This paper concerns the design of mechanisms for online scheduling in which agents bid for access to a re-usable resource such as processor time or wireless network access. Each agent is assumed to arrive and depart dynamically, and in the basic model require the resource for one unit of time. We seek mechanisms that are truthful in the sense that truthful revelation of arrival, departure and value information is a dominant strategy, and that are online in the sense that they make allocation decisions without knowledge of the future. First, we provide two characterizations for the class of truthful online allocation rules. The characterizations extend beyond the typical single-parameter settings, and formalize the role of restricted misreporting in reversing existing price-based characterizations. Second, we present an online auction for unit-length jobs that achieves total value that is 2-competitive with the maximum offline value. We prove that no truthful deterministic online mechanism can achieve a better competitive ratio. Third, we consider revenue competitiveness and prove that no deterministic truthful online auction has revenue that is constant-competitive with that of the offline Vickrey-Clarke-Groves (VCG) mechanism We provide a randomized online auction that achieves a competitive ratio of O(log h), where h is the ratio of maximum value to minimum value among the agents; this mechanism does not require prior knowledge of h. Finally, we generalize our model to settings with multiple re-usable goods and to agents with different job lengths.

We study a limited-supply online auction problem, in which an auctioneer has k goods to sell and bidders arrive and depart dynamically. We suppose that agent valuations are drawn independently from some unknown distribution and construct an adaptive auction that is nevertheless value-and time-strategyproof. For the k = 1 problem we have a strategyproof variant on the classic secretary problem. We present a 4-competitive (e-competitive) strategyproof online algorithm with respect to offline Vickrey for revenue (efficiency). We also show (in a model that slightly generalizes the assumption of independent valuations) that no mechanism can be better than 3/2-competitive (2-competitive) for revenue (efficiency). Our general approach considers a learning phase followed by an accepting phase, and is careful to handle incentive issues for agents that span the two phases. We extend to the k > 1 case, by deriving strategyproof mechanisms which are constant-competitive for revenue and efficiency. Finally, we present some strategyproof competitive algorithms for the case in which adversary uses a distribution known to the mechanism.

ABSTRACT:With random inputs, certain decision problems undergo a "phase transition." We prove similar behavior in an optimization context. Given a conjunctive normal form (CNF) formula F on n variables and with m k-variable clauses, denote by max F the maximum number of clauses satisfiable by a single assignment of the variables. (Thus the decision problem k-SAT is to determine if max F is equal to m.) With the formula F chosen at random, the expectation of max F is trivially bounded by (3/4)m ‫ޅ‬ max F m. We prove that for random formulas with m ϭ cn clauses: for constants c Ͻ 1, ‫ޅ‬ max F is cn Ϫ ⌰(1/n); for large c, it approaches ((3/4)c ϩ ⌰( ͌ c))n; and in the "window" c ϭ 1 ϩ ⌰(n Ϫ1/3 ), it is cn Ϫ ⌰(1). Our full results are more detailed, but this already shows that the optimization problem MAX 2-SAT undergoes a phase transition just as the 2-SAT decision problem does, and at the same critical value c ϭ 1. Most of our results are established without reference to the analogous propositions for decision 2-SAT, and can be used to reproduce them.We consider "online" versions of MAX 2-SAT, and show that for one version the obvious greedy algorithm is optimal; all other natural questions remain open. We can extend only our simplest MAX 2-SAT results to MAX k-SAT, but we conjecture a "MAX k-SAT limiting function conjecture" analogous to the folklore "satisfiability threshold conjecture," but open even for k ϭ 2. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. We also prove analogous results for random MAX CUT.

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Abstract-We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem.We present (2 − )-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier.

Abstract. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-ECSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log 4 n)-approximation algorithm for MPb-EC. This gives an O(log 4 n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an O( √ n) approximation algorithm for MPk-ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2 log 1−ε n ) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time. IntroductionWireless multihop networks are an important subject of study due to their extensive applications (see e.g., [8,24] performing network tasks while minimizing the power consumption of the radio transmitters in the network. In ad-hoc networks, a range assignment to radio transmitters means to assign a set of powers to mobile devices. We consider finding a range assignment for the nodes of a network such that the resulting communication network satisfies some prescribed properties, and such that the total power is minimized. Specifically, we consider "min-power" variants of three extensively studied "min-cost" problems: the b-Edge Cover problem and the k-Connected Spanning Subgraph Problem in undirected networks, and the k-Edge-Disjoint Paths problem in directed networks.In wired networks, generally we want to find a subgraph with the minimum cost instead of the minimum power. This is the main difference between the optimization problems for wired versus wireless networks. The power model for undirected graphs corresponds to static symmetric multi-hop ad-hoc wireless networks with omnidirectional transmitters. This model is justified and used in several other papers [3,4,14].An important network task is assuring high fault-tolerance ( [1-4, 11, 18]).The simplest version is when we require the network to be connected. In this case, the min-cost variant is just the min-cost spanning tree problem, while the min-power variant is NP-hard even in the Euclidean plane [9]. There are several localized and distributed heuristics to find the range assignment to keep the network connected [18,24,25]. Constant approximation guarantees for the min-power spanning tree problem are given in [4,14]. For general k, the bestPower Optimization for Connectivity Problems 3 previously known approximation ratio fo...

In this paper we study the prize-collecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prize-collecting versions of various problems. These problems are studied on a case by case basis by Bienstock et al. [5] by applying an LP-rounding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primal-dual schema (without any need to solve an LP). We fuse the primal-dual schema with Farkas lemma to obtain a combinatorial 3-approximation algorithm for the PrizeCollecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [12] for solving a special case of Kelly's problem [21] of pricing edges.We also consider the k-forest problem, a generalization of k-MST and k-Steiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the k-forest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n 1/6−ε ) for kforest requires substantially new ideas including improving the approximation factor O(n 1/3−ε ) for the notorious densest k-subgraph problem. We note that k-forest and prize-collecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former.