One of the classic problems in online decision-making is the secretary problem, where the goal is to hire the best secretary out of n rankable applicants or, in a natural extension, to maximize the probability of selecting the largest number from a sequence arriving in random order. Many works have considered generalizations of this problem where one can accept multiple values subject to a combinatorial constraint. The seminal work of Babaioff, Immorlica, Kempe, and Kleinberg (SODA'07, JACM'18) proposed the matroid secretary conjecture, suggesting that there exists an O(1)-competitive algorithm for the matroid constraint, and many works since have attempted to obtain algorithms for both general matroids and specific classes of matroids. The ultimate goal of these results is to obtain an e-competitive algorithm, and the strong matroid secretary conjecture states that this is possible for general matroids. One of the most important classes of matroids is the graphic matroid, where a set of edges in a graph is deemed independent if it contains no cycle. Given the rich combinatorial structure of graphs, obtaining algorithms for these matroids is often seen as a good first step towards solving the problem for general matroids. For matroid secretary, Babaioff et al. (SODA'07, JACM'18) first studied graphic matroid case and obtained a 16-competitive algorithm. Subsequent works have improved the competitive ratio, most recently to 4 by Soto, Turkieltaub, and Verdugo (SODA'18). In this paper, we break the 4-competitive barrier for the problem, obtaining a new algorithm with a competitive ratio of 3.95. For the special case of simple graphs (i.e., graphs that do not contain parallel edges) we further improve this to 3.77. Intuitively, solving the problem for simple graphs is easier as they do not contain cycles of length two. A natural question that arises is whether we can obtain a ratio arbitrarily close to e by assuming the graph has a large enough girth. We answer this question affirmatively, proving that one can obtain a competitive ratio arbitrarily close to e even for constant values of girth, providing further evidence for the strong matroid secretary conjecture. We further show that this bound is tight: for any constant g, one cannot obtain a competitive ratio better than e even if we assume that the input graph has girth at least g. To our knowledge, such a bound was not previously known even for simple graphs.

The delegated choice problem, introduced by Armstrong and Vickers (ECTA'10), involves a principal delegating a decision-making of selecting an element among n to an agent with potentially misaligned utility. To mitigate the agent's selfish behavior, the principal commits to acceptable sets and utilities in advance. Kleinberg and Kleinberg (EC'18) observed a novel connection to prophet inequality, a central problem in optimal stopping theory, stating that the delegated choice problem is equivalent to a version of prophet inequality with oblivious stopping rules in the single choice setting. This amplifies the significance of prophet inequality, originally cemented by its fruitful connection to numerous problems including posted pricing in the online auction for digital goods/multi-dimensional mechanisms, stochastic optimization, secretary problem, and Pandora's box. A fundamental question in this context is, whether the connection between prophet inequality and delegated choice follows the same path even under arbitrary combinatorial constraints.

We study the problem of delegated choice with inspection cost (DCIC), which is a variant of the delegated choice problem by Kleinberg and Kleinberg (EC'18) as well as an extension of the Pandora's box problem with nonobligatory inspection (PNOI) by Doval (JET'18). In our model, an agent may strategically misreport the proposed element's utility, unlike the standard delegated choice problem which assumes that the agent truthfully reports the utility for the proposed alternative. Thus, the principal needs to inspect the proposed element possibly along with other alternatives to maximize its own utility, given an exogenous cost of inspecting each element. Further, the delegation itself incurs a fixed cost, thus the principal can decide whether to delegate or not and inspect by herself.

We study the power of posted pricing mechanisms for Bayesian online optimization problems subject to combinatorial feasibility constraints. When the objective is to maximize social welfare, the problem is widely studied in the literature on prophet inequalities. While most (though not all) existing algorithms for prophet inequalities are implemented using a pricing mechanism, whether or not this can be done in general is unknown, and was formally left as an open question by Dutting, Feldman, Kesselheim, and Lucier (FOCS 2017, SICOMP 2020). Understanding the power and limitations of posted prices is important from a mechanism design perspective because any posted price mechanism is truthful, and is also interesting in its own right as it can guide future research on prophet inequalities.

We study contention resolution (CR) on a shared channel modelled as a game with selfish players. There are n agents and the adversary chooses some k smaller than n of them as players. Each participating player in a CR game has a packet to transmit. A transmission is successful if it is performed as the only one at a round. Each player aims to minimize its packet latency. We introduce the notion of adversarial equilibrium (AE), which incorporates adversarial selection of players. We develop efficient deterministic communication algorithms that are also AE. We characterize the price of anarchy in the CR games with respect to AE.

We study the Combinatorial Group Testing (CGT) problem in a novel game-theoretic framework, with a solution concept of Adversarial Equilibrium (AE). In this new framework, we have $n$ selfish agents corresponding to the elements of the universe $[n] =\{0,1,\ldots,n-1\}$ and a hidden set $K \subseteq [n]$ of active agents of size $|K| = k \ll n$. In each round of the game, each active agent decides if it is present in a query $Q \subseteq [n]$, and all agents receive feedback on $Q \cap K$. The goal of each active agent is to assure that its id could be learned from the feedback as early as possible. We present a comprehensive set of results in this new game, where we design and analyze adaptive algorithmic strategies of agents which are AE's. In particular, if $k$ is known to the agents, then we design adaptive AE strategies with provably near optimal learning time of $O(k \log(n/k))$. In the case of unknown $k$, we design an adaptive AE strategies with learning time of order $n^k$, and we prove a lower bound of $\Omega(n)$ on the learning time of any such algorithmic strategies. This shows a strong separations between the two models of known and unknown $k$, as well as between the classic CGT, i.e., without selfish agents, and our game theoretic CGT model.

We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. It is known for its inherent hardness of approximation. We focus on the well known minimum degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been very widely studied, where it is augmented with an advice that tells the greedy which minimum degree vertex to choose if it is not unique. Our main contribution is a new mathematical theory for the design of such greedy algorithms with efficiently computable advice and for the analysis of their approximation ratios. With this new theory we obtain the ultimate approximation ratio of 5/4 for greedy on graphs with maximum degree 3, which completely solves the open problem from the paper by Halldórsson and Yoshihara (1995). Our algorithm is the fastest currently known algorithm with this approximation ratio on such graphs. We also obtain a simple and short proof of the (D+2)/3-approximation ratio of any greedy on graphs with maximum degree D, the result proved previously by Halldórsson and Radhakrishnan (1994). We almost match this ratio by showing a lower bound of (D+1)/3 on the ratio of any greedy algorithm that can use any advice. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 6/5-approximation algorithm than the one currently known. We complement our positive, upper bound results with negative, lower bound results which prove that the problem of designing good advice for greedy is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on such previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of greedy advice is non-trivial.

The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level, our model comprises a graph whose nodes represent the agents, which can be thought of as the sources of pollution in the network. The edges between agents represent the effect of spread of pollution. The government who is the regulator, is responsible for the maximization of the social welfare and sets bounds on the levels of emitted pollution in both local areas as well as globally in the whole network. We first prove that the above optimization problem is NP-hard even on some special cases of graphs such as trees. We then turn our attention on the classes of trees and planar graphs which model realistic scenarios of the emitted pollution in water and air, respectively. We derive approximation algorithms for these two kinds of networks and provide deterministic truthful and truthful in expectation mechanisms. In some settings of the problem that we study, we achieve the best possible approximation results under standard complexity theoretic assumptions. Our approximation algorithm on planar graphs is obtained by a novel decomposition technique to deal with constraints on vertices. We note that no known planar decomposition techniques can be used here and our technique can be of independent interest. For trees we design a two level dynamic programming approach to obtain an FPTAS. This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multi-dimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size. We furthermore derive truthful in expectation mechanisms on general networks with bounded degree.

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $$\frac{e}{e-1}$$ . The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $$\frac{18}{13}$$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. By using a characterisation result of Bade, we show that any randomised mechanism that is a symmetrisation of a truthful, non-bossy and Pareto optimal mechanism has an improved lower bound of $$\frac{e}{e-1}$$ . Since our new mechanism is a symmetrisation of RSDM for strict preferences, it follows that this lower bound is tight. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.

Greedy algorithms are known to provide, in polynomial time, near optimal approximation guarantees for Combinatorial Auctions (CAs) with multidimensional bidders. It is known that truthful greedy-like mechanisms for CAs with multi-minded bidders do not achieve good approximation guarantees. In this work, we seek a deeper understanding of greedy mechanism design and investigate under which general assumptions, we can have efficient and truthful greedy mechanisms for CAs. Towards this goal, we use the framework of priority algorithms and weak and strong verification, where the bidders are not allowed to overbid on their winning set or on any subset of this set, respectively. We provide a complete characterization of the power of weak verification showing that it is sufficient and necessary for any greedy fixed priority algorithm to become truthful with the use of money or not, depending on the ordering of the bids. Moreover, we show that strong verification is sufficient and necessary to obtain a 2-approximate truthful mechanism with money, based on a known greedy algorithm, for the problem of submodular CAs in finite bidding domains. Our proof is based on an interesting structural analysis of the strongly connected components of the declaration graph.

Algorithmic Mechanism Design attempts to marry computation and incentives, mainly by leveraging monetary transfers between designer and selfish agents involved. This is principally because in absence of money, very little can be done to enforce truthfulness. However, in certain applications, money is unavailable, morally unacceptable or might simply be at odds with the objective of the mechanism. For example, in combinatorial auctions (CAs), the paradigmatic problem of the area, we aim at solutions of maximum social welfare but still charge the society to ensure truthfulness. Additionally, truthfulness of CAs is poorly understood already in the case in which bidders happen to be interested in only two different sets of goods. We focus on the design of incentive-compatible CAs without money in the general setting of k -minded bidders. We trade monetary transfers with the observation that the mechanism can detect certain lies of the bidders: i.e., we study truthful CAs with verification and without money. We prove a characterization of truthful mechanisms, which makes an interesting parallel with the well-understood case of CAs with money for single-minded bidders. We then give a host of upper bounds on the approximation ratio obtained by either deterministic or randomized truthful mechanisms when the sets and valuations are private knowledge of the bidders. (Most of these mechanisms run in polynomial time and return solutions with (nearly) best possible approximation guarantees.) We complement these positive results with a number of lower bounds (some of which are essentially tight) that hold in the easier case of public sets. We thus provide an almost complete picture of truthfully approximating CAs in this general setting with multi-dimensional bidders.

Classical online bipartite matching problem and its generalizations are central algorithmic optimization problems. The second related line of research is in the area of algorithmic mechanism design, referring to the broad class of house allocation or assignment problems. We introduce a single framework that unifies and generalizes these two streams of models. Our generalizations allow for arbitrary matroid constraints or knapsack constraints at every object in the allocation problem. We design and analyze approximation algorithms and truthful mechanisms for this framework. Our algorithms have best possible approximation guarantees for most of the special instantiations of this framework, and are strong generalizations of the previous known results.

We design and analyze deterministic truthful approximation mechanisms for multi-unit Combinatorial Auctions involving only a constant number of distinct goods, each in arbitrary limited supply. Prospective buyers (bidders) have preferences over multisets of items, i.e., for more than one unit per distinct good. Our objective is to determine allocations of multisets that maximize the Social Welfare. Our main results are for multi-minded and submodular bidders. In the first setting each bidder has a positive value for being allocated one multiset from a prespecified demand set of alternatives. In the second setting each bidder is associated to a submodular valuation function that defines his value for the multiset he is allocated. For multi-minded bidders, we design a truthful FPTAS that fully optimizes the Social Welfare, while violating the supply constraints on goods within factor (1+e), for any fixed e>0 (i.e., the approximation applies to the constraints and not to the Social Welfare). This result is best possible, in that full optimization is impossible without violating the supply constraints. For submodular bidders, we obtain a PTAS that approximates the optimum Social Welfare within factor (1+e), for any fixed e>0, without violating the supply constraints. This result is best possible as well. Our allocation algorithms are Maximal-in-Range and yield truthful mechanisms, when paired with Vickrey-Clarke-Groves payments.

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants.

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 the 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 multiobjective optimization problems. In this setting we are given a main objective function and a set of secondary objectives which are modeled via budget constraints. Multiobjective 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 multiobjective 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 multicriteria fully polynomial-time approximation schemes (FPTASs) for multiobjective optimization problems whose exact version admits a pseudopolynomial-time algorithm, as, for instance, the multibudgeted versions of minimum spanning tree, shortest path, maximum (perfect) matching, and matroid intersection. Our construction also applies to multidimensional knapsack and multiunit combinatorial auctions. Our FPTASs compute a $(1+\varepsilon)$-approximate solution violating each budget constraint by a factor $(1+\varepsilon)$. 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 nontruthful algorithms.

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader’s prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+ε)log m-approximation. This can be extended to provide a (1+ε)(log k+log m)-approximation for the general problem and k followers. The problem is also shown to be hard to approximate within $\mathcal{O}(\log^{\varepsilon}k + \log^{\varepsilon}m)$ for some ε>0. If followers have demands, the single-price algorithm provides an $\mathcal{O}(m^{2})$ -approximation, and the problem is hard to approximate within $\mathcal{O}(m^{\epsilon})$ for some ε>0. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex-cover, which is based on non-trivial max-flow and LP-duality techniques. This approach can be extended to provide constant-factor approximations for any constant number of followers.

We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products $\mathcal{P}$ based on a set of customer samples $\mathcal{C}$. We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor $\mathcal{O}(\log^{\varepsilon}|\mathcal{C}|)$ for some constant $\varepsilon>0$, unless $\mathrm{NP}\subseteq\mathrm{DTIME}(n^{\mathcal{O}(\log\log n)})$, thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within $\mathcal{O}(\ell^{\varepsilon})$, $\ell$ being an upper bound on the number of nonzero values per customer, and $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ assuming $\mathrm{NP}\nsubseteq\bigcap_{\delta>0}\mathrm{BPTIME}(2^{\mathcal{O}(n^{\delta})})$. These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model.

This paper deals with the design of efficiently computable incentive-compatible mechanisms for combinatorial optimization problems with single-minded agents each possibly having multiple private parameters. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone fully polynomial time approximation scheme (FPTAS). This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multiunit auctions. The best previous result for such auctions was a 2-appro-xi-ma-tion. In addition, we present a monotone PTAS for the generalized assignment problem with any constant number of private parameters per agent. The most efficient way to solve packing integer programs (PIPs) is linear programming–based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing, and multicast routing. Our primal-dual approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.

This paper studies - from the perspective of efficient computation - a type of competition that is widespread throughout the plant and animal kingdoms, higher education, politics and artificial contests. In this setting, an agent gains utility from his relative performance (on some measurable criterion) against other agents, as opposed to his absolute performance. We model this situation using ranking games in which each strategy corresponds to a level of competitiveness, and incurs an upfront cost that is higher for more competitive strategies. We study the Nash equilibria of these games, and polynomial-time algorithms for computing them. For games in which there is no tie between agents' levels of competitiveness we give a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give polynomial-time approximation schemes for two special cases: constant-sized set of strategies, and constant number of players. The latter result is improved to a fully polynomial-time approximation scheme when the constant number of players only compete to win the game, i.e. to be ranked first.

In a classic optimization problem the complete input data is 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, where we are given the 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. Our main contribution is showing that two of the main tools for the design of approximation algorithms for multi-objective optimization problems, namely approximate Pareto curves and Lagrangian relaxation, can lead to truthful approximation schemes. By exploiting the method of approximate Pareto curves, we devise truthful 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 technique applies also 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). For a relevant sub-class of the mentioned problems we also 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. This result is based on the Lagrangian relaxation method, in combination with our monotone guessing step and a random perturbation step (ensuring low expected running time in a way similar to the smoothed analysis of algorithms). All the mentioned results match the best known approximation ratios, which however are obtained by non-truthful algorithms.

We study the price of selfish routing in noncooperative networks like the Internet. In particular, we investigate the price of selfish routing using the price of anarchy (a.k.a. the coordination ratio) and other (e.g., bicriteria) measures in the recently introduced game theoretic parallel links network model of Koutsoupias and Papadimitriou. We generalize this model toward general, monotone families of cost functions and cost functions from queueing theory. A summary of our main results for general, monotone cost functions is as follows: 1. We give an exact characterization of all cost functions having a bounded/unbounded price of anarchy. For example, the price of anarchy for cost functions describing the expected delay in queueing systems is unbounded. 2. We show that an unbounded price of anarchy implies an extremely high performance degradation under bicriteria measures. In fact, the price of selfish routing can be as high as a bandwidth degradation by a factor that is linear in the network size. 3. We separate the game theoretic (integral) allocation model from the (fractional) flow model by demonstrating that even a very small or negligible amount of integrality can lead to a dramatic performance degradation. 4. We unify recent results on selfish routing under different objectives by showing that an unbounded price of anarchy under the min-max objective implies an unbounded price of anarchy under the average cost objective and vice versa. Our special focus lies on cost functions describing the behavior of Web servers that can open only a limited number of Transmission Control Protocol (TCP) connections. In particular, we compare the performance of queueing systems that serve all incoming requests with servers that reject requests in case of overload. Our analysis indicates that all queueing systems without rejection cannot give any reasonable guarantee on the expected delay of requests under selfish routing even when the injected load is far away from the capacity of the system. In contrast, Web server farms that are allowed to reject requests can guarantee a high quality of service for every individual request stream even under relatively high injection rates.

We study fundamental algorithmic questions concerning the complexity of market equilibria under perfect and imperfect information by means of a basic microeconomic game. Suppose a provider offers a service to a set of potential customers. Each customer has a particular demand of service and her behavior is determined by a utility function that is nonincreasing in the sum of demands that are served by the provider.Classical game theory assumes complete information : the provider has full knowledge of the behavior of all customers. We present a complete characterization of the complexity of computing optimal pricing strategies and of computing best/worst equilibria in this model. Basically, we show that most of these problems are inapproximable in the worst case but admit an FPAS in the average case. Our average case analysis covers large classes of distributions for customer utilities. We generalize our analysis to robust equilibria in which players change their strategies only when this promises a significant utility improvement.A more realistic model considers providers with incomplete information . Following the game theoretic framework of Bayesian games introduced by Harsanyi, the provider is aware of probability distributions describing the behavior of the customers and aims at estimating its expected revenue under best/worst equilibria. Somewhat counterintuitively, we obtain an FPRAS for the equilibria problem in the model with imperfect information although the problem with perfect information is inapproximable under the worst-case measures. In particular, the worst-case complexity of the considered problems increases with the precision of the available knowledge.

This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.

We investigate the price of selfish routing in non-cooperative networks in terms of the coordination and bicriteria ratios in the recently introduced game theoretic network model of Koutsoupias and Papadimitriou. We present the first thorough study of this model for general, monotone families of cost functions and for cost functionsm from Queueing Theory. Our main results can be summarized as follows.