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arxiv: 2602.17346 · v2 · pith:QCKY74JWnew · submitted 2026-02-19 · 💻 cs.DM · cs.DS· cs.LG

Partial Optimality in the Preordering Problem

David Stein , Jannik Irmai , Bjoern Andres This is my paper

Pith reviewed 2026-05-15 20:53 UTC · model grok-4.3

classification 💻 cs.DM cs.DScs.LG
keywords preorderingpartial optimalityNP-hard combinatorial optimizationbioinformaticssocial network analysispreorder maximization
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The pith

New partial optimality conditions decide more pairs that cannot appear in any optimal preorder.

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

The preordering problem asks for a reflexive and transitive relation on a finite set that maximizes the sum of given pairwise values for related pairs. This is NP-hard, so exact solutions remain limited to small instances. The paper contributes new conditions that identify additional pairs which cannot be related in any optimal preorder, together with efficient algorithms to test them. Experiments on real and synthetic data show these conditions raise the fraction of pairs for which the relation can be decided efficiently.

Core claim

Building on existing partial optimality techniques, the paper introduces new conditions and algorithms that correctly decide, for a larger fraction of pairs ab, that a ≰ b holds in every optimal preorder.

What carries the argument

Partial optimality conditions that certify pairs ab for which a cannot precede b in any optimal preorder, decided by efficient algorithms.

If this is right

  • A larger share of pairwise decisions can be fixed before invoking a full solver.
  • The fraction of undecided pairs shrinks on both real and synthetic instances.
  • The approach applies directly to preordering instances arising in bioinformatics and social network analysis.
  • The same algorithmic template can be used to certify the opposite relation a ≲ b for selected pairs.

Where Pith is reading between the lines

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

  • Exact solvers for the full preordering problem can start from a larger set of fixed decisions and therefore finish faster on moderate-sized instances.
  • The conditions may combine with existing branch-and-cut frameworks to reduce the size of the search tree.
  • Similar partial-optimality ideas could be tested on related problems such as ranking or clustering with order constraints.

Load-bearing premise

The new conditions are valid and the algorithms decide them without incorrectly ruling out pairs that do appear in some optimal preorder.

What would settle it

A concrete instance together with an optimal preorder that includes a pair the new conditions claim cannot appear, or an optimal preorder containing a pair the algorithm fails to decide.

Figures

Figures reproduced from arXiv: 2602.17346 by Bjoern Andres, David Stein, Jannik Irmai.

Figure 1
Figure 1. Figure 1: An instance of the preordering problem with the set V = {i, j, k, l, m} and the values c is defined by the table on the left. An optimal solution, i.e. a maximum value preorder ≲, is shown on the right with arrows from a to b indicating a ≲ b. The symmetric subset of the preorder is an equivalence relation and thus defines a partition, or clustering, of V (gray). The anti￾symmmetric subset well-defines a p… view at source ↗
Figure 2
Figure 2. Figure 2: In the example depicted above, must-join constraints (solid black arrows) and must-cut constraints (dashed black arrow) defined by a partial characteristic function of preorders, x˜ ∈ X˜V , imply additional must-join constraints (solid gray arrows) as well as additional must-cut constraints (dashed gray arrows). In this case, the closure clV x˜ is strictly more specific than x˜. Definition 4.3. Let V ̸= ∅ … view at source ↗
Figure 3
Figure 3. Figure 3: The map σij ◦ σδ(V \U,U) ◦ σδ(U′ ,V \U′) transforms the preorder depicted on the left to the preorder depicted on the right. and cuts. Moreover, it incorporates already-established par￾tial optimality. More efficiently decidable than the condition of Prop. 6.2 is the condition of the following specialization: Corollary 6.3. Let V ̸= ∅ finite, c ∈ R PV , xˆ ∈ Xˆ V , U ⊆ V and xˆ ∈ Xˆ V such that xˆ −1 (1) ∩… view at source ↗
Figure 4
Figure 4. Figure 4: Depicted above is an instance of the preordering problem with V = {p, q, r, s}, with non-zero values c shown as directed edges, and with zero values not shown. It witnesses the fact that Prop. 6.7 is not subsumed by Cor. 6.8; see Example 6.10. 7. Algorithms 7.1. Applying Prop. 6.2 We search for subsets U ⊆ V that satisfy the condition of Prop. 6.2 for a fixed pair ij ∈ PV by minimizing the rhs. of (4), i.e… view at source ↗
Figure 5
Figure 5. Figure 5: Shown above are the percentage of fixed variables (Row 1) and runtimes (Row 2) for applying Prop. 6.2, Cor. 6.3, Prop. 6.6, Prop. 6.7 (with Prop. 7.2) and Cor. 6.9 for b ∈ {0, 1} individually to instances of the synthetic dataset with respect to α ∈ [0, 1], |V | = 40 and pE ∈ {0.25, 0.50, 0.75}. 0 20 40 60 80 100 |dom ˆx| |PV | [%] pE = 0.25 pE = 0.50 pE = 0.75 0 0.5 1 10−1 101 103 α Runtimes [s] 0 0.5 1 α… view at source ↗
Figure 7
Figure 7. Figure 7: Shown above are the percentage of fixed variables (Row 1) and runtimes (Row 2) for applying Props. 6.2, 6.6 and 6.7 and Cors. 6.3 and 6.9 jointly to instances of the synthetic dataset with α ∈ {0.25, 0.65, 0.70, 0.75} and pE ∈ {0.25, 0.50, 0.75}. 0 100 200 0 20 40 60 80 100 |dom ˆx| |PV | [%] Prop. 6.2 0 100 200 Cor. 6.3 0 100 200 Cuts+Prop. 6.6 0 100 200 Cuts+Prop. 6.7 (Prop. 7.2) 0 100 200 Cor. 6.9 (b = … view at source ↗
Figure 8
Figure 8. Figure 8: Shown above are the percentage of fixed variables (Row 1) as well as the runtimes (Row 2) for applying Prop. 6.2, Cor. 6.3, Prop. 6.6, Prop. 6.7 (together with Prop. 7.2) and Cor. 6.9 for b ∈ {0, 1} individually as well as all condi￾tions jointly (Column 7) to instances of the Twitter dataset. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Preordering is a generalization of clustering and partial ordering with applications in bioinformatics and social network analysis. Given a finite set $V$ and a value $c_{ab} \in \mathbb{R}$ for every ordered pair $ab$ of elements of $V$, the preordering problem asks for a preorder $\lesssim$ on $V$ that maximizes the sum of the values of those pairs $ab$ for which $a \lesssim b$. Building on the state of the art in solving this NP-hard problem partially, we contribute new partial optimality conditions and efficient algorithms for deciding these conditions. In experiments with real and synthetic data, these new conditions increase, in particular, the fraction of pairs $ab$ for which it is decided efficiently that $a \not\lesssim b$ in an optimal preorder.

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 addresses the NP-hard preordering problem: given a set V and costs c_ab for each ordered pair, find a preorder ≲ maximizing the sum of c_ab over pairs satisfying a ≲ b. Building on prior reduction rules for partial optimality, the authors derive new partial optimality conditions together with efficient decision procedures. Experiments on real and synthetic instances show that the new conditions decide a strictly larger fraction of pairs ab as a ≰ b in every optimal preorder.

Significance. If the new conditions are valid and the algorithms decide them correctly, the work strengthens the toolkit for partial solving of the preordering problem. Larger reduction rates directly shrink the search space for exact solvers and are therefore practically relevant for the cited applications in bioinformatics and social-network analysis. The contribution is incremental but well-targeted; the experimental evidence of improved reduction power is the main source of value.

minor comments (2)
  1. [Section 5.1] The description of the synthetic data generator (Section 5.1) omits the precise distribution used for the cost matrix entries; adding the exact sampling procedure would improve reproducibility.
  2. [Definition 3.2] Notation for the new conditions (Definition 3.2) re-uses the symbol ≼ already employed for the preorder; a distinct symbol or explicit qualification would avoid momentary ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on partial optimality conditions for the preordering problem. We appreciate the recognition that the new conditions and decision procedures strengthen the toolkit for this NP-hard problem and that the experimental results demonstrate improved reduction power. The recommendation for minor revision is noted, and we will prepare a revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives new partial optimality conditions for the preordering problem by building directly on prior state-of-the-art reduction rules for the same NP-hard problem. The central contribution consists of valid conditions that correctly identify pairs ab for which a ≰ b must hold in every optimal preorder, together with efficient decision algorithms. These are validated by experiments showing an increased fraction of decided pairs on real and synthetic data. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the result to its own inputs appear in the derivation chain; the new conditions add independent content beyond the cited prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are identifiable. The work relies on standard definitions of preorders and NP-hardness.

axioms (1)
  • standard math The preordering problem is NP-hard.
    Invoked in the abstract as background for the partial solving approach.

pith-pipeline@v0.9.0 · 5435 in / 1064 out tokens · 23497 ms · 2026-05-15T20:53:25.864883+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    Jacob, J., Jentsch, M., Kostka, D., Bentink, S., and Spang, R

    URL https://openreview.net/forum ?id=cBsUnv7Cb3. Jacob, J., Jentsch, M., Kostka, D., Bentink, S., and Spang, R. Detecting hierarchical structure in molecular char- acteristics of disease using transitive approximations of directed graphs. Bioinformatics, 24(7):995–1001, 2008. doi: 10.1093/bioinformatics/btn056. Lange, J., Andres, B., and Swoboda, P. Combi...

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    Weller, M., Komusiewicz, C., Niedermeier, R., and Uhlmann, J

    doi: 10.11606/resimeusp.v3i3.74876. Weller, M., Komusiewicz, C., Niedermeier, R., and Uhlmann, J. On making directed graphs transitive. Jour- nal of Computer and System Sciences , 78(2):559–574,

    work page doi:10.11606/resimeusp.v3i3.74876

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    9 Partial Optimality in the Preordering Problem A

    doi: https://doi.org/10.1016/j.jcss.2011.07.001. 9 Partial Optimality in the Preordering Problem A. Proofs For any set V and any E′ ⊆ PV , abbreviate the transitive closure of E′ by tcl E′ := {pq ∈ PV | P (V,E ′)(p, q) ̸= ∅} . Proof of Prop. 3.2 Proof. Let x∗ ∈ X be an optimal solution to maxx∈X φ(x) such that x∗ /∈ X ′. Then, σ(x∗) is also optimal since ...

    work page doi:10.1016/j.jcss.2011.07.001 2011

  4. [4]

    6.2 is fulfilled for every ij ∈ δ(U, V \ U) as c− ij ≥ X pq∈δ(U,V \U )\ˆx−1(0) c+ pq = 0

    Thus, Prop. 6.2 is fulfilled for every ij ∈ δ(U, V \ U) as c− ij ≥ X pq∈δ(U,V \U )\ˆx−1(0) c+ pq = 0 . By Prop. 6.2, there exists an optimal solution x∗ ∈ XV [ˆx] such that x∗ ij = 0. By iteration over all ij ∈ δ(U, V \ U) \ ˆx−1(0), the claim follows. Proof of Lemma 6.4 Proof. Let x ∈ XV [ˆx], and ¯x = γij|1(x). If xij = 1, we have x′ = x. Now, we consid...

    work page

  5. [5]

    Since γij|1(x)ij = 1 for all x ∈ XV [ˆx], there is an optimal solu- tion x∗ to POPV,c[ˆx] such that x∗ ij = 1

    Now: φc(x′) − φc(x) = X pq∈P ′ 01\{ij} cpq x′ pq − xpq + cij + X pq∈P ′ 10 cpq x′ pq − xpq ≥ − X pq∈P ′ 01\{ij} c− pq + cij − X pq∈P ′ 10 c+ pq = − X pq∈P ′ 01 c− pq + c+ ij − X pq∈P ′ 10 c+ pq (5) ≥ 0 Thus, the map γij|1 is improving for POPV,c[ˆx]. Since γij|1(x)ij = 1 for all x ∈ XV [ˆx], there is an optimal solu- tion x∗ to POPV,c[ˆx] such that x∗ ij ...

    work page

  6. [6]

    For every pq ∈ δ(U, V \ U) we have x′ pq = 0 and thus pq /∈ P01[τ]

    Obviously, P01[τ] ∩ ˆx−1(1) = ∅. For every pq ∈ δ(U, V \ U) we have x′ pq = 0 and thus pq /∈ P01[τ]. For every pq ∈ δ(V \ U, U) such that for every r ∈ U it holds ˆxpr = 0 or yrq = 0 , we have that x′ pq = 0 by definition of τ, and therefore pq /∈ P01[τ]. Thus, P01[τ] ∩ δ(U) ⊆ P ′ 01. Secondly, we show that P10[τ] ∩ δ(U) ⊆ P ′

    work page

  7. [7]

    For every pq ∈ δ(V \ U, U) we have that xpq = 1 implies x′ pq = 1, by definition of τ

    Obviously, ˆx−1(0) ∩ P10[τ] = ∅. For every pq ∈ δ(V \ U, U) we have that xpq = 1 implies x′ pq = 1, by definition of τ. Therefore, δ(V \ U, U) ∩ P10[τ] = ∅. Thus, P10[τ] ∩ δ(U) ⊆ P ′ 10. (2) Let τ = τ y δ(V \U,U) and x′ = τ(x). Firstly, we show that P01[τ] ∩ δ(U) ⊆ P ′

    work page

  8. [8]

    For every pq ∈ δ(V \ U, U) we have x′ pq = 0 and thus, pq /∈ P01[τ]

    Obviously, P01[τ] ∩ ˆx−1(1) = ∅. For every pq ∈ δ(V \ U, U) we have x′ pq = 0 and thus, pq /∈ P01[τ]. For every pq ∈ δ(U, V \ U) such that for every r ∈ U it holds ypr = 0 or ˆxrq = 0 , we have that x′ pq = 0 by definition of τ, and therefore pq /∈ P01[τ]. Thus, P01[τ] ∩ δ(U) ⊆ P ′ 01. Secondly, we show that P10[τ] ∩ δ(U) ⊆ P ′

    work page

  9. [9]

    For every pq ∈ δ(U, V \ U) we have that xpq = 1 implies x′ pq = 1 by definition of τ

    Obviously, ˆx−1(0) ∩ P10[τ] = ∅. For every pq ∈ δ(U, V \ U) we have that xpq = 1 implies x′ pq = 1 by definition of τ. Therefore, δ(U, V \ U) ∩ P10[τ] = ∅. Thus, P10[τ] ∩ δ(U) ⊆ P ′ 10. (3) Let τ = τ y δ(U ) and x′ = τ(x). Firstly, we show that P01[τ] ∩ δ(U) ⊆ P ′

    work page

  10. [10]

    Therefore, P01[τ] ∩ δ(U) = ∅ ⊆ P ′ 01 = ∅

    For every pq ∈ δ(U) we have x′ pq = 0 and thus pq /∈ P01[τ]. Therefore, P01[τ] ∩ δ(U) = ∅ ⊆ P ′ 01 = ∅. Secondly, we show that P10[τ] ∩ δ(U) ⊆ P ′

    work page

  11. [11]

    Thus, P10[τ] ∩ δ(U) ⊆ P ′ 10

    Obviously, ˆx−1(0) ∩ P10[τ] = ∅. Thus, P10[τ] ∩ δ(U) ⊆ P ′ 10. Proof of Cor. 7.5 Proof. Let τ ∈ { τ y δ(U,V \U ), τ y δ(V \U,U), τ y δ(U )}, x ∈ XV [ˆx] and x′ = τ(x). By Lemma 7.3, we have that P ′′ 01 ∩ P ′′ 10 = ∅. 13 Partial Optimality in the Preordering Problem Moreover, for everypq ∈ δ(U) \(P ′′ 01 ∪ P ′′

    work page

  12. [12]

    Thus, for every pq ∈ P ′′ 01 we have that xpq − x′ pq ∈ {0, −1}, and for every pq ∈ P ′′ 10 we have that xpq − x′ pq ∈ {0, 1}

    it follows that pq /∈ P10[τ] ∪ P01[τ], and therefore x′ pq = xpq. Thus, for every pq ∈ P ′′ 01 we have that xpq − x′ pq ∈ {0, −1}, and for every pq ∈ P ′′ 10 we have that xpq − x′ pq ∈ {0, 1}. Therefore: X pq∈δ(U ) cpq xpq − x′ pq = X pq∈P ′′ 01 cpq(xpq − x′ pq) + X pq∈P ′′ 10 cpq(xpq − x′ pq) ≤ X pq∈P ′′ 01 c− pq + X pq∈P ′′ 10 c+ pq This concludes the p...

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  13. [13]

    Thus, for every pq ∈ P ′ 01 we have that xpq − x′ pq ∈ {0, −1}, and for every pq ∈ P ′ 10 we have that xpq − x′ pq ∈ {0, 1}

    it follows that pq /∈ P10[τ] ∪ P01[τ], and therefore x′ pq = xpq. Thus, for every pq ∈ P ′ 01 we have that xpq − x′ pq ∈ {0, −1}, and for every pq ∈ P ′ 10 we have that xpq − x′ pq ∈ {0, 1}. Therefore: X pq∈δ(U ) cpq xpq − x′ pq = X pq∈P ′ 01 cpq(xpq − x′ pq) + X pq∈P ′ 10 cpq(xpq − x′ pq) ≤ X pq∈P ′ 01 c− pq + X pq∈P ′ 10 c+ pq This concludes the proof. ...

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