Raising the bar for vertex cover: fixed-parameter tractability above a higher guarantee

Abstract

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 solved in time O*(15kμ), where the O* notation hides polynomial factors. 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 proven for a different parameter: namely, the excess kλ of the solution size over LP, the value of the linear programming relaxation of the standard LP formulation of Vertex Cover. Since LP ≥ MM for any graph, we have that kλ ≤ kμ for Yes instances. This is thus a stricter parameterization---the new parameter is, in general, smaller---and the running times carry over directly to the parameter kμ.We investigate an even stricter parameterization of Vertex Cover, namely the excess kλ of the solution size over the quantity (2LP -- MM). We ask: Given a graph G and kλ ∈ N as input, does G have a vertex cover of size at most (2LP -- MM) + kλ? The parameter is kλ. It can be shown that (2LP -- MM) is a lower bound on vertex cover size, and since LP ≥ MM we have that (2LP -- MM) ≥ LP, and hence that kλ ≤ kλ holds for Yes instances. Further, (kλ -- [EQUATION]) could be as large as (LP -- MM) and---to the best of our knowledge---this difference cannot be expressed as a function of kλ alone. These facts motivate and justify our choice of parameter: this is indeed a stricter parameterization whose tractability does not follow directly from known results.We show that Vertex Cover is fixed-parameter tractable for this stricter parameter k: We derive an algorithm which solves Vertex Cover in time O*(3k), thus pushing the envelope further on the parameterized tractability of Vertex Cover.