What is APX hardness?
A problem is said to be APX-hard if there is a PTAS reduction from every problem in APX to that problem, and to be APX-complete if the problem is APX-hard and also in APX.
How do you determine NP hardness?
To prove that problem A is NP-hard, reduce a known NP-hard problem to A. In other words, to prove that your problem is hard, you need to describe an ecient algorithm to solve a dierent problem, which you already know is hard, using an hypothetical ecient algorithm for your problem as a black-box subroutine.
Is TSP in APX?
In [23] it is proved that min ∆TSP is APX-hard (in other words, it cannot be solved by a polynomial time approximation schema, i.e., it is not approximable within standard-approximation ratio (1 + ε), for every constant ε > 0, unless P=NP).
Is TSP NP-hard?
It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.
What is metric TSP?
Metric TSP is a subcase of TSP where the Triangle Inequality holds. (This is always true when the weights on the edges are actual distances. Both TSP and Metric TSP are NP-hard problems, that is, there is no known polynomial-time algorithm for solving these problems, unless P=NP.
What happens if P NP?
If P equals NP, every NP problem would contain a hidden shortcut, allowing computers to quickly find perfect solutions to them. But if P does not equal NP, then no such shortcuts exist, and computers’ problem-solving powers will remain fundamentally and permanently limited.
Has anyone solved the traveling salesman problem?
Scientists in Japan have solved a more complex traveling salesman problem than ever before. The previous standard for instant solving was 16 “cities,” and these scientists have used a new kind of processor to solve 22 cities. They say it would have taken a traditional von Neumann CPU 1,200 years to do the same task.
Why is TSP a hard problem?
It is a well-known algorithmic problem in the fields of computer science and operations research. This means that TSP is classified as NP-hard because it has no “quick” solution and the complexity of calculating the best route will increase when you add more destinations to the problem.
Is metric TSP NP-hard?
Theorem: METRIC-TSP is NP-Complete: If G has a hamiltonian cycle, then there is a solution to the TSP problem. (This ham cycle gives a TST of length n). If there is a TST of length |V|, then every distance between successive points must be 1.