In this paper, we show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agent's secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agent's secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments.We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (QjjC max ), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, Qjj P C j (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for Qjj P w j C j (minimizing the weighted sum of completion times), we prove a lower bound of 2 p 3 for the best approximation ratio achievable by a truthful mechanism.

Mechanism design seeks algorithms whose inputs are provided by selfish agents who would lie if it were to their advantage. Incentive-compatible mechanisms compel the agents to tell the truth by making it in their self-interest to do so. Often, as in combinatorial auctions, such mechanisms involve the solution of NP-hard problems. Unfortunately, approximation algorithms typically destroy incentive compatibility. Randomized rounding is a commonly used technique for designing approximation algorithms. We devise a version of randomized rounding that is incentivecompatible, giving a truthful mechanism for combinatorial auctions with single parameter agents (e.g., "single minded bidders") that approximately maximizes the social value of the auction. We discuss two orthogonal notions of truthfulness for a randomized mechanism-truthfulness with high probability and in expectation-and give a mechanism that achieves both simultaneously.We consider combinatorial auctions where multiple copies of many different items are on sale, and each bidder i desires a subset Si. Given a set of bids, the problem of finding the allocation of items that maximizes total valuation is the well-known SetPacking problem. This problem is NP-hard, but for the case of items with many identical copies, the optimum can be approximated very well. To turn this approximation algorithm into a truthful auction mechanism, we overcome two problems: We show how to make the allocation algorithm monotone, and give a method to compute the appropriate payments efficiently.

We consider the problem of selecting a low-cost s - t path in a graph where the edge costs are a secret, known only to the various economic agents who own them. To solve this problem, Nisan and Ronen applied the celebrated Vickrey-Clarke-Groves (VCG) mechanism, which pays a premium to induce the edges so as to reveal their costs truthfully. We observe that this premium can be unacceptably high. There are simple instances where the mechanism pays Θ( n ) times the actual cost of the path, even if there is an alternate path available that costs only (1 + ϵ) times as much. This inspires the frugal path problem, which is to design a mechanism that selects a path and induces truthful cost revelation, without paying such a high premium. This article contributes negative results on the frugal path problem. On two large classes of graphs, including those having three node-disjoint s - t paths, we prove that no reasonable mechanism can always avoid paying a high premium to induce truthtelling. In particular, we introduce a general class of min function mechanisms, and show that all min function mechanisms can be forced to overpay just as badly as VCG. Meanwhile, we prove that every truthful mechanism satisfying some reasonable properties is a min function mechanism. Our results generalize to the problem of hiring a team to complete a task, where the analog of a path in the graph is a subset of the agents constituting a team capable of completing the task.

We introduce a new way of measuring and optimizing connectivity in conservation landscapes through time, accounting for both the biological needs of multiple species and the social and financial constraint of minimizing land area requiring additional protection. Our method is based on the concept of network flow; we demonstrate its use by optimizing protected areas in the Western Cape of South Africa to facilitate autogenic species shifts in geographic range under climate change for a family of endemic plants, the Cape Proteaceae. In 2005, P. Williams and colleagues introduced a novel framework for this protected area design task. To ensure population viability, they assumed each species should have a range size of at least 100 km2 of predicted suitable conditions contained in protected areas at all times between 2000 and 2050. The goal was to design multiple dispersal corridors for each species, connecting suitable conditions between time periods, subject to each species' limited dispersal ability, and minimizing the total area requiring additional protection. We show that both minimum range size and limited dispersal abilities can be naturally modeled using the concept of network flow. This allows us to apply well-established tools from operations research and computer science for solving network flow problems. Using the same data and this novel modeling approach, we reduce the area requiring additional protection by a third compared to previous methods, from 4593 km2 to 3062 km , while still achieving the same conservation planning goals. We prove that this is the best solution mathematically possible: the given planning goals cannot be achieved with a smaller area, given our modeling assumptions and data. Our method allows for flexibility and refinement of the underlying climate-change, species-habitat-suitability, and dispersal models. In particular, we propose an alternate formalization of a minimum range size moving through time and use network flow to achieve the revised goals, again with the smallest possible newly protected area (2850 km2). We show how to relate total dispersal distance to probability of successful dispersal, and compute a trade-off curve between this quantity and the total amount of extra land that must be protected.

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