We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. If there is an unlimited supply of items and agents are single minded then we show that finding the revenue maximizing envy free allocation pricing can be solved in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. We define an allocation pricing as if no agent wishes to replace her allocation with the union of the allocations of some set of other agents and her price with the sum of their prices. We show that it is -hard to decide if a given allocation pricing is multi envy free. We also show that revenue maximization multi envy free allocation pricing is hard. Furthermore, we give efficient algorithms and hardness results for various variants of the highway problem.
For the unlimited supply the problem is NP-hard (Briest and Kriesta @cite_11 ). Also the problem is given @math -approximation by @cite_3 . When the length of each interval is bounded by a constant or the valuation of each agent is bounded by a constant, @cite_13 give a fully polynomial time approximation scheme (FPTAS). If the intervals requested by different agents have a nested structure then an FPTAS is possible @cite_3 @cite_11 .
@cite_13 @cite_3 @cite_11
2159007113 2062453462
One of the most fundamental tasks of wireless sensor networks is to provide coverage of the deployment region. In this paper, we study the coverage of a line segment with a set of wireless sensors with adjustable coverage ranges. Each coverage range of a sensor is an interval centered at that sensor whose length is decided by the power the sensor chooses. The objective is to find a range assignment with the minimum cost. There are two variants of the optimization problem. In the discrete variant, each sensor can only choose from a finite set of powers while in the continuous variant, each sensor can choose power from a given interval. For the discrete variant of the problem, we present a polynomial-time exact algorithm. For the continuous variant of the problem, we develop constant-approximation algorithms when the cost for all sensors is proportional to rk for some constant k ≥ 1, where r is the covering radius corresponding to the chosen power. Specifically, if k = 1, we give a simple 1.25-approximation algorithm and a fully polynomial-time approximation scheme (FPTAS); if k > 1, we give a simple 2-approximation algorithm. We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)-approximation algorithm for pricing items to single-minded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [5] who achieve an O(k2) bound. For the case k = 2, where we obtain a 4-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations we on the edges, find prices pi ≥ 0 for the vertices to maximize Σ (pi + pj). e=(i,j):we ≥ pi + pj.We also improve the approximation of [11] from O(log m + log n) to O(log n), where m is the number of bidders and n is the number of items, for the "highway problem" in which all desired subsets are intervals on a line.Our approximation algorithms can be fed into the generic reduction of [2] to yield an incentive-compatible auction with nearly the same performance guarantees so long as the number of bidders is sufficiently large. In addition, we show how our algorithms can be combined with results of Blum and Hartline [3], [4], and Kalai and Vempala [13] to achieve good performance in the online setting, where customers arrive one at a time and each must be presented a set of item prices based only on knowledge of the customers seen so far. We consider a multi-agent scheduling problem on a single machine in which each agent is responsible for his own set of jobs and wishes to minimize the total weighted completion time of his own set of jobs. It is known that the unweighted problem with two agents is NP-hard in the ordinary sense. For this case, we can reduce our problem to a Multi-Objective Shortest-Path (MOSP) problem and this reduction leads to several results including Fully Polynomial Time Approximation Schemes (FPTAS). We also provide an efficient approximation algorithm with a reasonably good worst-case ratio. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.