## Abstract

In a classic optimization problem the complete input data is assumed to be known to the algorithm. This assumption may not be true anymore in optimization problems motivated by the Internet where part of the input data is private knowledge of independent selfish agents. The goal of algorithmic mechanism design is to provide (in polynomial time) a solution to the optimization problem and a set of incentives for the agents such that disclosing the input data is a dominant strategy for the agents. In case of NP-hard problems, the solution computed should also be a good approximation of the optimum.

In this paper we focus on mechanism design for multi-objective optimization problems. In this setting we are given a main objective function, and a set of secondary objectives which are modeled via budget constraints. Multi-objective optimization is a natural setting for mechanism design as many economical choices ask for a compromise between different, partially conflicting goals. The main contribution of this paper is showing that two of the main tools for the design of approximation algorithms for multi-objective optimization problems, namely approximate Pareto sets and Lagrangian relaxation, can lead to truthful approximation schemes.

By exploiting the method of approximate Pareto sets, we devise truthful deterministic and randomized multi-criteria FPTASs for multi-objective optimization problems whose exact version admits a pseudo-polynomial-time algorithm, as for instance the multi-budgeted versions of minimum spanning tree, shortest path, maximum (perfect) matching, and matroid intersection. Our construction also applies to multi-dimensional knapsack and multi-unit combinatorial auctions. Our FPTASs compute a (1 + ε)-approximate solution violating each budget constraint by a factor (1 + ε).

When feasible solutions induce an independence system, i.e., when subsets of feasible solutions are feasible as well, we present a PTAS (not violating any constraint), which combines the approach above with a novel monotone way to guess the heaviest elements in the optimum solution.

Finally, we present a universally truthful Las Vegas PTAS for minimum spanning tree with a single budget constraint, where one wants to compute a minimum cost spanning tree whose length is at most a given value L. This result is based on the Lagrangian relaxation method, in combination with our monotone guessing step and with a random perturbation step (ensuring low expected running time). This result can be derandomized in the case of integral lengths.

All the mentioned results match the best known approximation ratios, which are however obtained by non-truthful algorithms.

In this paper we focus on mechanism design for multi-objective optimization problems. In this setting we are given a main objective function, and a set of secondary objectives which are modeled via budget constraints. Multi-objective optimization is a natural setting for mechanism design as many economical choices ask for a compromise between different, partially conflicting goals. The main contribution of this paper is showing that two of the main tools for the design of approximation algorithms for multi-objective optimization problems, namely approximate Pareto sets and Lagrangian relaxation, can lead to truthful approximation schemes.

By exploiting the method of approximate Pareto sets, we devise truthful deterministic and randomized multi-criteria FPTASs for multi-objective optimization problems whose exact version admits a pseudo-polynomial-time algorithm, as for instance the multi-budgeted versions of minimum spanning tree, shortest path, maximum (perfect) matching, and matroid intersection. Our construction also applies to multi-dimensional knapsack and multi-unit combinatorial auctions. Our FPTASs compute a (1 + ε)-approximate solution violating each budget constraint by a factor (1 + ε).

When feasible solutions induce an independence system, i.e., when subsets of feasible solutions are feasible as well, we present a PTAS (not violating any constraint), which combines the approach above with a novel monotone way to guess the heaviest elements in the optimum solution.

Finally, we present a universally truthful Las Vegas PTAS for minimum spanning tree with a single budget constraint, where one wants to compute a minimum cost spanning tree whose length is at most a given value L. This result is based on the Lagrangian relaxation method, in combination with our monotone guessing step and with a random perturbation step (ensuring low expected running time). This result can be derandomized in the case of integral lengths.

All the mentioned results match the best known approximation ratios, which are however obtained by non-truthful algorithms.

Original language | English |
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Pages (from-to) | 1263- |

Journal | SIAM Journal on Computing |

Volume | 43 |

Issue number | 4 |

DOIs | |

Publication status | Published - 16 Jul 2014 |