Copula-Based Randomized Mechanisms for Truthful Scheduling on Two Unrelated Machines

Abstract

We design a Copula-based generic randomized truthful mechanism for scheduling on two unrelated machines with approximation ratio within [1.5852, 1.58606], offering an improved upper bound for the two-machine case. Moreover, we provide an upper bound 1.5067711 for the two-machine two-task case, which is almost tight in view of the lower bound of 1.506 for the scale-free truthful mechanisms [4]. Of independent interest is the explicit incorporation of the concept of Copula in the design and analysis of the proposed approximation algorithm. We hope that techniques like this one will also prove useful in solving other problems in the future.

New Bounds for Truthful Scheduling on Two Unrelated Selfish Machines

“…Our approach is to formulate a mathematical optimization problem for R n . This approach was not common until recently when several successful truthful or truthful in expectation mechanisms have been constructed using linear or nonlinear programs [2,3,7,15]. This paper continues the trend to combine optimization with mechanism design and has the following contributions: (13) in Corollary 3.…”

“…In the sequel we work with monotone, taskindependent, scale-free (denoted by MIS) task allocation algorithms. These algorithms provide good upper bounds on approximation ratios in scheduling [3,17,18,19,23]. Lu and Yu [17,18,19] present a way to construct a payment allocation procedure for MIS algorithms which results in truthful allocation mechanisms.…”

“…In numerical experiments we use uniform finite sets Table 2 shows the best obtained bounds and the k we use to compute these bounds. 1.505980 1.5068 [3] 1.5093 250 3 1.5076 1.5861 [3] 1.5238 50 4 1.5195 1.5628 20…”

“…Next, we compare our approach to the existing methods for upper bounds in [2,3,7,15] that use optimization. Our method for upper bounds generalizes the approach by Chen et al [3]. The generalization considers a broader class of algorithms and provides stronger upper bounds for n ≤ 4.…”

“…First, we present a result by Chen et al [3], which follows from Lu and Yu [19] Proposition 2. [ Chen et al [3]] For any given number of tasks n, P ∈ C n , and T ∈ R 2×n ++ ,…”