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

PODC 200/ Page 2/24 An algorithmic composition scheme preserving monotonicity 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

PODC 200/ Page 3/24 An algorithmic composition scheme preserving monotonicity 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

PODC 200/ Page 4/24 An algorithmic composition scheme preserving monotonicity 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

PODC 200/ Page 5/24 An algorithmic composition scheme preserving monotonicity A general monotonicity-preserving composition technique 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.

PODC 200/ Page 6/24 An algorithmic composition scheme preserving monotonicity A general monotonicity-preserving composition technique Composition Scheme of algorithms A 1 and A 2 1.let x 1 be the output returned by A 1 ; 2.use x 1 to create a suitable instance I for A 2 ; 3.let x 2 be the output returned by A 2 ; 4.let x be a solution built from x 1 and x 2 such that the work load assigned to any agent is the sum of the work loads assigned to it by A 1 and A 2 ; 5.return x

PODC 200/ Page 7/24 An algorithmic composition scheme preserving monotonicity A general monotonicity-preserving composition technique Properties of the proposed technique We show that if –A 1 is a SIM algorithm for a one-parameter problem and –A 2 is a monotone algorithm for a BD problem. Then the composition of A 1 and A 2 is a SIM algorithm We also show that if, in addition to previous hypothesis –A 1 and A 2 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

PODC 200/ Page 8/24 An algorithmic composition scheme preserving monotonicity Applications Using the presented techinique we design efficient approximate truthful mechanisms for several graph traversal problems: 1.Graphical TSP (approximation ratio 3/2) 2.Rural Postman Problem (approximation ratio 3/2) 3.Mixed Chinese Postman Problem (approximation ratio 2)

PODC 200/ Page 9/24 An algorithmic composition scheme preserving monotonicity Conclusions The presented technique: 1.provides a tool to design monotone algorithms 2.allows to compute efficiently the payments for the agents