Polynomial-time solvability for a constant number of agents with identical valuations

Determine whether cumulative maximum envy minimization (Temporal-EnvyMin) is polynomial-time solvable when the number of agents is constant and all agents have identical valuation functions; establish whether the authors’ conjecture that the answer is negative is correct.

Background

The paper studies Temporal-EnvyMin, in which a fixed sequence of indivisible goods arrives over time, each good must be allocated irrevocably, and the objective is to minimize the sum of the maximum pairwise envy over all rounds. The authors establish strong computational hardness for general instances, including NP-completeness and the nonexistence of constant-factor approximation algorithms under identical valuations and under binary valuations.

For a constant number of agents, the paper provides a dynamic program that is pseudopolynomial in the maximum item value and consequently gives polynomial-time solvability under suitable value bounds. Under identical integer valuations and a fixed number of agents, it also provides an FPTAS. The unresolved question is whether exact polynomial-time solvability holds for constant-agent instances with identical valuations, without imposing bounded or unary-encoded item values. The authors explicitly conjecture that the answer is negative.

References

That being said, an intriguing open question remains: is the problem polynomial-time solvable when the number of agents is constant and agents have identical valuation functions? We conjecture that the answer is `no'.

— Minimizing Cumulative Envy in Allocating a Sequence of Items  (2610.09843 - Goldberg et al., 7 Oct 2026) in Section 3, immediately after Corollary following Theorem 3.2 and before Theorem 3.3 (FPTAS result)