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The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from \(\frac{123}{122}\) ([9]) to \(\frac{117}{116}\). This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.

A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours, and a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph, the Colourful Components problem asks whether there exist at most p edges whose removal makes the graph colourful, and the Colourful Partition problem asks whether there exists a partition of the vertex set with at most p parts such that each part induces a colourful component. We study the problems on k-caterpillars (caterpillars with hairs of length at most k) and explore the boundary between polynomial and NP-complete cases. It is known that the problems are NP-complete on 2-caterpillars with unbounded maximum degree. We prove that both problems remain NP-complete on binary 4-caterpillars and on ternary 3-caterpillars. This answers an open question regarding the complexity of the problems on trees with maximum degree at most 5. On the positive side, we give a linear time algorithm for 1-caterpillars with unbounded degree, even if the backbone is a cycle, which outperforms the previous best complexity for paths and widens the class of graphs. Finally, we answer an open question regarding the complexity of Colourful Components on graphs with maximum degree at most 5. We show that the problem is NP-complete on 5-coloured planar graphs with maximum degree 4, and on 12-coloured planar graphs with maximum degree 3. Since the problem can be solved in polynomial-time on graphs with maximum degree 2, the results are the best possible with regard to the maximum degree.

The Dial-a-Ride problem may contain various constraints for pickup-delivery requests, such as time windows and ride time constraints. For a tour, given as a sequence of pickup and delivery stops, there exist polynomial time algorithms to find a schedule respecting these constraints, provided that there exists one. However, if no feasible schedule exists, the natural question is to find a schedule minimising constraint violations. We model a generic fixed-sequence scheduling problem, allowing lateness and ride time violations with linear penalty functions and prove its APX-hardness. We also present an approach leading to a polynomial time algorithm if only the time window constraints can be violated (by late visits). Then, we show that the problem can be solved in polynomial time if all the ride time constraints are bounded by a constant. Lastly, we give a polynomial time algorithm for the instances where all the pickups precede all the deliveries in the sequence of stops.

We consider the complexity of the firefighter problem where a budget of \({b \ge 1}\) firefighters are available at each time step. This problem is proved to be NP-complete even on trees of degree at most three and \({b = 1}\) [10] and on trees of bounded degree \(b+3\) for any fixed \(b \ge 2\) [3]. In this paper, we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any \(b \ge 1\). Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter “pathwidth” and “maximum degree” of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter “cluster vertex deletion”.