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Improved hardness of approximation for Stackelberg shortestpath pricing
, 2009
"... We consider the Stackelberg shortestpath pricing problem, which is defined as follows. Given a graph G with fixedcost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapes ..."
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We consider the Stackelberg shortestpath pricing problem, which is defined as follows. Given a graph G with fixedcost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest stpath in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortestpath pricing was known to be APXhard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2 − o(1). 1
Specializations and generalizations of the Stackelberg minimum spanning tree game
 In Proc. of the 6th Workshop on Internet and Network Economics (WINE), LNCS 6484
, 2010
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A capacitated cut and choose game
, 2015
"... We consider a two player zero sum game played between a cutter and a chooser consisting of two rounds. Every instance of the game has two parameters C < S known to both players. The cutter makes the first move, and cuts a cake of size S to an arbitrary number of pieces, each of size at most 1. Th ..."
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We consider a two player zero sum game played between a cutter and a chooser consisting of two rounds. Every instance of the game has two parameters C < S known to both players. The cutter makes the first move, and cuts a cake of size S to an arbitrary number of pieces, each of size at most 1. The chooser then replies by choosing a subset of the pieces whose total size is no greater than C. The chooser’s payoff is the total size of pieces captured, and maximizing his payoff involves solving the knapsack instance that results from the cutter’s move. We study the optimal minmax cutter strategy. While the set of possible cuts is infinite (as the size of the pieces are real numbers), we show that for every (S,C) an optimal cutter move exists, and that such a move may be found efficiently. Furthermore, although making an optimal reply is NPhard, the chooser is able to efficiently find a response achieving the game’s minmax value regardless of the cut chosen by the cutter.
Stackelberg Pricing is Hard to Approximate within 2 − epsilon
, 2009
"... Stackelberg Pricing Games is a twolevel combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decadeold shortest path version of this problem which is the first and most studied problem in this family. The game ..."
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Stackelberg Pricing Games is a twolevel combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decadeold shortest path version of this problem which is the first and most studied problem in this family. The game is played on a graph (representing a network) consisting of fixed cost edges and pricable or variable cost edges. The fixed cost edges already have some fixed price (representing the competitor’s prices). Our task is to choose prices for the variable cost edges. After that, a client will buy the cheapest path from a node s to a node t, using any combination of fixed cost and variable cost edges. The goal is to maximize the revenue on variable cost edges. In this paper, we show that the problem is hard to approximate within 2 − ǫ, improving the previous APXhardness result by Joret [to appear in Networks]. Our technique combines the existing ideas with a new insight into the price structure and its relation to the hardness of the instances.
Stackelberg Shortest Path Tree Game, Revisited
, 2012
"... Let G(V, E) be a directed graph with n vertices and m edges. The edges E of G are divided into two types: EF and EP. Each edge of EF has a fixed price. The edges of EP are the priceable edges and their price is not fixed a priori. Let r be a vertex of G. For an assignment of prices to the edges of E ..."
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Let G(V, E) be a directed graph with n vertices and m edges. The edges E of G are divided into two types: EF and EP. Each edge of EF has a fixed price. The edges of EP are the priceable edges and their price is not fixed a priori. Let r be a vertex of G. For an assignment of prices to the edges of EP, the revenue is given by the following procedure: select a shortest path tree T from r with respect to the prices (a tree of cheapest paths); the revenue is the sum, over all priceable edges e, of the product of the price of e and the number of vertices below e in T. Assuming that k = EP  ≥ 2 is a constant, we provide a data structure whose construction takes O(m+n log k−1 n) time and with the property that, when we assign prices to the edges of EP, the revenue can be computed in (log k−1 n). Using our data structure, we save almost a linear factor when computing the optimal strategy in the Stackelberg shortest paths tree game of [D. Bilò and L. Gualà and G. Proietti and P. Widmayer. Computational aspects of a 2Player Stackelberg shortest paths tree game. Proc. WINE 2008].