The travelling preacher, projection, and a lower bound for the stability number of a graph

AbstractThe coflow min–max equality is given a travelling preacher interpretation, and is applied to give a lower bound on the maximum size of a set of vertices, no two of which are joined by an edge

On approximately fair cost allocation in Euclidean TSP games

We consider the problem of allocating the cost of an optimal traveling salesman tour in a fair way among the nodes visited; in particular, we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality. We thereby use the model of TSP games in the sense of cooperative game theory. We give examples showing that the core of such games may be empty, even for the case of Euclidean distances. On the positive, we develop an LP-based allocation rule guaranteeing that no coalition pays more than alpha times its own cost, where alpha is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation, which is also known as the solution over the ''subtour polytope''. A well known conjecture states that alpha<=4/3. We also exhibit examples showing that this ratio cannot be improved below 4/3

Fair Cost Sharing Auction Mechanisms in Last Mile Ridesharing

With rapid growth of transportation demands in urban cities, one major challenge is to provide efficient and effective door-to-door service to passengers using the public transportation system. This is commonly known as the Last Mile problem. In this thesis, we consider a dynamic and demand responsive mechanism for Ridesharing on a non-dedicated commercial fleet (such as taxis). This problem is addressed as two sub problems, the first of which is a special type of vehicle routing problems (VRP). The second sub-problem, which is more challenging, is to allocate the cost (i.e. total fare) fairly among passengers. We propose auction mechanisms where we allow passengers to submit their willing payments. We show that our bidding model is budget-balanced, fairness-preserving, and most importantly, incentive-compatible. We also show how the winner determination problem can be solved efficiently. A series of experimental studies are designed to demonstrate the feasibility and efficiency of our proposed mechanisms

On approximately fair cost allocation in Euclidean TSP games

We consider the problem of allocating the cost of an optimal traveling salesman tour in a fair way among the nodes visited; in particular, we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality. We thereby use the model of TSP games in the sense of cooperative game theory. We give examples showing that the core of such games may be empty, even for the case of Euclidean distances. On the positive, we develop an LP-based allocation rule guaranteeing that no coalition pays more than ff times its own cost, where ff is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation, which is also known as the solution over the &quot;subtour polytope&quot;. A well known conjecture states that ff 4 3 . We also exhibit examples showing that this ratio cannot be improved below 4 3