Approximation algorithm

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to NP-hard optimization problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for Scheduling on Unrelated Parallel Machines. In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to NP-hard optimization problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for Scheduling on Unrelated Parallel Machines. The design and analysis of approximation algorithms crucially involves a mathematical proof certifying the quality of the returned solutions in the worst case. This distinguishes them from heuristics such as annealing or genetic algorithms, which find reasonably good solutions on some inputs, but provide no clear indication at the outset on when they may succeed or fail. There is widespread interest in theoretical computer science to better understand the limits to which we can approximate certain famous optimization problems. For example, one of the long-standing open questions in computer science is to determine whether there is an algorithm that outperforms the 1.5 approximation algorithm of Christofides to the Metric Traveling Salesman Problem. The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising mathematical connections and broadly applicable techniques to design algorithms for hard optimization problems. One well-known example of the former is the Goemans-Williamson algorithm for Maximum Cut which solves a graph theoretic problem using high dimensional geometry. A simple example of an approximation algorithm is one for the Minimum Vertex Cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find a vertex cover is to repeat the following process: find an uncovered edge, add both its endpoints to the cover, and remove all edges incident to either vertex from the graph. As any vertex cover of the input graph must use a distinct vertex to cover each edge that was considered in the process (since it forms a matching), the vertex cover produced, therefore, is at most twice as large as the optimal one. In other words, this is a constant factor approximation algorithm with an approximation factor of 2. Under the recent Unique Games Conjecture, this factor is even the best possible one. NP-hard problems vary greatly in their approximability; some, such as the Knapsack Problem, can be approximated within a multiplicative factor 1 + ϵ {displaystyle 1+epsilon } , for any fixed ϵ > 0 {displaystyle epsilon >0} , and therefore produce solutions arbitrarily close to the optimum (such a family of approximation algorithms is called a polynomial time approximation scheme or PTAS). Others are impossible to approximate within any constant, or even polynomial, factor unless P = NP, as in the case of the Maximum Clique Problem. Therefore, an important benefit of studying approximation algorithms is a fine-grained classification of the difficulty of various NP-hard problems beyond the one afforded by the theory of NP-completeness. In other words, although NP-complete problems may be equivalent (under polynomial time reductions) to each other from the perspective of exact solutions, the corresponding optimization problems behave very differently from the perspective of approximate solutions.