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Abstract-In this article we analyze the problem of optimal location of transportation hubs in Warsaw, namely the Park and Ride location problem (P&RP). We take into account the expected travel time using public transport between particular points of the trip. In the currently existing P&R system we have 14 hub locations, and in this case the maximum travel time exceeds 50 minutes. The P&R problem can be reduced to the centers location problem (in our particular approach -the dominating set problem, DS), which is an NP hard problem. In order to determine the optimal locations for P&R two methods: the greedy and the tabu search algorithms were chosen and implemented. According to the computational experiments for the travel time restriction to 50 minutes, we obtain the DS composed of 3 hubs, in contrast to the existing 14 elements. The analysis of the P&R location in time domain is presented in this article in the context of further development of the Warsaw public transportation network, which seems to be interesting.

Abstract-In this paper we consider a hub location problem in a real multimodal public transportation network. This problem is also known as the park-and-ride problem. Hubs stations are special facilities that serve as switches in such a network. In practice the set of hubs has a strategic importance, because all of the traffic that passes through the network can be controlled by these elements. From the theoretical point of view, the minimal hub problem is NP hard. Two different approaches to this problem are presented. The first group of methods bases on the greedy algorithms. In the second group the evolutionary strategy is used. The computational results for these algorithms proved a significant efficiency, what can be clearly expressed in terms of an input data reduction and also in quality measure values for the obtained solutions of this problem. I. INTRODUCTIONET US consider a real public transportation network. This network can be described as a graph [3], [8], [9]. In this graph, each tram/bus stop corresponds to one vertex and any two vertices are adjacent iff they belong to at least one common public transport line. Hence, the set of vertices along a route forms a connected subgraph of the entire graph. Each vertex in this graph is characterized/labeled by a set of numbers of tram/bus lines passing through it. LIn such a graph, a hub set is a subset of vertices, such that any two vertices are connected by a path whose vertices lie in it. Let us define the minimum hub set problem as the problem of finding the hub set of minimal cardinality for a given graph. This collection has a strategic importance for the transportation system, because all the traffic that passes through the network can be controlled by these vertices. Communication hubs are excellent candidates for park-andride (P&R, sometimes P+R) points ([4]) with good connections to the center and other parts of the city and this idea is a basis of presented work. This problem is well known and is commonly used in industry, in particular in such areas as transport, telecommunication, and distributed computing Let us observe that we can try to find the minimum hub set for a given network in two different ways. This problem would be directly reduced to the task of designation the minimum dominating set [6], [14], [15]. Alternatively, we have a possibility of creating a hypergraph, such that its set of vertices is the same as in the original network, and for each set of vertices, which correspond to stops lying along a tram/bus route, we form the hyperedge. In this case, we reduce the considered problem to the search for the minimal transversal in a constructed hypergraph. II. BASIC MATHEMATICAL DEFINITIONSThis section provides some basic notation. A graph is a representation of a set of objects, where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. More formally, a graph is an ordered pair G=(V,...