arXiv subjects

Elias Koutsoupias

Publications and source records attributed to Elias Koutsoupias.

At least 19 recordsLinked to original sources

arXiv2026

The $k$-server conjecture is true

The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it. Our proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal costs correspond to addition and multiplication of formal expressions, and each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival updates the representation via a change of basis and row replacement. The amortized analysis is based on a potential function defined in terms of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.

cs.DS↗

arXiv2026

Universally truthful mechanisms for scheduling

We consider universally truthful randomized mechanisms for the problem of scheduling $m$ jobs on $n$ unrelated machines. We prove a lower bound on the expected approximation ratio of every such mechanism whose probability distribution has discrete support. We show that no universally truthful randomized mechanism in this class can achieve approximation ratio smaller than $n/12 - o(n)$ with respect to the optimal makespan. We match this, up to a constant factor, by a mechanism with approximation ratio $n/2 + o(n)$.

cs.GT↗

arXiv2026

Complexity of Unambiguous Problems in $Σ^P_2$

Various practical problems within the class $Σ_{2}^P$ possess an unambiguity property, meaning that yes-instances correspond with a unique witness. The semantic class containing all unambiguous $Σ_{2}^P$ problems is denoted $UΣ_{2}^P$. Examples include the existence of (1) a dominating strategy in a game, (2) a Condorcet winner, (3) a strongly popular partition in hedonic games, and (4) a winner (source) in a tournament. The computational complexity of unambiguous problems is not well understood, leaving many questions unresolved. We address this gap in a broad complexity-theoretic sense; our main contributions consist of the following. - We identify three syntactic subclasses of $UΣ_{2}^P$ associated with general properties of problems that guarantee uniqueness: Polynomial Tournament Winner (PTW), Polynomial Condorcet Winner (PCW), and Polynomial Majority Argument (PMA). - We establish complexity upper and lower bounds for our proposed classes. In particular, we show that they are all contained in $S_2^P$ and are thus significantly easier than the immediate $Σ_{2}^P$ upper bound. - We characterize the complexity of various practical problems using this framework.

cs.CC↗

arXiv2025

One-dimensional vs. Multi-dimensional Pricing in Blockchain Protocols

Blockchain transactions consume diverse resources, foremost among them storage, but also computation, communication, and others. Efficiently charging for these resources is crucial for effective system resource allocation and long-term economic viability. The prevailing approach, one-dimensional pricing, sets a single price for a linear combination of resources. However, this often leads to under-utilization when resource capacities are limited. Multi-dimensional pricing, which independently prices each resource, offers an alternative but presents challenges in price discovery. This work focuses on the welfare achieved by these two schemes. We prove that multi-dimensional pricing is superior under stable blockchain conditions. Conversely, we show that one-dimensional pricing outperforms its multi-dimensional counterpart in transient states, exhibiting faster convergence and greater computational tractability. These results highlight a critical trade-off: while multi-dimensional pricing offers efficiency gains at equilibrium, its implementation incurs costs associated with system transitions. Our findings underscore the necessity for a deeper understanding of these transient effects before widespread adoption. Finally, we propose mechanisms that aim to mitigate some of these issues, paving the way for future research.

cs.GT↗

arXiv2025

Pool Formation in Oceanic Games: Shapley Value and Proportional Sharing

We study a game-theoretic model for pool formation in Proof of Stake blockchain protocols. In such systems, stakeholders can form pools as a means of obtaining regular rewards from participation in ledger maintenance, with the power of each pool being dependent on its collective stake. The question we are interested in is the design of mechanisms that suitably split rewards among pool members and achieve favorable properties in the resulting pool configuration. With this in mind, we initiate a non-cooperative game-theoretic analysis of the well known Shapley value scheme from cooperative game theory into the context of blockchains. In particular, we focus on the oceanic model of games, proposed by Milnor and Shapley (1978), which is suitable for populations where a small set of large players coexists with a big mass of rather small, negligible players. This provides an appropriate level of abstraction for pool formation processes among the stakeholders. We provide comparisons between the Shapley mechanism and the more standard proportional scheme, in terms of attained decentralization, via a Price of Stability analysis and in terms of susceptibility to Sybil attacks, i.e., the strategic splitting of a players' stake with the intention of participating in multiple pools for increased profit. Interestingly, while the widely deployed proportional scheme appears to have certain advantages, the Shapley value scheme, which rewards higher the most pivotal players, emerges as a competitive alternative, by being able to bypass some of the downsides of proportional sharing, while also not being far from optimal guarantees w.r.t. decentralization. Finally, we complement our study with some variations of proportional sharing, where the profit is split in proportion to a superadditive or a subadditive function of the stake, showing that the Shapley value scheme still maintains the same advantages.

cs.GT↗

arXiv2024

Balancing Participation and Decentralization in Proof-of-Stake Cryptocurrencies

Proof-of-stake blockchain protocols have emerged as a compelling paradigm for organizing distributed ledger systems. In proof-of-stake (PoS), a subset of stakeholders participate in validating a growing ledger of transactions. For the safety and liveness of the underlying system, it is desirable for the set of validators to include multiple independent entities as well as represent a non-negligible percentage of the total stake issued. In this paper, we study a secondary form of participation in the transaction validation process, which takes the form of stake delegation, whereby an agent delegates their stake to an active validator who acts as a stake pool operator. We study payment schemes that reward agents as a function of their collective actions regarding stake pool operation and delegation. Such payment schemes serve as a mechanism to incentivize participation in the validation process while maintaining decentralization. We observe natural trade-offs between these objectives and the total expenditure required to run the relevant payment schemes. Ultimately, we provide a family of payment schemes which can strike different balances between these competing objectives at equilibrium in a Bayesian game theoretic framework.

cs.GT↗

arXiv2023

Tiered Mechanisms for Blockchain Transaction Fees

Blockchain systems come with the promise of being inclusive for a variety of decentralized applications (DApps) that can serve different purposes and have different urgency requirements. Despite this, the transaction fee mechanisms currently deployed in popular platforms as well as previous modeling attempts for the associated mechanism design problem focus on an approach that favors increasing prices in favor of those clients who value immediate service during periods of congestion. To address this issue, we introduce a model that captures the traffic diversity of blockchain systems and a tiered pricing mechanism that is capable of implementing more inclusive transaction policies. In this model, we demonstrate formally that EIP-1559, the transaction fee mechanism currently used in Ethereum, is not inclusive and demonstrate experimentally that its prices surge horizontally during periods of congestion. On the other hand, we prove formally that our mechanism achieves stable prices in expectation and we provide experimental results that establish that prices for transactions can be kept low for low urgency transactions, resulting in a diverse set of transaction types entering the blockchain. At the same time, perhaps surprisingly, our mechanism does not necessarily sacrifice revenue since the lowering of the prices for low urgency transactions can be covered from high urgency ones due to the price discrimination ability of the mechanism.

cs.GT↗

arXiv2023

A proof of the Nisan-Ronen conjecture

Noam Nisan and Amir Ronen conjectured that the best approximation ratio of deterministic truthful mechanisms for makespan-minimization for $n$ unrelated machines is $n$. This work validates the conjecture.

cs.GT↗

arXiv2021

Incentives Against Power Grabs or How to Engineer the Revolution in a Pooled Proof of Stake System

Proof-of-Stake (PoS) blockchain systems, especially those that allow stakeholders to organize themselves in ``stake-pools'', have emerged as a compelling paradigm for the deployment of large scale distributed ledgers. A stake-pool operates a node that engages in the PoS protocol and potentially represents a large number of smaller stakeholders. While such pooled PoS operation is attractive from various angles, it also exhibits a significant shortcoming that, so far and to the best of our knowledge, has not been sufficiently understood or investigated. Pooled PoS operation, to be effective and not lead to sub-optimal dictatorial or cartel-like configurations, should enable the stakeholders to revoke and re-delegate their stake in a way that is aligned with their incentives. However, given that stake-pool operators are exactly those entities who determine what transactions are to be recorded in the ledger, they are quite likely to form a cartel and censor any transaction they want, such as those that attempt to adjust the current stake-pool lineup. In this way, a power grab takes place, where the stake-pool cartel perpetuates its control over the PoS system. We first model and observe formally the emergence of the above problem in pooled PoS systems, and then we describe an anti-censorship mechanism that takes advantage of the underlying cryptographic functions of the ledger and the nature of peer-to-peer networks to diffuse information without suppression. We provide a thorough game-theoretic analysis of this mechanism discovering various types of Nash equilibria which demonstrate that the ``revolution'', i.e., the strategic decision of pool members to withdraw support from a censoring cartel as well as the pool operators to step down, can be incentivized, under suitable and plausible conditions in the utility functions of the involved participants.

cs.GT↗

arXiv2021

On the Nisan-Ronen conjecture

The Nisan-Ronen conjecture states that no truthful mechanism for makespan-minimization when allocating $m$ tasks to $n$ unrelated machines can have approximation ratio less than $n$. Over more than two decades since its formulation, little progress has been made in resolving it and the best known lower bound is still a small constant. This work makes progress towards validating the conjecture by showing a lower bound of $1+\sqrt{n-1}$.

cs.GT↗

arXiv2021

Truthful allocation in graphs and hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, $k$-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the $L^p$-norm of the values of the players, a generalization of the makespan minimization that corresponds to $p=\infty$, and extend the results to any $p>0$.

cs.GT↗

arXiv2021

Towards the k-server conjecture: A unifying potential, pushing the frontier to the circle

The $k$-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a $k$-competitive deterministic algorithm for the $k$-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: $k=2$, $(k+1)$-point metrics, $(k+2)$-point metrics, the line metric, weighted star metrics, and $k=3$ in the Manhattan plane. The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving $k$-competitiveness of WFA for all these cases. We also use this potential to show $k$-competitiveness of WFA on multiray spaces and for $k=3$ on trees. While the DoubleCoverage algorithm was known to be $k$-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the $k$-server conjecture. To our major surprise, this is not the case, as we show (using connections to the $k$-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the $k$-server conjecture. On the other hand, if the $k$-server conjecture holds, then we believe it can be proved by a variant of our potential.

cs.DS↗

arXiv2020

Online Trading as a Secretary Problem

We consider the online problem in which an intermediary trades identical items with a sequence of n buyers and n sellers, each of unit demand. We assume that the values of the traders are selected by an adversary and the sequence is randomly permuted. We give competitive algorithms for two objectives: welfare and gain-from-trade.

cs.GT↗

arXiv2020

Reward Sharing Schemes for Stake Pools

We introduce and study reward sharing schemes (RSS) that promote the fair formation of {\em stake pools}\ in collaborative projects that involve a large number of stakeholders such as the maintenance of a proof-of-stake (PoS) blockchain. Our mechanisms are parameterized by a target value for the desired number of pools. We show that by properly incentivizing participants, the desired number of stake pools is a Nash equilibrium arising from rational play. Our equilibria also exhibit an efficiency / security tradeoff via a parameter that calibrates between including pools with the smallest cost and providing protection against Sybil attacks, the setting where a single stakeholder creates a large number of pools in the hopes to dominate the collaborative project. We then describe how RSS can be deployed in the PoS setting, mitigating a number of potential deployment attacks and protocol deviations that include censoring transactions, performing Sybil attacks with the objective to control the majority of stake, lying about the actual cost and others. Finally, we experimentally demonstrate fast convergence to equilibria in dynamic environments where players react to each other's strategic moves over an indefinite period of interactive play. We also show how simple reward sharing schemes that are seemingly more "fair", perhaps counterintuitively, converge to centralized equilibria.

cs.GT↗

arXiv2019

Fairness and Efficiency in DAG-based Cryptocurrencies

Bitcoin is a decentralised digital currency that serves as an alternative to existing transaction systems based on an external central authority for security. Although Bitcoin has many desirable properties, one of its fundamental shortcomings is its inability to process transactions at high rates. To address this challenge, many subsequent protocols either modify the rules of block acceptance (longest chain rule) and reward, or alter the graphical structure of the public ledger from a tree to a directed acyclic graph (DAG). Motivated by these approaches, we introduce a new general framework that captures ledger growth for a large class of DAG-based implementations. With this in hand, and by assuming honest miner behaviour, we (experimentally) explore how different DAG-based protocols perform in terms of fairness, i.e., if the block reward of a miner is proportional to their hash power, as well as efficiency, i.e. what proportion of user transactions a ledger deems valid after a certain length of time. Our results demonstrate fundamental structural limits on how well DAG-based ledger protocols cope with a high transaction load. More specifically, we show that even in a scenario where every miner on the system is honest in terms of when they publish blocks, what they point to, and what transactions each block contains, fairness and efficiency of the ledger can break down at specific hash rates if miners have differing levels of connectivity to the P2P network sustaining the protocol.

cs.CR↗

arXiv2019

Better Bounds for Online Line Chasing

We study online competitive algorithms for the \emph{line chasing problem} in Euclidean spaces $\reals^d$, where the input consists of an initial point $P_0$ and a sequence of lines $X_1,X_2,...,X_m$, revealed one at a time. At each step $t$, when the line $X_t$ is revealed, the algorithm must determine a point $P_t\in X_t$. An online algorithm is called $c$-competitive if for any input sequence the path $P_0, P_1,...,P_m$ it computes has length at most $c$ times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets $X_t$ are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was $28.1$, even in the plane. We significantly improve this bound, by providing a~$3$-competitive algorithm for any dimension $d$. We also improve the lower bound on the competitive ratio, from $1.412$ to $1.5358$.

cs.DS↗

arXiv2019

On the Nisan-Ronen conjecture for submodular valuations

We consider incentive compatible mechanisms for a domain that is very close to the domain of scheduling $n$ unrelated machines: the single exception is that the valuation of just one machine is submodular. For the scheduling problem with such cost functions, we give a lower bound of $Ω(\sqrt{n})$ on the approximation ratio of incentive compatible deterministic mechanisms. This is a strong information-theoretic impossibility result on the approximation ratio of mechanisms on relatively simple domains. The lower bound of the current work assumes no restriction on the mechanism side, but an expanded class of valuations, in contrast to previous general results on the Nisan-Ronen conjecture that hold for only special classes of mechanisms such as local, strongly monotone, and anonymous mechanisms. Our approach is based on a novel characterization of appropriately selected smaller instances that allows us to focus on particular type of algorithms (linear mechanisms), from which we extract a locality property that gives the lower bound.

cs.GT↗

arXiv2019

Blockchain Mining Games with Pay Forward

We study the strategic implications that arise from adding one extra option to the miners participating in the bitcoin protocol. We propose that when adding a block, miners also have the ability to pay forward an amount to be collected by the first miner who successfully extends their branch, giving them the power to influence the incentives for mining. We formulate a stochastic game for the study of such incentives and show that with this added option, smaller miners can guarantee that the best response of even substantially more powerful miners is to follow the expected behavior intended by the protocol designer.

cs.GT↗