Browsing by Subject "Shortest path problem"

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Open Access
Non-linear pricing by convex duality
(Elsevier, 2015) Pınar, M. Ç.
We consider the pricing problem of a risk-neutral monopolist who produces (at a cost) and offers an infinitely divisible good to a single potential buyer that can be of a finite number of (single dimensional) types. The buyer has a non-linear utility function that is differentiable, strictly concave and strictly increasing. Using a simple reformulation and shortest path problem duality as in Vohra (2011) we transform the initial non-convex pricing problem of the monopolist into an equivalent optimization problem yielding a closed-form pricing formula under a regularity assumption on the probability distribution of buyer types. We examine the solution of the problem when the regularity condition is relaxed in different ways, or when the production function is non-linear and convex. For arbitrary type distributions, we offer a complete solution procedure.
Open Access
On envy-free perfect matching
(Elsevier, 2019) Arbib, C.; Karaşan, Oya Ekin; Pınar, Mustafa
Consider a situation in which individuals –the buyers –have different valuations for the products of a given set. An envy-free assignment of product items to buyers requires that the items obtained by every buyer be purchased at a price not larger than his/her valuation, and each buyer’s welfare (difference between product value and price) be the largest possible. Under this condition, the problem of finding prices maximizing the seller’s revenue is known to be APX -hard even for unit-demand bidders (with several other inapproximability results for different variants), that is, when each buyer wishes to buy at most one item. Here, we focus on Envy-free Complete Allocation, the special case where a fixed number of copies of each product is available, each of the n buyers must get exactly one product item, and all the products must be sold. This case is known to be solvable in O(n4) time. We revisit a series of results on this problem and, answering a question found in Leonard (1983), show how to solve it in O(n3) time by connections to perfect matchings and shortest paths.