  Parameterization above (or below) a guarantee is a successful concept in
parameterized algorithms. The idea is that many computational problems admit
``natural'' guarantees bringing to algorithmic questions whether a better
solution (above the guarantee) could be obtained efficiently. The above
guarantee paradigm has led to several exciting discoveries in the areas of
parameterized algorithms and kernelization. We argue that this paradigm could
bring forth fresh perspectives on well-studied problems in approximation
algorithms. Our example is the longest cycle problem. One of the oldest results
in extremal combinatorics is the celebrated Dirac's theorem from 1952. Dirac's
theorem provides the following guarantee on the length of the longest cycle:
for every 2-connected n-vertex graph G with minimum degree \delta(G)\leq n/2,
the length of a longest cycle L is at least 2\delta(G). Thus, the ``essential''
part in finding the longest cycle is in approximating the ``offset'' k = L - 2
\delta(G). The main result of this paper is the above-guarantee approximation
theorem for k. Informally, the theorem says that approximating the offset k is
not harder than approximating the total length L of a cycle. In other words,
for any (reasonably well-behaved) function f, a polynomial time algorithm
constructing a cycle of length f(L) in an undirected graph with a cycle of
length L, yields a polynomial time algorithm constructing a cycle of length
2\delta(G)+\Omega(f(k)).
