We prove that certain problems naturally arising in knot theory are NP–hard or NP–complete. These the of obtaining one diagram from another a link bounded number Reidemeister moves, determining whether has an unlinking splitting k <mml:annotation enc...

the travelling salesmen problem (tsp) is one of the most important and famous combinational optimization problems that aim to find the shortest tour. in this problem, the salesman starts to move from an arbitrary place called depot and after visiting all nodes, finally comes back to depot. solving this problem seems hard because program statement is simple and leads this problem belonging to np...

the traveling salesman problem (tsp) is a well-known combinatorial optimization problem and holds a central place in logistics management. the tsp has received much attention because of its practical applications in industrial problems. many exact, heuristic and metaheuristic approaches have been proposed to solve tsp in recent years. in this paper, a modified ant colony optimization (maco) is ...

A number of results in hamiltonian graph theory are of the form P1 implies P2, where P1 is a property of graphs that is NP-hard and P2 is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chvátal-Erdös Theorem, which states that every graph G with α ≤ κ is hamiltonian. Here κ is the vertex connectivity of G and α is the cardinality of a largest set of i...

A generally-accepted minimum requirement for an algorithm to be considered ‘efficient’ is that its running time is polynomial: O(nc) for some constant c, where n is the size of the input.1 Researchers recognized early on that not all problems can be solved this quickly, but we had a hard time figuring out exactly which ones could and which ones couldn’t. there are several so-called NP-hard prob...