Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee

The standard parameterization of the VERTEX COVER problem (Given an undirected graph G and k ∈ N as input, does G have a vertex cover of size at most k?) has the solution size k as the parameter. The following more challenging parameterization of VERTEX COVER stems from the observation that the size MM of a maximum matching of G lower-bounds the size of any vertex cover of G: Does G have a vertex cover of size at most MM + kµ? The parameter is the excess kµ of the solution size over the lower bound MM. Razgon and O’Sullivan (ICALP 2008) showed that this above-guarantee parameterization of VERTEX COVER is fixed-parameter tractable and can be solved1 in time O⋆ (15kµ ). This was first improved to O⋆ (9kµ ) (Raman et al., ESA 2011), then to O⋆ (4kµ ) (Cygan et al., IPEC 2011, TOCT 2013), then to O⋆ (2.618kµ ) (Narayanaswamy et al., STACS 2012) and finally to the current best bound O⋆ (2.3146kµ ) (Lokshtanov et al., TALG 2014). The last two bounds were in fact prov...

• Type: Journal
• Year: 2016
• Where: SODA
• Authors: Shivam Garg, Geevarghese Philip
• Keywords: Algorithms, SODA 2016