Tight bounds for fractional and randomized truthful scheduling

Georgios Anastasiadis; George Christodoulou; Elias Koutsoupias; Annamaria Kovacs; Conrad Schecker

AI summary · arXiv: 2611.06642v1 · cs.GT

Tight bounds for fractional and randomized truthful scheduling

Georgios Anastasiadis, George Christodoulou, Elias Koutsoupias, Annamaria Kovacs, Conrad Schecker

Original paper Submitted to arXiv on 04 Nov. 2026 License: NONEXCLUSIVE-DISTRIB 1.0 Also in: fr

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The essentials

The paper proves a tight lower bound of (n+1)/2 for truthful fractional scheduling and gives a universally truthful randomized mechanism with ratio n/2+o(n).

Based on the abstract only: check the details in the original paper.

Question

Can randomized truthfulness achieve a sublinear approximation ratio for scheduling on unrelated machines? What is the optimal approximation ratio for truthful fractional scheduling?

Method

The authors analyze truthful fractional mechanisms for scheduling on n unrelated machines, comparing their performance with the optimal integral makespan. They establish a lower bound and present a universally truthful mechanism, then use the relationship between truthful-in-expectation mechanisms and truthful fractional mechanisms to extend the lower bound.

Findings

Result Every deterministic truthful fractional mechanism on n unrelated machines has worst-case approximation ratio at least (n+1)/2, matching the known upper bound.
See the passage in the paper
We prove that every deterministic truthful fractional mechanism on $n$ unrelated machines has worst-case approximation ratio at least $(n+1)/2$, even against the optimal integral makespan.
Result The lower bound also applies to truthful-in-expectation and universally truthful mechanisms.
See the passage in the paper
Since the expected allocations and payments of a truthful-in-expectation mechanism define a truthful fractional mechanism, the same lower bound applies to all truthful-in-expectation mechanisms and, in particular, to all universally truthful mechanisms.
Result The authors give a universally truthful mechanism with approximation ratio n/2+o(n).
See the passage in the paper
On the positive side, we give a universally truthful mechanism with approximation ratio $n/2+o(n)$.
Authors’ interpretation The authors conclude that randomization improves the optimal deterministic approximation ratio by an asymptotic factor of two, but cannot do better.
See the passage in the paper
Thus, randomization improves on the optimal deterministic approximation ratio by an asymptotic factor of two, and no more.

Limitations

The stated results concern scheduling on unrelated machines and do not establish bounds for other scheduling settings.

Why it matters

The results settle the optimal approximation ratio for truthful fractional scheduling and establish that randomized truthful scheduling has optimal leading constant 1/2. They show that randomization improves over deterministic scheduling by an asymptotic factor of two, but cannot yield a sublinear approximation.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: truthful scheduling unrelated machines fractional allocation approximation ratio randomized mechanisms

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Comprehensive Summary
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Original abstract

The paper studies truthful scheduling on unrelated machines. Each machine may have a different processing time for each task, and the objective is to assign tasks to minimize the makespan—the time when the last machine finishes. Because machines may misreport their processing times, the paper asks how well a mechanism can schedule tasks while making truthful reporting advantageous, and whether randomization can substantially improve the approximation guarantee. The main technical result concerns fractional scheduling, where a task may be split among machines. The authors prove that every deterministic truthful fractional mechanism on $n$ unrelated machines has a worst-case approximation ratio of at least $(n+1)/2$, measured against the optimal integral makespan. This matches the known upper bound, establishing an optimal ratio of exactly $(n+1)/2$ for truthful fractional scheduling. Notably, the result holds even though the mechanism may allocate tasks fractionally while the benchmark is an integral schedule. The abstract extends this lower bound to randomized mechanisms by observing that the expected allocations and payments of a truthful-in-expectation mechanism form a truthful fractional mechanism. The fractional lower bound therefore also applies to truthful-in-expectation mechanisms, including universally truthful mechanisms, which randomize over individually truthful mechanisms. Under these truthfulness requirements, randomization cannot yield a sublinear approximation ratio: the lower bound grows linearly with the number of machines. The paper also presents a universally truthful mechanism with an approximation ratio of $n/2+o(n)$. Together with the lower bound, this establishes an asymptotically optimal leading constant of $1/2$ for randomized truthful scheduling. The abstract thus claims that randomization improves the optimal deterministic approximation ratio by an asymptotic factor of two, but cannot improve it beyond this scale. The available material consists only of the title, authors, and abstract; the full text is unavailable under the paper’s license. Accordingly, this summary reports the stated results and the high-level connection between fractional and randomized mechanisms, but does not describe the proof techniques, construction details, unstated assumptions, or technical caveats. The abstract does not specify the universally truthful mechanism, derive its approximation ratio, or present the instances used to establish the lower bound. These details would be needed for a fuller assessment of the methods and their scope.

Truthful Scheduling Has a Sharp Price

Scheduling jobs across machines sounds straightforward: assign each job so that the machine that finishes last completes as soon as possible. But the problem changes when machines have private information. A machine may know how long it takes to process each job, while a scheduling mechanism does not. If machines can benefit by misreporting those processing times, the mechanism must account for strategic behavior as well as efficiency.

“Tight bounds for fractional and randomized truthful scheduling,” by Georgios Anastasiadis, George Christodoulou, Elias Koutsoupias, Annamaria Kovacs, and Conrad Schecker, studies how well such mechanisms can perform. Its central message is that truthful scheduling has a substantial, precisely quantified cost—and that randomization can reduce this cost, but not eliminate it.

The scheduling problem

The setting is one of unrelated machines: each machine can take a different amount of time to process each task. The objective is to minimize the makespan, meaning the time at which the last machine finishes. To make good assignments, a mechanism needs accurate information about processing times. Yet when machines report those times themselves, the reports may be strategic.

Truthfulness requires that a machine does best by reporting its information honestly, rather than gaining an advantage through a false report. The paper asks what approximation guarantee is possible when scheduling must satisfy this requirement. An approximation ratio compares the mechanism’s makespan with the optimal integral schedule: a ratio of 2, for example, means the mechanism’s result is at most twice the benchmark.

A tight bound for fractional scheduling

The main result concerns deterministic truthful fractional mechanisms. In fractional scheduling, a task may be split across multiple machines instead of being assigned entirely to one. Even with this added flexibility, the authors prove that every such mechanism has a worst-case approximation ratio of at least (n+1)/2 on n machines, when compared with the optimal integral makespan.

This lower bound matches a known upper bound. Together, they establish an optimal ratio of exactly (n+1)/2 for deterministic truthful fractional scheduling. The comparison is notable: the mechanism is allowed to produce fractional assignments, but its performance is measured against an integral schedule. The result says that fractional allocations do not let deterministic truthful mechanisms beat this benchmark.

Why the result reaches randomized mechanisms

The paper also connects fractional scheduling to randomization. A truthful-in-expectation mechanism may randomize its allocations and payments, while requiring that truthful reporting maximizes a machine’s expected outcome. As described in the abstract, the expected allocations and payments of such a mechanism form a truthful fractional mechanism. The fractional lower bound therefore applies to truthful-in-expectation mechanisms as well.

Universally truthful mechanisms are covered too. These mechanisms randomize over individually truthful mechanisms, so each mechanism selected by the randomization is truthful on its own. The paper’s stated result means that, under these truthfulness requirements, randomization cannot produce a sublinear approximation ratio: the lower bound still grows linearly with the number of machines.

Randomization helps, but only up to a point

The lower bound is paired with an upper bound. The authors present a universally truthful mechanism with approximation ratio n/2 + o(n). This matches the lower bound’s leading term as the number of machines grows, establishing an asymptotically optimal leading constant of one-half for randomized truthful scheduling.

In broad terms, this is an asymptotic factor-of-two improvement over the optimal deterministic ratio, whose leading term is also one-half of n but whose exact expression is (n+1)/2. The gain is meaningful, but limited: randomization improves the scale of the guarantee only within the bounds established by the paper. It does not make truthful scheduling nearly as efficient as unrestricted optimization when the number of machines is large.

What the abstract establishes—and what it leaves open

The results provide a clear headline: deterministic truthful fractional scheduling has a tight exact bound, while randomized truthful scheduling has an asymptotically tight bound. The link between expected allocations and truthful fractional mechanisms is the stated reason the lower bound extends to randomized settings.

The available material is limited to the paper’s title, authors, and abstract. It does not provide the proof techniques, the construction of the universally truthful mechanism, or the specific instances used for the lower bound. Nor does it specify further assumptions or technical caveats. Those details would be necessary to assess how the arguments work and how broadly the results apply.

Even at the level of the abstract, the paper draws a useful boundary around the role of randomization. It can improve the leading constant compared with deterministic truthful scheduling, yet truthfulness still imposes a cost that scales with the number of machines. For this problem, strategic reliability and near-optimal makespan cannot both be achieved without compromise.

Created on 09 Nov. 2026

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Tight bounds for fractional and randomized truthful scheduling

The essentials

Question

Method

Findings

Limitations

Why it matters

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Definitions

Truthful Scheduling Has a Sharp Price

The scheduling problem

A tight bound for fractional scheduling

Why the result reaches randomized mechanisms

Randomization helps, but only up to a point

What the abstract establishes—and what it leaves open

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