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)).

翻译：参数化算法中，基于保证值的参数化（高于或低于保证值）是一个成功概念。其核心思想是，许多计算问题存在“自然”保证值，从而引出一个算法性问题：能否高效获得优于保证值的解？这一范式已在参数化算法与核化领域催生出多项激动人心的发现。我们论证，该范式有望为近似算法中已被充分研究的问题提供新视角。以最长圈问题为例——极值组合学最古老的成果之一，是1952年著名的迪拉克定理。该定理对最长圈长度给出如下保证：对于任意最小度δ(G)≤n/2的2连通n顶点图G，最长圈长度L至少为2δ(G)。因此，求解最长圈问题时，“本质”部分在于逼近偏移量k = L - 2δ(G)。本文主要成果是关于k的超越保证近似定理。非正式地说，该定理表明逼近偏移量k并不比逼近圈总长度L更困难。换言之，对任意行为合理的函数f，若存在多项式时间算法可在包含长度为L的圈的无向图中构造长度为f(L)的圈，则必然存在多项式时间算法可构造长度为2δ(G)+Ω(f(k))的圈。