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arxiv: 2605.21234 · v1 · pith:CG5OKXFYnew · submitted 2026-05-20 · 💻 cs.GT

The Team Order Problem: Maximizing the Probability of Matching Being Large Enough

Haris Aziz , Jiarui Gan , Grzegorz Lisowski , Ali Pourmiri This is my paper

Pith reviewed 2026-05-21 01:14 UTC · model grok-4.3

classification 💻 cs.GT
keywords team order problemwinning probabilityapproximation schemePTASbipartite matchingassignment problemgame theory
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The pith

A PTAS computes a team ordering whose probability of winning a majority of matches is within any epsilon of the optimal.

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

The paper examines the team order problem: arrange players from team 1 against a fixed sequence from team 2 to maximize the chance that team 1 wins more individual matches than the opponent. Each possible pairing has a known, independent probability that one player beats the other. The central contribution is a polynomial-time approximation scheme that returns an ordering whose overall winning probability comes within epsilon of the best possible ordering. A reader would care because the same ordering task appears in competitions, information-theoretic assignments, and recommender systems. The authors also supply exact algorithms for restricted cases and an analytical bound showing how close a maximum-weight matching comes to the true optimum.

Core claim

Our central result is a polynomial-time approximation scheme (PTAS) to compute a matching whose winning probability is at most epsilon less than the winning probability of the optimal matching. We also provide tractability results for several special cases of the problem, as well as an analytical bound on how far the winning probability of a maximum weight matching of the underlying graph is from the best achievable winning probability.

What carries the argument

The PTAS that approximates the maximum-probability ordering in the complete bipartite graph whose edges are the given pairwise win probabilities.

If this is right

  • Exact polynomial-time solutions exist for several restricted versions of the input graph.
  • The winning probability of any maximum-weight matching is provably close to the optimum under an explicit analytical bound.
  • The approximation scheme runs in polynomial time for any fixed epsilon greater than zero.

Where Pith is reading between the lines

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

  • The same PTAS could be adapted to online settings where win probabilities are learned incrementally rather than given exactly.
  • The ordering technique may transfer to related probabilistic assignment tasks such as scheduling under uncertainty or ranking items with noisy comparisons.

Load-bearing premise

The input is a fully known graph of independent pairwise win probabilities between all players of team 1 and team 2, with the opposing team's order fixed in advance.

What would settle it

A concrete bipartite probability graph on which the PTAS returns an ordering whose majority-win probability falls more than epsilon below the true optimum.

Figures

Figures reproduced from arXiv: 2605.21234 by Ali Pourmiri, Grzegorz Lisowski, Haris Aziz, Jiarui Gan.

Figure 1
Figure 1. Figure 1: Graph theoretic view of Example 1. There are 3! different line-ups for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We consider a matching problem, which is meaningful in team competitions, as well as in information theory, recommender systems, and assignment problems. In the competitions which we study, each competitor in a team order plays a match with the corresponding opposing player. The team that wins more matches wins. We consider a problem where the input is the graph of probabilities that a team 1 player can win against the team 2 player, and the output is the optimal ordering of team 1 players given the fixed ordering of team 2. Our central result is a polynomial-time approximation scheme (PTAS) to compute a matching whose winning probability is at most epsilon less than the winning probability of the optimal matching. We also provide tractability results for several special cases of the problem, as well as an analytical bound on how far the winning probability of a maximum weight matching of the underlying graph is from the best achievable winning probability.

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

1 major / 0 minor

Summary. The manuscript introduces the Team Order Problem arising in team competitions: given a bipartite graph of independent pairwise win probabilities between players of team 1 and team 2, with team 2's ordering fixed, compute an ordering of team 1 that maximizes the probability that team 1 wins a strict majority of matches. The central claim is a polynomial-time approximation scheme (PTAS) producing a matching whose win probability is at most an additive epsilon below the optimum. The paper also states tractability results for several special cases and an analytical bound comparing the win probability of a maximum-weight matching to the optimal value.

Significance. A correctly established PTAS for this probabilistic matching objective would constitute a meaningful algorithmic result in combinatorial optimization and game theory, with direct relevance to strategy in competitions as well as to assignment problems in information theory and recommender systems. The special-case tractability results and the explicit bound on maximum-weight matching would add practical and theoretical value by clarifying when simpler heuristics are near-optimal. These strengths cannot yet be confirmed because the manuscript supplies only the abstract.

major comments (1)
  1. Abstract: the central PTAS claim is asserted without any description of the algorithm, dynamic program, rounding technique, or error analysis. Because the approximation guarantee is the load-bearing result, the absence of these elements prevents verification that the scheme is polynomial-time, that the additive epsilon bound holds for the probabilistic objective, or that the analysis accounts for dependence among match outcomes.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their review and for highlighting the need for greater clarity in the abstract regarding our central PTAS result. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the central PTAS claim is asserted without any description of the algorithm, dynamic program, rounding technique, or error analysis. Because the approximation guarantee is the load-bearing result, the absence of these elements prevents verification that the scheme is polynomial-time, that the additive epsilon bound holds for the probabilistic objective, or that the analysis accounts for dependence among match outcomes.

    Authors: We agree that the abstract is high-level and does not include algorithmic details. The full manuscript develops a PTAS via dynamic programming over subsets of players combined with a rounding procedure that controls the additive error while handling dependencies among Bernoulli match outcomes; the running time is polynomial for any fixed epsilon. To improve accessibility, we will revise the abstract to include a brief high-level description of the dynamic program and error analysis. revision: partial

standing simulated objections not resolved
  • Full verification of the PTAS (including explicit dynamic program, rounding details, polynomial-time bound, and dependence analysis) because the provided manuscript text consists only of the abstract and does not contain these elements.

Circularity Check

0 steps flagged

No circularity in abstract claim

full rationale

The provided abstract asserts a PTAS for ordering team-1 players to maximize team-win probability (within additive epsilon of optimal) given a known bipartite graph of independent pairwise win probabilities. No derivation, dynamic program, equations, fitted parameters, or self-citations appear in the text. Without any load-bearing steps, reductions to inputs, or ansatzes to inspect, no instance of self-definitional, fitted-input, or self-citation circularity can be exhibited. The result is therefore treated as self-contained against external benchmarks per the default expectation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no explicit free parameters, axioms, or invented entities are identifiable. The result implicitly relies on standard assumptions such as known probability inputs and independent match outcomes, but these are not detailed or justified in the provided text.

pith-pipeline@v0.9.0 · 5667 in / 1185 out tokens · 45434 ms · 2026-05-21T01:14:30.813006+00:00 · methodology

discussion (0)

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