Cost-optimal Planning, Delete Relaxation, Approximability, and Heuristics

Cost-optimal planning is a very well-studied topic within planning, and it has proven to be computationally hard both in theory and in practice. Since cost-optimal planning is an optimisation problem, it is natural to analyse it through the lens of approximation. An important reason for studying cost-optimal planning is heuristic search; heuristic functions that guide the search in planning can often be viewed as algorithms solving or approximating certain optimisation problems. Many heuristic functions (such as the ubiquitious h+ heuristic) are based on delete relaxation, which ignores negative effects of actions. Planning for instances where the actions have no negative effects is often referred to as monotone planning. The aim of this article is to analyse the approximability of cost-optimal monotone planning, and thus the performance of relevant heuristic functions. Our findings imply that it may be beneficial to study these kind of problems within the framework of parameterised complexity and we initiate work in this direction.

The Parameterised Complexity of Computing the Maximum Modularity of a Graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be $$\textsf {NP}$$ NP -complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to $$\textsf {FPT}$$ FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

Parameterised Enumeration for Modification Problems

Recently, Creignou et al. (Theory Comput. Syst. 2017), introduced the class Delay FPT into parameterised complexity theory in order to capture the notion of efficiently solvable parameterised enumeration problems. In this paper, we propose a framework for parameterised ordered enumeration and will show how to obtain enumeration algorithms running with an FPT delay in the context of general modification problems. We study these problems considering two different orders of solutions, namely, lexicographic order and order by size. Furthermore, we present two generic algorithmic strategies. The first one is based on the well-known principle of self-reducibility and is used in the context of lexicographic order. The second one shows that the existence of a neighbourhood structure among the solutions implies the existence of an algorithm running with FPT delay which outputs all solutions ordered non-decreasingly by their size.

A Parameterised Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. In this paper, we analyse the runtime of some evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem and the generalised travelling salesperson problem in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl ( 2012 ) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) evolutionary algorithm working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the problem can be solved in fixed-parameter time with the global structure representation. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other’s hard instances very efficiently. For the generalised travelling salesperson problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) evolutionary algorithm working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem.