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Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

Jean Cardinal, Xavier Goaoc, Sarah Wajsbrot

TL;DR

This work studies hitting and covering problems for continuous affine families of convex polyhedra, linking them to robust optimization's two-level finite adaptability. By establishing a hitting-covering duality and leveraging lifting techniques, the authors recast adaptability questions as geometric hitting/covering problems and apply quantifier-elimination to obtain strongly polynomial algorithms when key parameters are fixed. They also develop a one-parameter, fixed-dimension hitting-set algorithm via parametric search on linear programming, achieving strongly quadratic to linear-time performance in realistic regimes. The results yield new tractability for left-hand-side uncertainty in finite adaptability and provide practical algorithms for robust optimization scenarios with affine dependencies. Overall, the paper unifies geometric hitting theory with robust optimization, delivering concrete complexity bounds and methods for otherwise nonlinear, intractable problems.

Abstract

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

TL;DR

This work studies hitting and covering problems for continuous affine families of convex polyhedra, linking them to robust optimization's two-level finite adaptability. By establishing a hitting-covering duality and leveraging lifting techniques, the authors recast adaptability questions as geometric hitting/covering problems and apply quantifier-elimination to obtain strongly polynomial algorithms when key parameters are fixed. They also develop a one-parameter, fixed-dimension hitting-set algorithm via parametric search on linear programming, achieving strongly quadratic to linear-time performance in realistic regimes. The results yield new tractability for left-hand-side uncertainty in finite adaptability and provide practical algorithms for robust optimization scenarios with affine dependencies. Overall, the paper unifies geometric hitting theory with robust optimization, delivering concrete complexity bounds and methods for otherwise nonlinear, intractable problems.

Abstract

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

Paper Structure

This paper contains 28 sections, 15 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hitting affine families of polyhedra
  3. Context: robust optimization and finite adaptability
  4. Our results
  5. Background and related works
  6. Complexity of finite adaptability
  7. Related notions in computational geometry.
  8. Relation between hitting/covering and finite adaptability
  9. The hitting-covering duality
  10. Direct correspondences
  11. A simpler correspondence via lifting
  12. Polynomial complexity bounds
  13. Semi-algebraic sets and first-order formulas
  14. Application to $k$-adaptability
  15. Hitting a one-parameter family of polyhedra
  16. ...and 13 more sections

Key Result

Theorem 1

For every constant $k$, $\ell$, $\delta$, $p$, there is an algorithm that solves the decision version of the $k$-adaptability Problem eq:fa on $m$ constraints, with $\Omega$ a polyhedron in $\mathbb{R}^p$ given as an intersection of $v$ halfspaces, in time strongly polynomial in $m$ and $v$.

Figures (1)

  • Figure 1: Two examples of affine families of polygons together with a hitting set (the black dots); For readability, we represent the polygon $P(\omega)$ for only a finite subset of $\omega$ in the domain.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 4: basu:hal-01083587
  • ...and 17 more