# Volume computation for sparse boolean quadric relaxations

## TL;DR

This paper extends volume computation methods for convex relaxations of boolean quadric polytopes to structured sparse graphs, providing algorithms, formulas, and asymptotic behaviors, with implications for understanding relaxation quality.

## Contribution

It introduces efficient algorithms and closed-form formulas for volume calculations of relaxations on sparse graphs, expanding prior work from complete graphs to more complex structures.

## Key findings

- Closed-form volume expressions for stars, paths, and cycles.
- Efficient volume computation algorithm for graphs with bounded tree width.
- Cycle relaxations closely approximate the original polytope, with potential extensions to more complex graphs.

## Abstract

Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as the (full) correlation polytope) $\mathscr{P}(G)$ of the complete graph $G=K_n$. We extend this work to structured sparse graphs, giving: (i) an efficient algorithm for $vol(\mathscr{Q}(G))$ when $G$ has bounded tree width, (ii) closed-form expressions (and asymptotic behaviors) for $vol(\mathscr{Q}(G))$ for all stars, paths, and cycles, and (iii) a closed-form expression for $vol(\mathscr{P}(G))$ for all cycles. Further, we demonstrate that when $G$ is a cycle, the simple relaxation $\mathscr{Q}(G)$ is a very close model for the much more complicated $\mathscr{P}(G)$. Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series-parallel graphs.