On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem

by   Elisabeth Gaar, et al.

One of many different hierarchies towards the stability number of a graph is the exact subgraph hierarchy (ESH). On the first level it starts to compute the Lovász theta function as a semidefinite program (SDP) with a matrix variable of order n+1 and n+m+1 constraints, where n is the number of vertices and m is the number of edges of a graph G. On the k-th level of the ESH it adds all exact subgraph constraints (ESC) for subgraphs of G with k vertices to the SDP. These ESCs make sure that the submatrix of the matrix variable corresponding to the subgraphs are in the appropriate polytopes. In order to exploit the ESH computationally one only includes ESCs for certain wisely chosen subgraphs. In this paper we introduce a variant of the ESH that starts with an alternative SDP to compute the Lovász theta function with a matrix variable of order n and only m+1 constraints. We show that it makes sense to include the ESCs into this SDP and build the compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems favorable as the SDP is smaller. However, we prove that the bounds obtained with the ESH are always at least as good as those of the CESH. In computations sometimes they are significantly better. We also introduce scaled ESCs (SESCs), which are a more natural way to include exactness constraints into the smaller SDP and we prove that including an SESC is equivalent to including an ESC for every subgraph.


page 1

page 2

page 3

page 4


The Babylonian Graph

The Babylonian graph B has the positive integers as vertices and connect...

Second-order moments of the size of randomly induced subgraphs of given order

For a graph G and a positive integer c, let M_c(G) be the size of a subg...

Reconfiguring spanning and induced subgraphs

Subgraph reconfiguration is a family of problems focusing on the reachab...

Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments

Given a graph G, the NP-hard Maximum Planar Subgraph problem asks for a ...

Superlinear Lower Bounds for Distributed Subgraph Detection

In the distributed subgraph-freeness problem, we are given a graph H, an...

Pruning Bayesian Networks for Efficient Computation

This paper analyzes the circumstances under which Bayesian networks can ...

Better Fewer but Better: Community Search with Outliers

Given a set of vertices in a network, that we believe are of interest fo...

Please sign up or login with your details

Forgot password? Click here to reset