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arxiv: 2604.25033 · v1 · submitted 2026-04-27 · 🧮 math.OC

A polynomial-time solvable class of sparse box-constrained polynomial optimization problems

Aida Khajavirad This is my paper

Pith reviewed 2026-05-08 02:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords polynomial optimizationbox constraintshypergraph sparsitytreewidthpolynomial-time algorithmsunit hypercubebinary optimization
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The pith

A sparsity condition on the hypergraph of a polynomial allows its minimization over the unit hypercube to be solved in polynomial time.

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

The paper establishes a new sufficient condition under which minimizing a multivariate polynomial over the unit hypercube becomes solvable in time polynomial in the input size. By modeling the polynomial via a hypergraph that captures variable interactions, the approach locates a subset of variables that must take binary values at any optimal solution. The remaining continuous variables can then be removed locally whenever they sit inside small, weakly connected blocks. This reduction produces a binary polynomial optimization problem whose structure inherits bounded treewidth and therefore admits an efficient algorithm. The result widens the known tractable class beyond purely binary problems to include certain mixed continuous cases.

Core claim

We establish a sufficient condition based on the hypergraph representation of the polynomial under which the box-constrained optimization problem over the unit hypercube can be solved in polynomial time. The condition identifies variables attaining binary values at optimality and permits local elimination of the remaining continuous variables when they appear in small, weakly coupled blocks, thereby reducing the original problem to a structured binary optimization instance that inherits efficient solvability from bounded-treewidth results.

What carries the argument

The hypergraph representation of the polynomial, which encodes sparsity and enables identification of binary-attaining variables together with local elimination of continuous variables in small weakly coupled blocks.

If this is right

  • Any box-constrained polynomial meeting the hypergraph sparsity condition becomes solvable in polynomial time.
  • The classical bounded-treewidth tractability result for binary polynomial optimization extends directly to this mixed continuous setting.
  • Continuous variables can be removed without losing polynomial-time guarantees provided they occur in sufficiently small and weakly coupled blocks.
  • The reduction yields an explicit algorithmic pipeline: detect the binary subset, eliminate local continuous blocks, then solve the resulting binary instance.

Where Pith is reading between the lines

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

  • The same hypergraph reduction technique could be tested on polynomials with additional linear or quadratic constraints to see whether tractability survives.
  • Checking the condition in practice reduces to finding a suitable vertex cover or block decomposition on the hypergraph, which may itself admit fast heuristics.
  • Instances arising from engineering design or machine-learning loss functions could be preprocessed to test whether they fall into this solvable class.

Load-bearing premise

The polynomial admits a hypergraph representation whose sparsity structure permits identification of a subset of variables attaining binary values at optimality together with local elimination of the remaining continuous variables when they appear in small, weakly coupled blocks.

What would settle it

Construct an explicit polynomial whose hypergraph satisfies the stated sparsity condition yet whose minimum over the unit hypercube cannot be computed in time polynomial in the input encoding length.

read the original abstract

We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which the problem can be solved in time polynomial in the encoding length of the input. Our approach identifies a subset of variables that attain binary values at optimality and shows how the remaining continuous variables can be eliminated locally when they appear in small, weakly coupled blocks, yielding a reduction to a structured binary optimization problem that can be solved efficiently. Our result extends the classical tractability result for binary polynomial optimization, namely, that problems with bounded treewidth are solvable in polynomial time, to box-constrained polynomial optimization.

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 / 3 minor

Summary. The paper studies minimization of a multivariate polynomial over the unit hypercube. Representing the polynomial via a hypergraph and exploiting sparsity, it establishes a sufficient condition for polynomial-time solvability in the input encoding length. The approach identifies variables attaining binary values at optimality, eliminates remaining continuous variables locally within small weakly coupled blocks, and reduces the problem to a bounded-treewidth binary polynomial optimization instance known to be tractable. This extends the classical bounded-treewidth tractability result from binary to box-constrained polynomial optimization.

Significance. If the claimed reduction holds under the stated sparsity condition, the result meaningfully enlarges the class of box-constrained polynomial problems known to be solvable in polynomial time by leveraging hypergraph structure and dynamic programming. It provides a clean theoretical bridge between continuous and discrete sparse optimization, with potential algorithmic implications for nonconvex problems where variables decouple into binary and locally eliminable continuous blocks.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should include an explicit, self-contained statement of the sufficient sparsity condition (e.g., a formal definition of the hypergraph property that guarantees the binary-attaining subset and local elimination).
  2. [Abstract] Clarify whether the polynomial-time bound is with respect to the encoding length alone or also depends on the treewidth parameter; if the latter, state the dependence explicitly.
  3. [§3 or §4] Add a small illustrative example (e.g., a quadratic or cubic polynomial on 4–6 variables) showing the hypergraph, the identified binary variables, and the elimination step before the reduction to the binary instance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The referee's description accurately captures our main result: a sparsity condition on the hypergraph that permits identifying variables attaining binary values at optimality, followed by local elimination of the remaining continuous variables in small weakly coupled blocks, reducing the problem to a bounded-treewidth binary polynomial optimization instance. We appreciate the recognition that this extends the classical tractability result and provides a bridge between continuous and discrete sparse optimization.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central claim establishes a sufficient condition for polynomial-time solvability by representing the objective via a hypergraph, identifying a subset of variables that attain binary values at optimality, and performing local elimination of remaining continuous variables in small weakly coupled blocks. This yields a reduction to a bounded-treewidth binary polynomial optimization instance. The argument invokes classical graph-theoretic tractability results for bounded-treewidth problems and standard hypergraph sparsity representations rather than any self-definitional mapping, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or step reduces to its own inputs by construction; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the ability to represent any polynomial as a hypergraph and on the existence of an efficient algorithm for bounded-treewidth binary polynomial optimization; both are standard in the literature and not invented here.

axioms (2)
  • domain assumption Any multivariate polynomial admits a hypergraph representation whose edges correspond to monomials.
    Invoked to exploit sparsity structure for the reduction.
  • standard math Binary polynomial optimization problems with bounded treewidth are solvable in polynomial time.
    Classical result used as the target of the reduction.

pith-pipeline@v0.9.0 · 5405 in / 1286 out tokens · 32062 ms · 2026-05-08T02:30:28.205961+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry . Springer, 2006

    work page 2006

  2. [2]

    H. L. Bodlaender and A. Koster. Combinatorial optimization on graphs of bounded treewidth. The Computer Journal, 51(3):255–269, 2008

    work page 2008

  3. [3]

    Boros and P.L

    E. Boros and P.L. Hammer. Pseudo-Boolean optimization. Discrete applied mathematics, 123(1):155–225, 2002

    work page 2002

  4. [4]

    Capelli, A

    F. Capelli, A. Del Pia, and S. Di Gregorio. A knowledge compilation take on binary polynomial optimization. arXiv preprint arXiv:2311.00149 , 2023. 14

    work page arXiv 2023

  5. [5]

    Chandrasekaran, N

    V. Chandrasekaran, N. Srebro, and P. Harsha. Complexity of inference in graphical models. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence UAI’08 , 2008

    work page 2008

  6. [6]

    Crama, P

    Y. Crama, P. Hansen, and B. Jaumard. The basic algorithm for pseudo-Boolean programming revisited. Discrete Applied Mathematics, 29(2–3):171–185, 1990

    work page 1990

  7. [7]

    Del Pia and S

    A. Del Pia and S. Di Gregorio. On the complexity of binary polynomial optimization over acyclic hypergraphs. Algorithmica, 85:2189–2213, 2023

    work page 2023

  8. [8]

    Del Pia and A

    A. Del Pia and A. Khajavirad. A polyhedral study of binary polynomial programs. Mathematics of Operations Research, 42(2):389–410, 2017

    work page 2017

  9. [9]

    Del Pia and A

    A. Del Pia and A. Khajavirad. Beyond hypergraph acyclicity: limits of tractability for pseudo-boolean opti- mization. arXiv preprint arxiv:2410.23045 , 2024

    work page arXiv 2024

  10. [10]

    Del Pia and A

    A. Del Pia and A. Khajavirad. The pseudo-boolean polytope and polynomial-size extended formulations for binary polynomial optimization. Mathematical Programming, 212(1):717–761, 2025

    work page 2025

  11. [11]

    S. S. Dey and A. Khajavirad. A second-order cone representable class of nonconvex quadratic programs. Math- ematical Programming, 2026

    work page 2026

  12. [12]

    Gr¨ otschel, L

    M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver. Geometric algorithms and combinatorial optimization , volume 2. Springer Science & Business Media, 2012

    work page 2012

  13. [13]

    Khajavirad

    A. Khajavirad. Tight semidefinite programming relaxations for sparse box-constrained quadratic programs. arXiv preprint arXiv:2601.18545 , 2026

    work page arXiv 2026

  14. [14]

    Nesterov and A

    Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming . SIAM, 1994

    work page 1994

  15. [15]

    J. Renegar. On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. preliminaries. the geometry of semi-algebraic sets. the decision problem for the existential theory of the reals. Journal of symbolic computation , 13(3):255–299, 1992

    work page 1992

  16. [16]

    Schrijver

    A. Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, Series B , 80(2):346–355, 2000

    work page 2000

  17. [17]

    S. P. Tarasov and L. G. Khachiyan. Polynomial solvability of convex quadratic programming. In Doklady akademii nauk, volume 248, pages 1049–1051. Russian Academy of Sciences, 1979. 15

    work page 1979