Interval Scheduling Games

TL;DR

The paper studies a color-based interval scheduling game where a machine configures colors over a horizon $[0,T)$ to cover jobs, with each job $j$ having length $p_j$, weight $w_j$, and color $c_j$, controlled by the corresponding color-player. A polynomial-time dynamic program computes the machine’s optimal configuration for a given strategy profile, while social optima are poly-time for fixed $c$ but NP-hard in general, connected to Knapsack via reductions. NE existence/computation and the dynamics of BR updates are analyzed across classes including ${ t G}_{single}$ and ${ t G}_{unit}$, yielding tight PoA and PoS bounds and highlighting when equilibria can be inefficient or fail to exist. The results provide both structural insights and practical algorithms relevant to real-time wireless beam scheduling and other resource-allocation settings where color-based parallel processing is allowed.

Abstract

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.

Figures (6)

• Figure 1: A game $G$ with $c=2$ that has no NE. Intervals are labeled by their weights. Bold intervals are covered by the machine.
• Figure 2: The non-covered jobs $a$, $b$, and the sets $X,Y$. The jobs of $J_{in}(\sigma)$ are bold.
• Figure 3: The stability of $\sigma$ implies that point $p_i$ is within a busy interval.
• Figure 4: A game $G \in {\mathcal{G}}_{prop}$ with $c=2$ that has no NE.
• Figure 5: A game with $c=2$ and PoS$= \frac{2}{1+\delta} =2-\epsilon$.

Theorems & Definitions (48)

• Example 2.1
• Theorem 3.1
• proof
• Claim 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 5.1