Brief Annoucement: An algorithm composition scheme preserving monotonicity

1. Brief Annoucement: An algorithm composition scheme preserving monotonicity • Davide Bilò ETH Zürich, Switzerland • Luca Forlizzi Università dell'Aquila, Italy • Luciano Gualà Università di Roma "Tor Vergata", Italy • Guido Proietti Università dell'Aquila & IASI-CNR, Italy • Work partially supported by the ResearchProject GRID.IT, funded by the Italian Ministry of Education,University and Research

2. Introduction • An algorithmic mechanism design problem can be thought as a classic well-formulated optimization problem, but where part of the input is retained by selfish agents • Agents have to be incentivized to disclose to the system their secret data through suitable payments • A mechanism is a pair M = (A, p), where A is an algorithm that, given an instance of the problem and given (possibly false) pieces of information provided by the agents, returns a feasible solution, and p is a scheme which describes the payments provided to the agents • A mechanism is truthful if its payments guarantee that agents are not encouraged to lie

3. One-parameter Mechanisms • Two well known classes of truthful mechanisms: • Vickrey-Clarke-Groves mechanisms, for utilitarian problems (i.e. such that the measure of any feasible solution coincides with the sum of all the agents’ contributions) • One-parameter mechanisms, for problems where the information held by each agent can be expressed throughout a single value. Given the information obtained by the agents, the algorithm of a one-parameter mechanism assigns a work load to each agent, which is a measure of the amount of work incurred by the agent in the computed solution • Many classic optimization problems with selfish agents fall within the class of one-parameter problems

4. Monotone algorithms for one-parameter problems • A nice property of one-parameters problems: whenever an algorithm for the problem enjoys a property known as monotonicity, it is known how to design a payment scheme which ensures truthfulness. • Intuitively, an algorithm (for a minimization problem) is said to be monotone when the work load assigned to each agent is not increasing with respect to the agent’s bid (assuming all others bids remain fixed). • Unfortunately, known algorithms for many classical optimization problems, often turn out to be non-monotone. • No general technique is known to establish the monotonicity of an algorithm, or to monotonize it •  Our contribution: a composition scheme preserving monotonicity

5. Def. 1: An algorithm A is said to be Step-Integral Monotone (SIM) if A is monotone, and the work load function of each agent is a non-negative integer-valued function. Def. 2: A binary demand (BD) problem is a one-parameter problem in which the work load of each agent can be either 0 or 1. A general monotonicity-preserving composition technique

6. Composition Scheme of algorithms A1 and A2 let x1 be the output returned by A1; use x1 to create a suitable instance I for A2; let x2 be the output returned by A2; let x be a solution built from x1 and x2 such that the work load assigned to any agent is the sum of the work loads assigned to it by A1 and A2; return x A general monotonicity-preserving composition technique

7. A general monotonicity-preserving composition technique • Properties of the proposed technique • We show that if • A1 is a SIM algorithm for a one-parameter problem and • A2 is a monotone algorithm for a BD problem. Then the composition of A1 and A2 is a SIM algorithm • We also show that if, in addition to previous hypothesis • A1 and A2 are polynomial-time algorithms • the payments for the BD problem can be computed in polynomial time Then the payments for the problem solved by the composed algorithm can be computed in polynomial time

8. Applications • Using the presented techinique we design efficient approximate truthful mechanisms for several graph traversal problems: • Graphical TSP (approximation ratio 3/2) • Rural Postman Problem (approximation ratio 3/2) • Mixed Chinese Postman Problem (approximation ratio 2)

9. Conclusions • The presented technique: • provides a tool to design monotone algorithms • allows to compute efficiently the payments for the agents