On profit-maximizing envy-free pricing
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Single-minded unlimited supply pricing on sparse instances
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A quasi-PTAS for unsplittable flow on line graphs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximation algorithms and online mechanisms for item pricing
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
On Hardness of Pricing Items for Single-Minded Bidders
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Combination Can Be Hard: Approximability of the Unique Coverage Problem
SIAM Journal on Computing
On Profit-Maximizing Pricing for the Highway and Tollbooth Problems
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
A quasi-PTAS for profit-maximizing pricing on line graphs
ESA'07 Proceedings of the 15th annual European conference on Algorithms
A sublogarithmic approximation for highway and tollbooth pricing
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
On the hardness of pricing loss-leaders
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The complexity of making unique choices: approximating 1-in-k SAT
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Pricing on paths: a PTAS for the highway problem
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Near-optimal pricing in near-linear time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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In this paper we focus on problems characterized by an input n-node tree and a collection of subpaths. Motivated by the fact that some of these problems admit a very good approximation (or even a poly-time exact algorithm) when the input tree is a path, we develop a decomposition theorem of trees into paths. Our decomposition allows us to partition the input problem into a collection of O(loglogn) subproblems, where in each subproblem either the input tree is a path or there exists a hitting set F of edges such that each path has a non-empty, small intersection with F. When both kinds of subproblems admit constant approximations, our method implies an O(loglogn) approximation for the original problem. We illustrate the above technique by considering two natural problems of the mentioned kind, namely Uniform Tree Tollbooth and Unique Tree Coverage. In Uniform Tree Tollbooth each subpath has a budget, where budgets are within a constant factor from each other, and we have to choose non-negative edge prices so that we maximize the total price of subpaths whose budget is not exceeded. In Unique Tree Coverage each subpath has a weight, and the goal is to select a subset X of edges so that we maximize the total weight of subpaths containing exactly one edge of X. We obtain O(loglogn) approximation algorithms for both problems. The previous best approximations are O(logn/loglogn) by Gamzu and Segev [ICALP'10] and O(logn) by Demaine et al. [SICOMP'08] for the first and second problem, respectively, however both previous results were obtained for much more general problems with arbitrary budgets (weights).