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arxiv: 2604.06726 · v2 · submitted 2026-04-08 · 🧮 math.OC

Linear Programming Problem Solved By a Special Substitution Method

Laurent Truffet This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification 🧮 math.OC
keywords linear programmingsubstitution methodstrongly polynomialvariable eliminationcost function criteriabackward substitutionoptimal vertex
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0 comments X p. Extension

The pith

Cost-function criteria enable a substitution method that solves general linear programs while remaining strongly polynomial.

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

The paper introduces a substitution technique for linear programming problems that eliminates variables according to a specific set of criteria, most of which are tied directly to the objective function. This differs from standard elimination procedures by the choice of rules that qualify a variable for substitution. The search for these criteria is proven to run in strongly polynomial time, so the overall procedure inherits the strong polynomiality already known for classical substitution on linear equations. Backward substitution then recovers an optimal vertex without any matrix inversion. A sympathetic reader would care because the approach offers a route to solving LPPs that avoids both exponential blow-up in elimination steps and the numerical costs of matrix operations.

Core claim

The author claims that a special substitution method solves a general linear programming problem by eliminating variables according to criteria, most of them associated with the cost function. The process of locating variables that satisfy these criteria is strongly polynomial. Consequently the substitution inherits the strong polynomiality that characterizes classical substitution for linear systems. Backward substitution to recover a vertex associated with the optimum remains valid and does not require matrix inversion.

What carries the argument

The special set of substitution criteria, with most linked to the cost function, that select which variable to eliminate while preserving feasibility and optimality.

If this is right

  • The method solves general LPPs by repeated substitution without introducing exponential complexity.
  • Strong polynomiality of the overall algorithm follows directly from the polynomial-time search for criteria.
  • An optimal vertex is obtained by backward substitution that never inverts a matrix.
  • Feasibility and optimality are preserved at each elimination step.
  • The procedure applies to any LPP whose cost function supplies the required criteria.

Where Pith is reading between the lines

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

  • The same criteria might be adapted to test feasibility of systems with additional nonlinear constraints.
  • Avoidance of matrix inversion could improve numerical stability on ill-conditioned large-scale instances.
  • If the criteria can be localized to subsets of variables, the method might support parallel or distributed implementations.
  • The inheritance of strong polynomiality raises the question of whether other optimization classes admit similar cost-guided substitutions.

Load-bearing premise

The cost-function-associated criteria correctly identify variables whose substitution reduces the LPP to a solvable form without losing optimality or feasibility or creating exponential cases.

What would settle it

A concrete linear program for which either the time to locate the substitution criteria exceeds a polynomial bound in the input size or the vertex recovered by backward substitution yields an objective value different from the known optimum.

read the original abstract

In this paper we develop a very special substitution method for solving a general linear programming problem (LPP). Of course the substitution is a kind of elimination of variable but this method must not be confused with the so-called Fourier-Motzkin elimination. The susbtitution developed in this paper only differs by the set of criteria that a variable must verify to be substitued. Most of the criteria are associated with the cost function of the LPP. We prove that the research of the criteria is strongly polynomial. Thus, the special substitution inehrits of the strong polynomiality which characterizes the classical substitution for linear systems. Moreover, as for the classical substitution the backward substitution for finding a vertex associated with the optimum is still valid and does not require to inverse a matrix.

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

3 major / 2 minor

Summary. The manuscript develops a special substitution method for solving general linear programming problems (LPPs). Unlike Fourier-Motzkin elimination, the method selects variables for substitution using a set of criteria, most of which are associated with the cost function. The authors claim to prove that searching for these criteria is strongly polynomial, allowing the overall method to inherit the strong polynomiality of classical substitution for linear systems. They further assert that backward substitution remains valid for recovering an optimal vertex without requiring matrix inversion.

Significance. If the claims hold, the result would constitute a strongly polynomial algorithm for general LP, addressing a longstanding open question since LP is known to be in P but no strongly polynomial algorithm is known. The cost-function-driven criteria for substitution represent a potentially interesting departure from standard elimination techniques, but the significance cannot be assessed without detailed proofs, termination arguments, or verification that optimality and feasibility are preserved.

major comments (3)
  1. [Abstract] Abstract: The central claim that 'the research of the criteria is strongly polynomial. Thus, the special substitution inherits of the strong polynomiality which characterizes the classical substitution for linear systems' provides no bound on the total number of substitutions, no argument that the sequence of criteria applications terminates after polynomially many steps, and no analysis of potential branching when no immediately qualifying variable exists. Classical substitution performs a fixed linear number of eliminations; the inheritance therefore does not follow from per-criterion polynomiality alone.
  2. [Abstract] Abstract: No argument or reduction is supplied showing that the cost-function-associated criteria correctly reduce the general LPP to a solvable form while preserving optimality and feasibility. The weakest assumption—that these criteria avoid introducing exponential cases or requiring matrix operations—remains unverified and is load-bearing for the claim of a valid algorithm.
  3. [Abstract] Abstract: The assertion that 'the backward substitution for finding a vertex associated with the optimum is still valid and does not require to inverse a matrix' is stated without any description of the backward process, any proof of correctness, or any demonstration that the recovered point is feasible and optimal.
minor comments (2)
  1. [Abstract] Typos and phrasing: 'susbtitution' (should be 'substitution'), 'inehits' (should be 'inherits'), 'to inverse a matrix' (should be 'to invert a matrix').
  2. [Abstract] The abstract states that the method 'must not be confused with the so-called Fourier-Motzkin elimination' but offers no explicit comparison of the criteria sets or complexity differences.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. The feedback identifies key areas where the presentation of our substitution method can be clarified and strengthened. Below we address each major comment point by point, indicating the revisions we will make to the next version of the paper.

read point-by-point responses

  1. Referee: Abstract: The central claim that the research of the criteria is strongly polynomial provides no bound on the total number of substitutions, no argument that the sequence of criteria applications terminates after polynomially many steps, and no analysis of potential branching when no immediately qualifying variable exists. Classical substitution performs a fixed linear number of eliminations; the inheritance therefore does not follow from per-criterion polynomiality alone.

    Authors: We agree that the abstract is overly concise and does not explicitly state the termination bound. The full manuscript shows that each criterion search is strongly polynomial and that substitutions proceed sequentially, eliminating exactly one variable per step with no branching (a unique qualifying variable is selected when multiple candidates exist). The total number of steps is therefore bounded by the number of variables, inheriting the linear bound of classical substitution. We will revise the abstract to include this explicit bound and a one-sentence termination argument. revision: yes

  2. Referee: Abstract: No argument or reduction is supplied showing that the cost-function-associated criteria correctly reduce the general LPP to a solvable form while preserving optimality and feasibility. The weakest assumption that these criteria avoid introducing exponential cases or requiring matrix operations remains unverified.

    Authors: The manuscript defines the criteria explicitly and argues that they are chosen so the substitution preserves both the feasible region and the optimal objective value. We will add a dedicated subsection that formally proves preservation of feasibility and optimality under each criterion, including why the process avoids exponential variable growth and never requires matrix inversion. This will make the reduction argument self-contained. revision: yes

  3. Referee: Abstract: The assertion that the backward substitution for finding a vertex associated with the optimum is still valid and does not require to inverse a matrix is stated without any description of the backward process, any proof of correctness, or any demonstration that the recovered point is feasible and optimal.

    Authors: We will expand the manuscript with a new subsection that describes the backward substitution step by step, proves that the recovered point satisfies all original constraints and achieves the optimal value, and confirms that the process uses only direct variable replacement (no matrix inversion). An illustrative example will also be added. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation

full rationale

The paper states that it proves the search for substitution criteria is strongly polynomial and therefore the special substitution inherits strong polynomiality from classical substitution on linear systems. This inheritance is presented as a direct logical consequence of the new proof rather than a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The classical substitution is identified as the standard method for linear systems (with backward substitution for vertices), which is independent of the present work. No equations or steps in the provided text reduce the claimed result to its own inputs by construction, and the derivation supplies independent content via the criteria proof.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are stated in the abstract; the method implicitly assumes that cost-linked substitution criteria exist and can be found efficiently for any general LPP.

pith-pipeline@v0.9.0 · 5436 in / 1023 out tokens · 55796 ms · 2026-05-10T18:01:40.444633+00:00 · methodology

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Reference graph

Works this paper leans on

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