The Price of Information in Combinatorial Optimization

Abstract

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay.Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility?A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of n independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.

Beating Greedy for Stochastic Bipartite Matching

“…Once we have a query-commit algorithm for multi-point distributions, we can basically get a price-of-information algorithm by a clean reduction [Sin18]: for each edge e with weight given by the random variable X e and probing cost π e , let τ e be the solution to the equation E[max{(X e − τ e ), 0}] = π e , and let Y e = min(X e , τ e ) be a new random variable. Now run the query-commit algorithm with Y e , and whenever the algorithm queries any copy of the edge e, we probe e, and we only pay π e the first time we probe that edge.…”

“…, X em ). Let M be the collection of all valid bipartite matchings in G. The following lemma is due to Singla [Sin18].…”

“…The analysis of the expected PoI utility of Approx-PoI consists of two parts. First we use the same technique used by Singla [Sin18] to show that we can analyze the expected utility using Y e variables and no query costs instead of using X e variables with query costs. Next we reproduce almost the same analyis by Ehsani et al [EHKS18] to prove the approximation guarantee.…”

Online Pandora’s Boxes and Bandits

“…A classic model for information learning costs is the Pandora's box problem, attributed to Weitzman (Weitzman. 1979), which has the following form.…”

“…2016), who also present additional applications, including to auctions. Very recently Singla (Singla. 2018) generalizes the approach of Kleinberg et al (Robert Kleinberg and Weyl.…”