arXiv subjects

Tami Tamir

Publications and source records attributed to Tami Tamir.

8 recordsLinked to original sources

arXiv2026

Job-Scheduling Games with Time-Dependent Processing Times

Job-scheduling games have traditionally assumed fixed processing times. However, in many realistic environments, ranging from cyber-security response to high-frequency trading, a task's duration depends on its starting time. We study job-scheduling games with time-dependent processing times, where job lengths are linear functions of their start times, exhibiting either positive deterioration (increasing length) or negative deterioration (decreasing length). We analyze these games under various coordination mechanisms and priority policies. By introducing the concept of delay-averse agents, we provide a unifying framework to characterize equilibrium existence. For delay-averse jobs, we show that stability is maintained and pure Nash equilibria (NE) can be computed efficiently. In contrast, for non-delay-averse jobs, we demonstrate that a NE may not exist, and prove that deciding its existence is NP-complete, even on identical machines - a fundamental departure from classical coordination mechanisms. Regarding equilibrium inefficiency, we show that the Price of Anarchy (PoA) can be significantly higher than in environments with fixed processing times. To mitigate this, we propose and analyze three coordination mechanisms: SBPT (Shortest Basic Processing Time), which reduces the PoA in games with positive deterioration to a constant, and SDR (Smallest Deterioration Rate) and LBDR (Largest Basic-Deterioration Ratio) for negative deterioration, which achieve tight constant PoA bounds of $2$ and $\max\{\frac{e}{e-1}, 2-\frac{1}{m}\}$, respectively. Our results bridge the gap between centralized time-dependent scheduling and decentralized game-theoretic analysis.

cs.GT↗

arXiv2026

Interval Scheduling Games with Color-Based Concurrent Jobs

We consider a game-theoretic variant of an interval scheduling problem. Every job is associated with a length, a weight, and a color. Each player controls all the jobs of a specific color, and needs to decide on a processing interval for each of its jobs. Jobs of the same color can be processed simultaneously by the machine. A job is covered if the machine is configured to its color during its whole processing interval. The goal of the machine is to maximize the sum of weights of all covered jobs, and the goal of each player is to place its jobs such that the sum of weights of covered jobs from its color is maximized. The study of this game is motivated by several applications like antenna scheduling for wireless networks. We first show that given a strategy profile of the players, the machine scheduling problem can be solved in polynomial time. We then study the game from the players' point of view. We analyze the existence of Nash equilibria, its computation, and inefficiency. We distinguish between instances of the classical interval scheduling problem, in which every player controls a single job, and instances in which color sets may include multiple jobs.

cs.GT↗

arXiv2025

Bipartite Matching with Pair-Dependent Bounds

Let $G=(U \cup V, E)$ be a bipartite graph, where $U$ represents jobs and $V$ represents machines. We study a new variant of the bipartite matching problem in which each job in $U$ can be matched to at most one machine in $V$, and the number of jobs that can be assigned to a machine depends on the specific jobs matched to it. These pair-dependent bounds reflect systems where different jobs have varying tolerance for congestion, determined by the specific machine they are assigned to. We define a bipartite PD-matching as a set of edges $M \subseteq E$ that satisfies these job-to-machine tolerance constraints. This variant of matching extends well-known matching problems, however, despite its relevance to real-world systems, it has not been studied before. We study bipartite PD-matchings with the objective of maximizing the matching size. As we show, the problem exhibits significant differences from previously studied matching problems. We analyze its computational complexity both in the general case and for specific restricted instances, presenting hardness results alongside optimal and approximation algorithms.

cs.DS↗

arXiv2025

Coordination Mechanisms with Rank-Based Utilities

In classical job-scheduling games, each job behaves as a selfish player, choosing a machine to minimize its own completion time. To reduce the equilibria inefficiency, coordination mechanisms are employed, allowing each machine to follow its own scheduling policy. In this paper we study the effects of incorporating rank-based utilities within coordination mechanisms across environments with either identical or unrelated machines. With rank-based utilities, players aim to perform well relative to their competitors, rather than solely minimizing their completion time. We first demonstrate that even in basic setups, such as two identical machines with unit-length jobs, a pure Nash equilibrium (NE) assignment may not exist. This observation motivates our inquiry into the complexity of determining whether a given game instance admits a NE. We prove that this problem is NP-complete, even in highly restricted cases. In contrast, we identify specific classes of games where a NE is guaranteed to exist, or where the decision problem can be resolved in polynomial time. Additionally, we examine how competition impacts the efficiency of Nash equilibria, or sink equilibria if a NE does not exist. We derive tight bounds on the price of anarchy, and show that competition may either enhance or degrade overall performance.

cs.GT↗

arXiv2023

Scheduling Games with Machine-Dependent Priority Lists

We consider a scheduling game on parallel related machines, in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We prove that it is NP-hard to decide if a pure Nash equilibrium exists and characterize four classes of instances in which a pure Nash equilibrium is guaranteed to exist. For each of these classes, we give an algorithm that computes a Nash equilibrium, we prove that best-response dynamics converge to a Nash equilibrium, and we bound the inefficiency of Nash equilibria with respect to the makespan of the schedule and the sum of completion times. In addition, we show that although a pure Nash equilibrium is guaranteed to exist in instances with identical machines, it is NP-hard to approximate the best Nash equilibrium with respect to both objectives.

cs.GT↗

arXiv2020

The Efficiency of Best-Response Dynamics

Best response (BR) dynamics is a natural method by which players proceed toward a pure Nash equilibrium via a local search method. The quality of the equilibrium reached may depend heavily on the order by which players are chosen to perform their best response moves. A {\em deviator rule} $S$ is a method for selecting the next deviating player. We provide a measure for quantifying the performance of different deviator rules. The {\em inefficiency} of a deviator rule $S$ is the maximum ratio, over all initial profiles $p$, between the social cost of the worst equilibrium reachable by $S$ from $p$ and the social cost of the best equilibrium reachable from $p$. This inefficiency always lies between $1$ and the {\em price of anarchy}. We study the inefficiency of various deviator rules in network formation games and job scheduling games (both are congestion games, where BR dynamics always converges to a pure NE). For some classes of games, we compute optimal deviator rules. Furthermore, we define and study a new class of deviator rules, called {\em local} deviator rules. Such rules choose the next deviator as a function of a restricted set of parameters, and satisfy a natural condition called {\em independence of irrelevant players}. We present upper bounds on the inefficiency of some local deviator rules, and also show that for some classes of games, no local deviator rule can guarantee inefficiency lower than the price of anarchy.

cs.GT↗

arXiv2014

Approximate Strong Equilibrium in Job Scheduling Games

A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is predominantly used in the analysis of multiagent systems. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit stability against coalitional deviations, in that the benefits from a joint deviation are bounded. In this sense, NE approximates strong equilibrium. Coalition formation is a key issue in multiagent systems. We provide a framework for quantifying the stability and the performance of various assignment policies and solution concepts in the face of coalitional deviations. Within this framework we evaluate a given configuration according to three measures: (i) IR_min: the maximal number alpha, such that there exists a coalition in which the minimal improvement ratio among the coalition members is alpha, (ii) IR_max: the maximal number alpha, such that there exists a coalition in which the maximal improvement ratio among the coalition members is alpha, and (iii) DR_max: the maximal possible damage ratio of an agent outside the coalition. We analyze these measures in job scheduling games on identical machines. In particular, we provide upper and lower bounds for the above three measures for both NE and the well-known assignment rule Longest Processing Time (LPT). Our results indicate that LPT performs better than a general NE. However, LPT is not the best possible approximation. In particular, we present a polynomial time approximation scheme (PTAS) for the makespan minimization problem which provides a schedule with IR_min of 1+epsilon for any given epsilon. With respect to computational complexity, we show that given an NE on m >= 3 identical machines or m >= 2 unrelated machines, it is NP-hard to determine whether a given coalition can deviate such that every member decreases its cost.

cs.GT↗

arXiv2011

Approximations for Monotone and Non-monotone Submodular Maximization with Knapsack Constraints

Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to $d$ knapsack constraints, where $d$ is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $α$-approximation algorithm for the continuous relaxation implies a randomized $(α- \eps)$-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to $1/4- \eps$, for any $\eps > 0$, and to obtain a nearly optimal $(1-e^{-1}-\eps)-$approximation ratio for the monotone case, for any $\eps>0$. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.

cs.DS↗