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Dual Bounded Generation: Polynomial, Second-order Cone and Positive Semidefinite Matrix Inequalities

Khaled Elbassioni

TL;DR

This work extends the dual-bounded generation framework for monotone inequality systems beyond linearity to polynomial, second-order cone (SOC), and positive semidefinite (PSD) inequalities, establishing bounds that yield (quasi-)polynomial-time incremental enumeration of maximal feasible vectors. Central to the results are traction-based bounds for real-valued and separable/quasi-separable functions, Möbius-inversion techniques, and finite-cell arguments that bound the size of the dual minimal-infeasible family $\mathcal{I}(\mathcal{F})$, enabling efficient enumeration under well-specified parameter regimes. The paper also demonstrates concrete applications, including fair allocation with Nash welfare, chance-constrained knapsacks and coverings, and quantum hypergraph covers, illustrating the practical impact of the bounds under realistic structural assumptions. Open questions remain on polynomial-time enumeration in broader settings, independence from $t$ and $R$, and fixed-dimension SOC/PSD cases, guiding future research on dual-bounded generation for complex inequality classes.

Abstract

In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.

Dual Bounded Generation: Polynomial, Second-order Cone and Positive Semidefinite Matrix Inequalities

TL;DR

This work extends the dual-bounded generation framework for monotone inequality systems beyond linearity to polynomial, second-order cone (SOC), and positive semidefinite (PSD) inequalities, establishing bounds that yield (quasi-)polynomial-time incremental enumeration of maximal feasible vectors. Central to the results are traction-based bounds for real-valued and separable/quasi-separable functions, Möbius-inversion techniques, and finite-cell arguments that bound the size of the dual minimal-infeasible family , enabling efficient enumeration under well-specified parameter regimes. The paper also demonstrates concrete applications, including fair allocation with Nash welfare, chance-constrained knapsacks and coverings, and quantum hypergraph covers, illustrating the practical impact of the bounds under realistic structural assumptions. Open questions remain on polynomial-time enumeration in broader settings, independence from and , and fixed-dimension SOC/PSD cases, guiding future research on dual-bounded generation for complex inequality classes.

Abstract

In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.
Paper Structure (14 sections, 16 theorems, 78 equations)

This paper contains 14 sections, 16 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Main results.
  3. Some Applications
  4. Fair allocation of goods.
  5. Chance-constrained multi-dimensional knapsack inequalities.
  6. Chance-constrained covering binary programs.
  7. Quantum hypergraph covers.
  8. Supermodular Inequalities
  9. Products of Affine Functions
  10. Separable Monotone Functions and Polynomial Inequalities
  11. Second-order Cone Inequalities
  12. Positive Semidefinite Matrix Inequalities
  13. Some Open Questions
  14. Omitted Proofs

Key Result

Proposition 1

A function $f:\mathcal{C}\to\mathbb R$ is supermodular if and only if, for any $j\in[n]$, for any $z\in\mathcal{C}_j\setminus\{c_j\}$, and for any $\mathbf x\in\mathcal{C}_1\times\ldots\times\mathcal{C}_{j-1}\times\{z\}\times\mathcal{C}_{j+1}\times\ldots\times\mathcal{C}_k$, the difference is monotone in $\mathbf x$.

Theorems & Definitions (29)

  • Proposition 1
  • Theorem 1
  • Lemma 2: Lemma 11 in BEGK-DAM03
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['t1']}
  • Corollary 1
  • Proposition 2
  • Theorem 4
  • Proposition 3
  • ...and 19 more