Markakis & Samaris Solve Fair Division with Conflict Graphs
New algorithm guarantees EF1 allocations even when items can't be bundled together.
A new paper from Evangelos Markakis and Michalis Samaris (arXiv:2606.13083) addresses the challenge of fairly allocating indivisible goods when certain items cannot be placed in the same bundle—modeled as a conflict graph where edges represent conflicts. The problem is known to be hard: EF1 (envy-free up to one good) allocations may not exist under general constraints. The authors parameterize the analysis by the maximum degree Δ of the conflict graph, offering a refined understanding of when fair allocations are possible.
For agents with ordered or tiered valuations, they derive a tight bound on the number of items m that guarantees a complete EF1 allocation. This bound depends on n (number of agents) and Δ, and is exact when Δ > 2n/3. They also give an approximation algorithm when m exceeds this bound. For general additive valuations—a more challenging case—they focus on instances where m ≤ 2n, using a clever combination of Round Robin and matching theory to provide almost complete characterization. The work leverages classical matching results to push the frontier of fair division under constraints, with implications for resource allocation in multi-agent systems, cloud computing, and any domain where item incompatibilities matter.
- Tight existence bound for EF1 with ordered valuations: m ≤ (n-1)(Δ+1) for Δ > 2n/3
- Approximation algorithm provided when number of items exceeds the tight bound
- For general additive valuations, EF1 guaranteed when m ≤ 2n using Round Robin with matchings
Why It Matters
Practical fair allocation algorithms for real-world scenarios with item conflicts (e.g., scheduling, spectrum allocation, cloud resources).