In set-system auctions , there are several overlapping teams of agents, and a task that can be completed by any of these teams. The buyer's goal is to hire a team and pay as little as possible. Recently, Karlin, Kempe and Tamir introduced a new definition of frugality ratio for this setting. Informally, the frugality ratio is the ratio of the total payment of a mechanism to perceived fair cost. In this paper, we study this together with alternative notions of fair cost, and how the resulting frugality ratios relate to each other for various kinds of set systems. We propose a new truthful polynomial-time auction for the vertex cover problem (where the feasible sets correspond to the vertex covers of a given graph), based on the local ratio algorithm of Bar-Yehuda and Even. The mechanism guarantees to find a winning set whose cost is at most twice the optimal. In this situation, even though it is NP-hard to find a lowest-cost feasible set, we show that local optimality of a solution can be used to derive frugality bounds that are within a constant factor of best possible. To prove this result, we use our alternative notions of frugality via a bootstrapping technique, which may be of independent interest.
Vertex-cover auctions have been studied in the past by Talwar @cite_9 and Calinescu @cite_3 . Both of these papers are based on the definition of frugality ratio used in @cite_5 ; as mentioned before, this means that their results only apply to bipartite graphs. Talwar @cite_9 shows that the frugality ratio of VCG is at most @math . However, since finding the cheapest vertex cover is an NP-hard problem, the VCG mechanism is computationally infeasible. The first (and, to the best of our knowledge, only) paper to investigate polynomial-time truthful mechanisms for vertex cover is @cite_3 . That paper studies an auction that is based on the greedy allocation algorithm, which has an approximation ratio of @math . While the main focus of @cite_3 is the more general set cover problem, the results of @cite_3 imply a frugality ratio of @math for vertex cover. Our results improve on those of @cite_9 as our mechanism is polynomial-time computable, as well as on those of @cite_3 , as our mechanism has a better approximation ratio, and we prove a stronger bound on the frugality ratio; moreover, this bound also applies to the mechanism of @cite_3 .
@cite_5 @cite_9 @cite_3
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In a STACS 2003 paper, Talwar analyzes the overpayment the VCG mechanism incurs for ensuring truthfulness in auctions. Among other results, he studies k-Set Cover (given a universe U and a collection of sets S 1 , S 2 , ? , S m , each having a cost c ( S i ) and at most k elements of U, find a minimum cost subcollection - a cover - whose union equals U) and shows that the payment of the VCG mechanism is at most k ? c ( OPT ' ) , where OPT ' is the best cover disjoint from the optimum cover OPT . The VCG mechanism requires finding an optimum cover. For k ? 3 , k-Set Cover is known to be NP-hard, and thus truthful mechanisms based on approximation algorithms are desirable. We show that the payment incurred by two mechanisms based on approximation algorithms (including the Greedy algorithm) is bounded by ( k - 1 ) c ( OPT ) + k ? c ( OPT ' ) . The same approximation algorithms have payment bounded by k ( c ( OPT ) + c ( OPT ' ) ) when applied to more general set systems, which include k-Polymatroid Cover, a problem related to Steiner Tree computations. If q is such that an element in a k-Set Cover instance appears in at most q sets, we show that the total payment based on our algorithms is bounded by q ? k 2 times the total payment of the VCG mechanism. The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1 n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube 0, 1 n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3 2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4. We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. This mechanism introduces a way to implement interesting truthful outcomes in settings where monetary payments are not an option. For a multi-dimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1 e ≈ 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. In addition to this, we uncover a connection between the Partial Allocation mechanism and VCG-based mechanism design. We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded. A vertex cover of a graph is a subset of vertices such that each edge has at least one endpoint in the subset. Determining the minimum vertex cover is a well-known NP-complete problem in a sequential setting. Several techniques, e.g., depth-first search, a local ratio theorem, and semidefinite relaxation, have given good approximation algorithms. However, some of them cannot be applied to a distributed setting, in particular self-stabilizing algorithms. Thus only a 2-approximation solution based on a self-stabilizing maximal matching has been obviously known until now. In this paper we propose a new self-stabilizing vertex cover algorithm that achieves (2–1 Δ)-approximation ratio, where Δ is the maximum degree of a given network. We first introduce a sequential (2–1 Δ)-approximation algorithm that uses a maximal matching with the high-degree-first order of vertices. Then we present a self-stabilizing algorithm based on the same idea, and show that the output of the algorithm is the same as that of the sequential one.