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arxiv: 2402.04353 · v2 · submitted 2024-02-06 · 💻 cs.GT

Fair Interval Scheduling of Indivisible Chores

Sarfaraz Equbal , Rohit Gurjar , Yatharth Kumar , Swaprava Nath , Raghuvansh Saxena , Rohit Vaish This is my paper

Pith reviewed 2026-05-24 03:57 UTC · model grok-4.3

classification 💻 cs.GT
keywords fair chore allocationinterval schedulingEF1 fairnessmonotone valuationsinterval graphsmaximal schedulesconflict constraints
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The pith

Polynomial-time algorithm computes an EF1 and maximal schedule for two agents on interval chore graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a polynomial-time algorithm that finds a schedule of indivisible chores satisfying envy-freeness up to one chore (EF1) and maximality for two agents with monotone valuations when conflicts form an interval graph. The method relies on a coloring technique for interval graphs. For identical additive valuations on path graphs, an EF1 and maximal schedule exists for any number of agents via a reduction to the cycle-plus-triangles theorem. Efficient algorithms are also given for four or more agents under dichotomous valuations. Stronger combinations such as EFX with maximality or EF1 with Pareto optimality are shown to be impossible.

Core claim

For two agents with monotone valuations whose chore conflicts form an interval graph, there exists a polynomial-time algorithm to compute an EF1 and maximal schedule using a coloring technique on the interval graph.

What carries the argument

A coloring technique in interval graphs that produces an EF1 maximal schedule.

If this is right

  • For two agents, EF1 and maximal schedules are computable in polynomial time when conflicts form an interval graph.
  • Existence of EF1 and maximal schedules holds for any number of agents with identical valuations when conflicts form a path graph.
  • An efficient algorithm exists for four or more agents with dichotomous valuations.
  • EFX along with maximality cannot be achieved, nor can EF1 along with Pareto optimality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The coloring technique may extend to other structured conflict graphs.
  • These scheduling results could guide practical systems assigning time-windowed tasks.
  • The separation between achievable and impossible fairness-efficiency pairs highlights limits for general conflict structures.

Load-bearing premise

The conflict constraints between chores form an interval graph.

What would settle it

An instance with two agents and monotone valuations on an interval graph where no EF1 maximal schedule exists or cannot be found in polynomial time.

Figures

Figures reproduced from arXiv: 2402.04353 by Raghuvansh Saxena, Rohit Gurjar, Rohit Vaish, Sarfaraz Equbal, Swaprava Nath, Yatharth Kumar.

Figure 1
Figure 1. Figure 1: Summary of our results. The arrows denote logical implications between fairness and efficiency notions. The positive and negative results are shown in green and red, respectively. No. of agents Path graph Interval graph n = 2 ✓ for monotone valuations ✓ for monotone valuations (Theorem 1) (Theorem 3) arbitrary n ✓ for identical dichotomous ✓ for identical valuations and valuations and n ≥ 4 bounded compone… view at source ↗
Figure 2
Figure 2. Figure 2: (a) A scheduling instance and (b) its conflict graph. Conflict graph. Given any CSP instance ⟨A, C, T , V ⟩, we will find it convenient to define its conflict graph G = (C, E) as follows [Hummel and Hetland, 2022, Chiarelli et al., 2023]: The set of vertices is the set of chores, and for any pair of vertices ci , cj ∈ C, there is an undirected edge {ci , cj} if and only if ci and cj overlap; see [PITH_FUL… view at source ↗
Figure 3
Figure 3. Figure 3: The instance used in Example 4 (left) and its conflict graph and valuations (right). Suppose agent a1 goes first in the round robin algorithm. Then the induced schedule is X1 = {c1, c3, c5, c7} and X2 = {c4, c2, c6, c8}. Note that v2(X2 \ c) < v2(X1) for every c ∈ X2, implying that the schedule fails EF1. The reason why round robin fails EF1 in the scheduling model is that after picking c4 in the first rou… view at source ↗
Figure 4
Figure 4. Figure 4: Illustrating the coloring technique on a path graph. • R1 = {ch : h is odd} and B1 = {ch : h is even}. • For any i ∈ {2, 3, . . . , m − 1}, – Ri contains {ch : 1 ≤ h < i, h is even} ∪ {ch : i < h ≤ k, h is odd}, – Bi contains {ch : 1 ≤ h < i, h is odd} ∪ {ch : i < h ≤ k, h is even}, – and the chore ci is unassigned ci+1. • Rk = {ch : h is even} and Bk = {ch : h is odd}. Feasibility: It is easy to see that … view at source ↗
Figure 5
Figure 5. Figure 5: Pictorial representation of first and the last schedule of the sequence bundle, while unmarked chores will retain their unassigned status. Feasibility and maximality of X0 : According to the definition of a marked chore, any chore ch can overlap with at most two other marked chores, namely, ch−1 and ch+1 . If h is even, then h − 1 and h + 1 must be odd, and vice versa. Therefore, in the X0 schedule, it fol… view at source ↗
Figure 6
Figure 6. Figure 6: Visual illustration of all five scenarios along with the reassignment guidelines for unsupported chores. Letters with stock represent the color previously assigned to the chore, while the color mentioned above it reflects the assignment after reassignment. Now, consider the case when Xm−i has unsupported chores from Ui . Then from, induction hypothesis for Claim 2, we know that ci , ci−1, and ci−2 are in t… view at source ↗
Figure 7
Figure 7. Figure 7: Illustrating the weighted round-robin algorithm when the number of agents is even. The large and small squares denote the heavy and light chores, respectively. The strongly (respectively, lightly) shaded squares denote the chores that are originally present (respectively, the dummy chores). Thick borders around the squares denote that in that round, all agents receive an original chore, while the squares w… view at source ↗
Figure 8
Figure 8. Figure 8: Illustrating the weighted round-robin algorithm when the number of agents is odd. The drawing convention is the same as described in the caption of [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
read the original abstract

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents with identical additive valuations, we show the existence of an EF1 and maximal schedule when the constraints constitute a path graph. This result uses a reduction to the ``cycle-plus-triangles'' theorem. Using different techniques, we provide an efficient algorithm for finding such a schedule when there are four or more agents and the valuations are further assumed to be dichotomous. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies fair scheduling of indivisible chores subject to interval-based conflict constraints. Its central claim is a polynomial-time algorithm that computes an EF1 and maximal schedule for two agents with monotone valuations on arbitrary interval graphs, obtained via a coloring technique on interval graphs. It further shows existence of EF1 and maximal schedules for any number of agents with identical additive valuations when conflicts form a path graph (via reduction to the cycle-plus-triangles theorem), gives an efficient algorithm for four or more agents under dichotomous valuations, and proves impossibility results for EFX+maximality and for EF1+ Pareto optimality.

Significance. If the main algorithmic result holds, the work supplies an efficient method for simultaneously achieving EF1 fairness and maximality in a natural constrained fair-division setting; the interval-graph coloring technique is noted as potentially reusable. The reduction to the cycle-plus-triangles theorem supplies a non-trivial link between fair division and classical graph theory.

minor comments (2)
  1. [Abstract] Abstract: the two-agent result is stated for monotone valuations while the path-graph result uses additive valuations; a brief clarification of the relationship (or whether monotonicity is strictly weaker) would aid readability.
  2. [Abstract] Abstract: the path-graph result is stated only as existence; it would be useful to indicate explicitly whether a polynomial-time algorithm is also obtained or whether the result is non-constructive.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the algorithmic contribution and the connection to the cycle-plus-triangles theorem, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central result is a constructive polynomial-time algorithm for EF1+maximal schedules on interval graphs for two agents, obtained by applying standard interval-graph coloring to produce the schedule. The path-graph existence result invokes the external cycle-plus-triangles theorem. No derivation step reduces by definition or construction to its own inputs, no fitted parameters are relabeled as predictions, and all cited theorems are external rather than self-referential. The paper is self-contained against the stated graph-theoretic assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard results from graph theory; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Interval graphs admit efficient coloring
    Invoked to obtain the two-agent polynomial-time algorithm.
  • standard math Cycle-plus-triangles theorem holds for the relevant graphs
    Used in the reduction for existence on path graphs.

pith-pipeline@v0.9.0 · 5789 in / 1266 out tokens · 26066 ms · 2026-05-24T03:57:33.963906+00:00 · methodology

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Reference graph

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