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A Knowledge Compilation Take on Binary Polynomial Optimization

Florent Capelli, Alberto Del Pia, Silvia Di Gregorio

TL;DR

This paper draws a novel connection between BPO and the field of Knowledge Compilation, which allows to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations.

Abstract

The Binary Polynomial Optimization (BPO) problem is defined as the problem of maximizing a given polynomial function over all binary points. The main contribution of this paper is to draw a novel connection between BPO and the field of Knowledge Compilation. This connection allows us to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations. In particular, for instances of BPO with hypergraphs that are either $β$-acyclic or with bounded incidence treewidth, we obtain strongly polynomial algorithms for BPO, and extended formulations of polynomial size for the corresponding multilinear polytopes. The generality of our technique allows us to obtain the same type of results for extensions of BPO, where we enforce extended cardinality constraints on the set of binary points, and where variables are replaced by literals. We also obtain strongly polynomial algorithms for the variant of the above problems where we seek $k$ best feasible solutions, instead of only one optimal solution. Computational results show that the resulting algorithms can be significantly faster than current state-of-the-art.

A Knowledge Compilation Take on Binary Polynomial Optimization

TL;DR

This paper draws a novel connection between BPO and the field of Knowledge Compilation, which allows to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations.

Abstract

The Binary Polynomial Optimization (BPO) problem is defined as the problem of maximizing a given polynomial function over all binary points. The main contribution of this paper is to draw a novel connection between BPO and the field of Knowledge Compilation. This connection allows us to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations. In particular, for instances of BPO with hypergraphs that are either -acyclic or with bounded incidence treewidth, we obtain strongly polynomial algorithms for BPO, and extended formulations of polynomial size for the corresponding multilinear polytopes. The generality of our technique allows us to obtain the same type of results for extensions of BPO, where we enforce extended cardinality constraints on the set of binary points, and where variables are replaced by literals. We also obtain strongly polynomial algorithms for the variant of the above problems where we seek best feasible solutions, instead of only one optimal solution. Computational results show that the resulting algorithms can be significantly faster than current state-of-the-art.
Paper Structure (21 sections, 32 theorems, 28 equations, 7 figures)

This paper contains 21 sections, 32 theorems, 28 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Binary Polynomial Optimization and Boolean Maximization Problem
  4. Binary Polynomial Optimization Problem.
  5. Boolean Maximization Problem.
  6. BPO as Boolean Maximization Problem.
  7. Encoding BPO as CNF formulas
  8. CNF formulas.
  9. A first encoding.
  10. A CNF encoding preserving treewidth
  11. Treewidth.
  12. A CNF encoding preserving $\beta$-acyclicity
  13. Solving BPO using Knowledge Compilation
  14. Beyond BPO
  15. Adding Cardinality Constraints
  16. ...and 6 more sections

Key Result

Theorem 1

There is a strongly polynomial time algorithm to solve the BPO problem, provided that $H$ is $\beta$-acyclic, or the incidence treewidth of $H$ is bounded by $\log(\mathop{\mathrm{poly}}\nolimits(|V|,|E|))$. Furthermore, under the same assumptions, there exists a polynomial-size extended formulation

Figures (7)

  • Figure 1: Structure of the paper.
  • Figure 2: Hypergraph $G$ representing $x_1 x_2 x_3 + x_4 x_5 + x_2 x_3 x_4 x_5 x_6$.
  • Figure 3: Hypergraph representing first CNF construction.
  • Figure 4: Hypergraph and CNF for $\beta$-acyclic preserving encoding.
  • Figure 5: A deterministic DNNF and the Boolean function it computes. Observe that each $\wedge$-node mentions disjoint variables ($x$ or $y$ or $z$) and the each $\vee$-node is deterministic: the bottom one because they differ on the value of $z$ and the top one because they differ on the value assigned to $x$ and $y$.
  • ...and 2 more figures

Theorems & Definitions (50)

  • Theorem 1: Tractability of BPO
  • Theorem 2: Tractability of BPO$_\text{L}$
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Example 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 40 more