[Paper Review] Extension complexity of stable set polytopes of bipartite graphs

Manuel Aprile, Yuri Faenza+3 more|arXiv (Cornell University)|Feb 28, 2017

Advanced Graph Theory Research10 references4 citations

TL;DR

This paper establishes improved upper and lower bounds on the extension complexity of stable set polytopes for bipartite graphs. It proves an upper bound of $ O(n^2 / /log n) $ using rectangle covers and centered rectangles in a geometric construction, and a lower bound of $ \Omega(n \log n) $ for incidence graphs of finite projective planes, demonstrating superlinear extension complexity for the first time in this class of graphs.

ABSTRACT

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $\mathsf{xc} (\mathsf{STAB}(G))$ is $Ω(n \log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.

Motivation & Objective

• To improve the known upper and lower bounds on the extension complexity of stable set polytopes for bipartite graphs.
• To determine whether the trivial bounds $ n \leq \mathsf{xc}(\mathsf{STAB}(G)) \leq n + m $ can be significantly tightened.
• To construct explicit families of bipartite graphs where the extension complexity exceeds linear growth in the number of vertices.
• To explore the limitations of current techniques, such as fooling sets and rectangle covers, in deriving stronger lower bounds.

Proposed method

• The authors use rectangle covers and centered rectangles in a geometric framework to construct compact extensions of the stable set polytope.
• They define a hierarchical tree structure $ T(q+1) $ to organize vertices and lines, enabling recursive construction of covering rectangles.
• For each center point $ \mathsf{c} $, they combine rectangles from multiple lines using a summation operation that preserves coverage while reducing total count.
• They apply a reduction technique that merges multiple rectangles with the same center into a single maximal rectangle, reducing the total number of rectangles by a factor of $ q $.
• The method leverages properties of finite projective planes to construct graphs with high fooling set size, enabling strong lower bounds.
• They use the concept of special entries in the slack matrix and prove that all such entries are covered by a carefully constructed set of centered rectangles.

Experimental results

Research questions

• RQ1 Can the upper bound of $ n + m $ on the extension complexity of $ \mathsf{STAB}(G) $ for bipartite graphs be improved for dense graphs?
• RQ2 Is there a family of bipartite graphs for which the extension complexity grows faster than linearly in $ n $, the number of vertices?
• RQ3 What is the maximum possible size of a fooling set in the edge-vs-stable-set matrix of a 3-regular bipartite graph?
• RQ4 Can the current rectangle cover and fooling set techniques be extended to yield super-linear lower bounds for other graph classes?
• RQ5 What structural properties of bipartite graphs lead to high extension complexity in their stable set polytopes?

Key findings

• The extension complexity of $ \mathsf{STAB}(G) $ for any bipartite graph $ G $ with $ n $ vertices is at most $ O(n^2 / \log n) $, improving the trivial $ n + m $ bound when $ m = \Omega(n^2) $.
• For incidence graphs of finite projective planes, the extension complexity is $ \Omega(n \log n) $, the first known superlinear lower bound for stable set polytopes of bipartite graphs.
• There exist 3-regular bipartite graphs whose edge-vs-stable-set matrix admits a fooling set of size equal to the number of edges, indicating high complexity.
• The construction of rectangle covers using centered rectangles and hierarchical tree structures allows a reduction in the number of rectangles from $ O(nq\log q) $ to $ O(n\log n) $, enabling the improved upper bound.
• The paper shows that the current approach using rectangle covers and fooling sets cannot yield a lower bound better than $ \Omega(n \log n) $, indicating a need for new techniques to improve further.