Mechanism design without money has a rich history in social choice literature. Due to the strong impossibility theorem by Gibbard and Satterthwaite, exploring domains in which there exist dominant strategy mechanisms is one of the central questions in the field. We propose a general framework, called the generalized packing problem (\gpp), to study the mechanism design questions without payment. The \gpp\ possesses a rich structure and comprises a number of well-studied models as special cases, including, e.g., matroid, matching, knapsack, independent set, and the generalized assignment problem. We adopt the agenda of approximate mechanism design where the objective is to design a truthful (or strategyproof) mechanism without money that can be implemented in polynomial time and yields a good approximation to the socially optimal solution. We study several special cases of \gpp, and give constant approximation mechanisms for matroid, matching, knapsack, and the generalized assignment problem. Our result for generalized assignment problem solves an open problem proposed in \cite{DG10}. Our main technical contribution is in exploitation of the approaches from stable matching, which is a fundamental solution concept in the context of matching marketplaces, in application to mechanism design. Stable matching, while conceptually simple, provides a set of powerful tools to manage and analyze self-interested behaviors of participating agents. Our mechanism uses a stable matching algorithm as a critical component and adopts other approaches like random sampling and online mechanisms. Our work also enriches the stable matching theory with a new knapsack constrained matching model.

Abstract. A searcher is tasked with exploring a graph with edge lengths and vertex weights, starting from a designated vertex. Initially, only the starting vertex is considered explored. At each step, the searcher adds an edge to the solution, connecting an unexplored vertex to an explored one. The time required to add an edge equals its length. The objective is to minimize the weighted sum of exploration times for all vertices. We demonstrate that this problem is hard to approximate and present algorithms with improved approximation guarantees. Specifically, we provide a [Formula: see text]-approximation for any [Formula: see text] for the general case. On instances where the vertex weights are binary, we achieve a [Formula: see text]-approximation. Finally, we develop a polynomial-time approximation scheme for Euclidean graphs. Previously, only an 8-approximation was known for all these cases.

We study the selection of agents based on mutual nominations, a theoretical problem with many applications from committee selection to AI alignment. As agents both select and are selected, they may be incentivized to misrepresent their true opinion about the eligibility of others to influence their own chances of selection. Impartial mechanisms circumvent this issue by guaranteeing that the selection of an agent is independent of the nominations cast by that agent. Previous research has established strong bounds on the performance of impartial mechanisms, measured by their ability to approximate the number of nominations for the most highly nominated agents. We study to what extent the performance of impartial mechanisms can be improved if they are given a prediction of a set of agents receiving a maximum number of nominations. Specifically, we provide bounds on the consistency and robustness of such mechanisms, where consistency measures the performance of the mechanisms when the prediction is correct and robustness its performance when the prediction is incorrect. For the general setting where up to $k$ agents are to be selected and agents nominate any number of other agents, we give a mechanism with consistency $1-O\big(\frac{1}{k}\big)$ and robustness $1-\frac{1}{e}-O\big(\frac{1}{k}\big)$. For the special case of selecting a single agent based on a single nomination per agent, we prove that $1$-consistency can be achieved while guaranteeing $\frac{1}{2}$-robustness. A close comparison with previous results shows that (asymptotically) optimal consistency can be achieved with little to no sacrifice in terms of robustness.

Abstract We introduce a framework for incremental–decremental maximization that captures the gradual transformation or renewal of infrastructures. In our model, an initial solution is transformed one element at a time and the utility of an intermediate solution is given by the sum of the utilities of the transformed and untransformed parts. We propose a simple randomized algorithm and a more sophisticated deterministic algorithm, both of which find an order in which to transform the elements while maintaining a large utility during all stages of transformation, relative to an optimal solution for the current stage. More specifically, our algorithms yield competitive solutions for utility functions of bounded curvature and/or generic submodularity ratio, and, in particular, for submodular functions and functions satisfying the gross substitutes property. Our results show that incremental–decremental maximization is substantially more difficult than incremental maximization.

Abstract. We settle the complexity of computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions [Formula: see text] as we show that the computation is [Formula: see text]-complete. To prove that the problem is contained in [Formula: see text], we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique for this method is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix where each entry of the Laplacian is a Laplacian again. Using the properties of this matrix allows us to recompute efficiently the Laplacian after the support of the equilibrium changes by matrix pivot operations. These insights give rise to a path following formulation for computing an equilibrium where states correspond to supports that are feasible for some demands and neighboring supports are feasible for increased or decreased flow demands. A closer investigation of the block Laplacian system further allows us to orient the states giving rise to unique predecessor and successor states, thus putting the problem into [Formula: see text]. For the [Formula: see text]-hardness, we reduce from computing an approximate equilibrium of a bimatrix win-lose game. As a byproduct of our reduction we further show that computing a multiclass Wardrop equilibrium with class-dependent affine cost functions is [Formula: see text]-complete as well. As another byproduct of our [Formula: see text]-completeness proof, we obtain an algorithm that computes a continuum of equilibria parametrized by the players’ flow demand. For player-specific costs, the continuum may involve several increases and decreases of the demand and yields an algorithm that runs in polynomial space. For games with player-independent costs, only demand increases are necessary, yielding an algorithm computing all equilibria as a function of the flow demand that runs in time polynomial in the output.

We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial $(α,μ)$-approximation is possible, i.e., a solution that with budget $B+α$ for all $B \in \mathbb{R}_{\geq 0}$ is a multiplicative $μ$-approximation compared to the optimum solution with budget $B$. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a $(χ,1)$-approximation, where $χ$ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is $(γ,2)$-competitive where $γ$ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a $(γ,3)$-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a $(3χ,8)$-approximation and, more generally, a $\smash{\bigl((4\ell - 1)χ, \frac{2^{\ell + 2}}{2^{\ell}-1}\bigr)}$-approximation for every fixed $\ell \in \mathbb{N}$.

We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links is equipped with deterministic capacities and stochastic travel times where the latter depend on a realized scenario. A continuum of flow particles arrives at the system at a constant rate. A system operator knows the realization of the scenario and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized scenario, and decide on a link to use. Inflow into a link exceeding the link's capacity builds up in a queue that increases the travel time on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected travel time along all links with positive inflow is equal at every point in time. We provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant $ε>0$. The algorithm solves a Langrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute also the optimal signals. It uses a different cell decomposition technique together with a piece-wise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.

Abstract. We consider the problem of maximizing a fractionally subadditive function under an increasing knapsack constraint. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most [Formula: see text], under the assumption that the values of singleton sets are in the range [Formula: see text], and we give a lower bound of [Formula: see text] on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of [Formula: see text] and an upper bound of [Formula: see text] for the incremental maximization of classical flows with capacities in [Formula: see text] which is tight for the unit capacity case.

A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a (2e+ε)-approximation for any ε > 0. For the case that all vertices have unit weight, we provide a 2e-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an 8-approximation was known.

We consider the problem of finding an incremental solution to a cardinality-constrained maximization problem that not only captures the solution for a fixed cardinality, but also describes how to gradually grow the solution as the cardinality bound increases. The goal is to find an incremental solution that guarantees a good competitive ratio against the optimum solution for all cardinalities simultaneously. The central challenge is to characterize maximization problems where this is possible, and to determine the best-possible competitive ratio that can be attained. A lower bound of 2.18 and an upper bound of φ + 1 ≈ 2.618 are known on the competitive ratio for monotone and accountable objectives [Bernstein et al., Math. Prog., 2022], which capture a wide range of maximization problems. We introduce a continuization technique and identify an optimal incremental algorithm that provides strong evidence that φ+1 is the best-possible competitive ratio. Using this continuization, we obtain an improved lower bound of 2.246 by studying a particular recurrence relation whose characteristic polynomial has complex roots exactly beyond the lower bound. Based on the optimal continuous algorithm combined with a scaling approach, we also provide a 1.772-competitive randomized algorithm. We complement this by a randomized lower bound of 1.447 via Yao’s principle.

We give new bounds for the single-nomination model of impartial selection, a problem proposed by Holzman and Moulin (Econometrica, 2013). A selection mechanism, which may be randomized, selects one individual from a group of n based on nominations among members of the group; a mechanism is impartial if the selection of an individual is independent of nominations cast by that individual, and α-optimal if under any circumstance the expected number of nominations received by the selected individual is at least α times that received by any individual. In a many-nominations model, where individuals may cast an arbitrary number of nominations, the so-called permutation mechanism is 1/2-optimal, and this is best possible. In the single-nomination model, where each individual casts exactly one nomination, the permutation mechanism does better and prior to this work was known to be 67/108-optimal but no better than 2/3-optimal. We show that it is in fact 2/3-optimal for all n. This result is obtained via tight bounds on the performance of the mechanism for graphs with maximum degree Δ, for any Δ, which we prove using an adversarial argument. We then show that the permutation mechanism is not best possible; indeed, by combining the permutation mechanism, another mechanism called plurality with runner-up, and some new ideas, 2105/3147-optimality can be achieved for all n. We finally give new upper bounds on α for any α-optimal impartial mechanism. They improve on the existing upper bounds for all n ≥ 7 and imply that no impartial mechanism can be better than 76/105-optimal for all n; they do not preclude the existence of a (3/4 − ε)-optimal impartial mechanism for arbitrary ε > 0 if n is large.

We study the complexity of computing equilibria in binary public goods games on undirected graphs. In such a game, players correspond to vertices in a graph and face a binary choice of performing an action, or not. Each player's decision depends only on the number of neighbors in the graph who perform the action and is encoded by a per-player binary pattern. We show that games with decreasing patterns (where players only want to act up to a threshold number of adjacent players doing so) always have a pure Nash equilibrium and that one is reached from any starting profile by following a polynomially bounded sequence of best responses. For non-monotonic patterns of the form 10k10* (where players want to act alone or alongside k + 1 neighbors), we show that it is NP-hard to decide whether a pure Nash equilibrium exists. We further investigate a generalization of the model that permits ties of varying strength: an edge with integral weight w behaves as w parallel edges. While, in this model, a pure Nash equilibrium still exists for decreasing patters, we show that the task of computing one is PLS-complete.

We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a K-approximate equilibrium can always be guaranteed, where K (approximately equal to 1.1974) is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a K-approximate equilibrium. The factor K is tight, meaning that there is an instance that does not admit an A-approximate equilibrium for any A < K. Thus A = K is the smallest possible value of A such that the existence of an A-approximate equilibrium can be guaranteed for any instance of the considered game. Secondly, we focus on approximate equilibria of a given fixed instance. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible A among all A-approximate equilibria of the given instance.

Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. We give a deterministic mechanism with an additive performance guarantee of O(n(1+κ)/2) in a setting with n individuals where each individual casts O(nκ) nominations, where κ∈[0,1]. For κ=0, i.e. when each individual casts at most a constant number of nominations, this bound is O(√n). This matches the best-known guarantee for randomized mechanisms and a single nomination. For κ=1 the bound is O(n). This is trivial, as even a mechanism that never selects provides an additive guarantee of n-1. We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.

It is a well-known fact that selfish behavior degrades the performance of traffic networks. Various measures have been proposed in the literature as a remedy for the inefficiency of traffic equilibria (such as road tolls or network design techniques). However, it often seems impractical and/or politically undesirable that these measures get implemented to a substantial extent.

In the Symmetric Rendezvous Search on the Line with Unknown Initial Distance, two identical agents are placed on the real line with their distance, the other's location, and their orientation unknown to them. Moving along the line at unit speed and executing the same randomized search strategy, the agents' goal is to meet up as early as possible. The expected meeting time obviously depends on the unknown initial distance and orientations. The quality of a randomized search strategy is thus measured by its competitive ratio, that is, the ratio of the expected meeting time and the earliest possible meeting time (half the initial distance). We present a class of successively refined randomized search strategies together with a rigorous mathematical analysis of their continuously improved competitive ratios. These strategies all rely on the basic idea of performing an infinite sequence of steps of geometrically increasing size in random directions, always returning to the agent's initial position before starting the next step. In addition, our more refined strategies use two novel ideas. First, remembering their past random choices, the agents randomly choose the direction of the next step in a Markov-chain-like manner. Second, choosing the next few random directions in advance, each agent may combine consecutive steps in the same direction into one longer step. As our main result, we show that this combination of looking into the past as well as into the future leads to a substantially improved competitive ratio of 13.93 compared to the previously best known bound of 24.85 (Ozsoyeller et al. 2013).

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium. For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

Abstract We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are $$\mathsf {APX}$$ APX -hard to approximate and present constant-factor approximation algorithms based upon two different algorithmic techniques: a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation, and a greedy strategy. We further show that a combination of these techniques can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of these algorithms for problem instances arising in the context of real-world gas transport networks.

We settle the complexity of computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions le,i(x) = ae,ix + be,i showing that it is PPAD-complete. To prove that the problem is contained in PPAD, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix. This leads to a path following formulation where states correspond to supports that are feasible for some demands and neighboring supports are feasible for increased or decreased flow demands. A closer investigation of the block Laplacian system allows to orient the states giving rise to unique predecessor and successor states thus putting the problem into PPAD. For the PPAD-hardness, we reduce from computing an approximate equilibrium of a bimatrix win-lose game. As a byproduct of our reduction we further show that computing a multiclass Wardrop equilibrium with class-dependent affine cost functions is PPAD-complete as well. As a byproduct of our PPAD-completeness proof, we obtain an algorithm that computes all equilibria parametrized by the players' flow demands. For player-specific costs, this computation may require several increases and decreases of the demands leading to an algorithm that runs in polynomial space but exponential time. For player-independent costs only demand increases are necessary. If the coefficients be,i are in general position, this yields an algorithm computing all equilibria as a function of the flow demand running in time polynomial in the output.

We develop an algorithm that computes for a given undirected or directed network with flow-dependent piece-wise linear edge cost functions all Wardrop equilibria as a function of the flow demand. Our algorithm is based on Katzenelson's homotopy method for electrical networks. The algorithm uses a bijection between vertex potentials and flow excess vectors that is piecewise linear in the potential space and where each linear segment can be interpreted as an augmenting flow in a residual network. The algorithm iteratively increases the excess of one or more vertex pairs until the bijection reaches a point of non-differentiability. Then, the next linear region is chosen in a simplex-like pivot step and the algorithm proceeds. We first show that this algorithm correctly computes all Wardrop equilibria in undirected single-commodity networks along the chosen path of excess vectors. We then adapt our algorithm to also work for discontinuous cost functions which allows to model directed edges and/or edge capacities. Our algorithm is output-polynomial in non-degenerate instances where the solution curve never hits a point where the cost function of more than one edge becomes non-differentiable. For degenerate instances we still obtain an output-polynomial algorithm computing the linear segments of the bijection by a convex program. The latter technique also allows to handle multiple commodities.

We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n ) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n ) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n ) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n -vertex graph using O (log log n ) pebbles, even when n is not known a priori . The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O (log log n ) agents with constant memory can explore any n -vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n ) when we allow each agent to have at most O ((log n ) 1 -ε) bits of memory for any ε > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n ) pebbles to explore any n -vertex graph.

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into supervertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding supervertices remain at distance at least $\varphi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x)=x/\alpha-\beta$, where $\alpha\geq 1$ and $\beta\geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (nonoptimal) contractions despite our hardness results.

Wardrop equilibria in nonatomic congestion games are in general inefficient as they do not induce an optimal flow that minimizes the total travel time. Network tolls are a prominent and popular way to induce an optimum flow in equilibrium. The classical approach to find such tolls is marginal cost pricing which requires the exact knowledge of the demand on the network. In this paper, we investigate under which conditions demand-independent optimum tolls exist that induce the system optimum flow for any travel demand in the network. We give several characterizations for the existence of such tolls both in terms of the cost structure and the network structure of the game. Specifically we show that demand-independent optimum tolls exist if and only if the edge cost functions are shifted monomials as used by the Bureau of Public Roads. Moreover, non-negative demand-independent optimum tolls exist when the network is a directed acyclic multi-graph. Finally, we show that any network with a single origin-destination pair admits demand-independent optimum tolls that, although not necessarily non-negative, satisfy a budget constraint.

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. \nIn this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. \nFormally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance d, the corresponding super-vertices remain at distance at least \\varphi(d) in the contracted graph, where \\varphi is a tolerance function bounding the permitted distance distortion. \nWe present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions \\varphi(x)=x/\\alpha-\\beta, where \\alpha \\geq 1 and \\beta \\geq 0 are arbitrary real-valued parameters. \nSpecifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. \nFurther we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.

We study general resource allocation problems with a diseconomy of scale. Given a finite set of commodities that request certain resources, the cost of each resource grows superlinearly with the demand for it, and our goal is to minimize the total cost of the resources. In large systems with limited coordination, it is natural to consider local dynamics where in each step a single commodity switches its allocated resources whenever the new solution after the switch has smaller total cost over all commodities. This yields a deterministic and polynomial time algorithm with approximation factor arbitrarily close to the locality gap, i.e., the worst case ratio of the cost of a local optimal and a global optimal solution. For costs that are polynomials with non-negative coefficients and maximal degree d, we provide a locality gap for weighted problems that is tight for all values of d. For unweighted problems, the locality gap asymptotically matches the approximation guarantee of the currently best known centralized algorithm [Makarychev, Srividenko FOCS14] but only requires local knowledge of the commodities.

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio $\varphi\approx1.618$. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items in the beginning and try to pack the items in this order, without changing the order later on. We give efficient algorithms computing these policies. On the other hand, we show that, for any $\alpha>1$, the problem of deciding whether a given universal policy achieves a factor of $\alpha$ is ${\mathsf{coNP}}$-complete. If $\alpha$ is part of the input, the same problem is shown to be ${\mathsf{coNP}}$-complete for items with unit densities. Finally, we show that it is ${\mathsf{coNP}}$-hard to decide, for given $\alpha$, whether a set of items admits a universal policy with factor $\alpha$, even if all items have unit densities.

We consider the fundamental problem of exploring an undirected and initially unknown graph by an agent with little memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that ⊝(log n) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the memory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent ℴ(log log n) distinguishable pebbles and bits of memory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, Ω(log log n) distinguishable pebbles are necessary for exploration.

We study mechanisms that select members of a set of agents based on nominations by other members and that are impartial in the sense that agents cannot influence their own chance of selection. Prior work has shown that deterministic mechanisms for selecting any fixed number of agents are severely limited, whereas randomization allows for the selection of a single agent that in expectation receives at least 1 / 2 of the maximum number of nominations. The bound of 1 / 2 is in fact best possible subject to impartiality. We prove here that the same bound can also be achieved deterministically by sometimes but not always selecting a second agent. We then show a separation between randomized mechanisms that make exactly two or up to two choices, and give upper and lower bounds on the performance of mechanisms allowed more than two choices. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing their fixed demand. The disutility of a used only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large‐scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 262–285 2015

We study a fundamental problem in social choice theory, the selection of a member of a set of agents based on impartial nominations by agents from that set. Studied previously by Alon et al. [Proceedings of TARK, 2011, pp. 101--110] and by Holzman and Moulin [Econometrica, 81 (2013), pp. 173--196], this problem arises when representatives are selected from within a group or when publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. This is best possible subject to impartiality and resolves a conjecture of Alon et al. Further results are given for the case where some agent receives many nominations and the case where each agent casts at least one nomination.

We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of alpha-approximate pure Nash equilibria and the convergence of alpha-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor alpha for a given class of cost functions. For example for concave cost functions the factor is at most 3/2, for quadratic cost functions it is at most 4/3, and for polynomial cost functions of maximal degree d it is at at most d + 1. For games with two players we obtain tight bounds which are as small as for example 1.054 in the case of quadratic cost functions.

We study the problem of selecting a member of a set of agents based on impartial nominations by agents from that set. The problem was studied previously by Alon et al. and by Holzman and Moulin and has important applications in situations where representatives are selected from within a group or where publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. Subject to impartiality, this is best possible.

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items and try to pack the items in this order, independent of the observations made while packing. We give efficient algorithms computing these policies. On the other hand, we show that, for any a>1, the problem of deciding whether a given universal policy achieves a factor of a is coNP-complete. If a is part of the input, the same problem is shown to be coNP-complete for items with unit densities. Finally, we show that it is coNP-hard to decide, for given a, whether a set of items admits a universal policy with factor a, even if all items have unit densities.

We initiate the study of congestion games with variable demands where the (variable) demand has to be assigned to exactly one subset of resources. The players' incentives to use higher demands are stimulated by non-decreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of non-cooperative games captures many elements of real-world applications, it has not been studied in this generality, to our knowledge, in the past.