels attracting the search. Our study brings new insights and tools to the study of phase transitions in combinatorial optimisation

Local Optima Networks for the Permutation Flowshop Scheduling Problem: Makespan vs. Total Flow Time

Local Optima Networks were proposed to understand the structure of combinatorial landscapes at a coarse-grained level. We consider a compressed variant of such networks with features that are meaningful for the study of search difficulty in the context of local search. In particular, we investigate different landscapes of the Permutation Flowshop Scheduling Problem. The insert and 2-exchange neighbourhoods are considered, and two different objective functions are taken into account: the makespan and the total flow time. The aim is to analyse the network features in order to find differences between the landscape structures, giving insights about which features impact algorithm performance. We evaluate the correlation between landscape properties and the performance of an Iterated Local Search algorithm. Visualisation of the network structure is also given, where evident differences between the makespan and total flow time are observed

Rigidity and flexibility of biological networks

The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl

Coarse-Grained Barrier Trees of Fitness Landscapes

Recent literature suggests that local optima in fitness landscapes are clustered, which offers an explanation of why perturbation-based metaheuristics often fail to find the global optimum: they become trapped in a sub-optimal cluster. We introduce a method to extract and visualize the global organization of these clusters in form of a barrier tree. Barrier trees have been used to visualize the barriers between local optima basins in fitness landscapes. Our method computes a more coarsely grained tree to reveal the barriers between clusters of local optima. The core element is a new variant of the flooding algorithm, applicable to local optima networks, a compressed representation of fitness landscapes. To identify the clusters, we apply a community detection algorithm. A sample of 200 NK fitness landscapes suggests that the depth of their coarse-grained barrier tree is related to their search difficulty

Tunnelling Crossover Networks for the Asymmetric TSP

Local optima networks are a compact representation of fitness landscapes that can be used for analysis and visualisation. This paper provides the first analysis of the Asymmetric Travelling Salesman Problem using local optima networks. These are generated by sampling the search space by recording the progress of an existing evolutionary algorithm based on the Generalised Asymmetric Partition Crossover. They are compared to networks sampled through the Chained Lin-Kernighan heuristic across 25 instances. Structural differences and similarities are identified, as well as examples where crossover smooths the landscape

Anatomy of the Local Optima Level in Combinatorial Optimisation

Many situations in daily life represent complex combinatorial optimisation problems. These include issues such as efficient fuel consumption, nurse scheduling, or distribution of humanitarian aid. There are many algorithms that attempt to solve these problems but the ability to understand their likely performance on a given problem is still lacking. Fitness landscape analysis identifies some of the reasons why metaheuristic algorithms behave in a particular way. The Local Optima Network (LON) model, proposed in 2008, encodes local optima connectivity in fitness landscapes. In this approach, nodes are local optima and edges encode transitions between these optima. A LON provides a static image of the dynamics of algorithm-problem inter- play. Analysing these structures provides insights into the reactions between optimisation problems and metaheuristic search algorithms. This thesis proposes that analysis of the local optima space of combinatorial fitness landscapes encoded using a LON provides important information concerning potential search algorithm performance. It considers the question as to whether or not features of LONs can contribute to explaining or predicting the outcome of trying to optimise an associated combinatorial problem. Topological landscape features of LONs are proposed, analysed and compared. Benchmark and novel problem instances are studied; both types of problem are sampled and in some cases exhaustively-enumerated such that LONs can be extracted for analysis. Investigations into the nature and biases of LON construction algorithms are conducted and compared. Contributions include aligning fractal geometry to the study of LONs; proposals for novel ways to compute fractal dimension from these structures; comparing the power of different LON construction algorithms for explaining algorithm performances; and analysing the interplay between algorithmic operations and infeasible regions in the local optima space using LONs as a tool. Throughout the thesis, large scale structural patterns in fitness landscapes are shown to be strongly linked with metaheuristic algorithm performance. This includes arrangements of local optima funnel structures; spatial and geometric complexity in the LON (measured by their fractal dimensionality) and fitness levels in the space of local optima. These features are demonstrated to have explanatory or predictive ability with respect to algorithm performance for the underlying combinatorial problems. The results presented here indicate that large topological patterns in fitness landscapes are important during metaheuristic search algorithm design. In many cases they are incontrovertibly linked to the success of the algorithm. These results indicate that use of the suggested fitness landscape measures would be highly beneficial when considering the design of search algorithms for a given problem domain

Networks in molecular evolution

Networks are a common theme at all levels of molecular evolution: Networks of metastable states and their connecting saddle points determine structure and folding kinetics of biopolymers. Neutral networks in sequence space explain the evolvability of both nucleic acids and polypeptides by linking Darwinian selection with neutral drift. Interacting replicators, be they simple molecules or highly complex mammals, form intricate ecological networks that are crucial for their survival. Chemical reactions are collected in extensive metabolic networks by means of specific enzymes; both the enzymes and the chemical reaction network that they govern undergoes evolutionary changes. Networks of gene regulation, protein-protein interaction, and cell signaling form the physico-chemical basis for morphogenesis and development. The nervous systems of higher animals form another distinct level of network architecture. We are beginning to understand the structure and function of each of the individual levels in some detail. Yet, their interplay largely remains still in the dark