To describe sat, a very important problem in complexity theory to describe two more classes of problems. Np complete is a complexity class which represents the set of all problems x in np for which it is possible to reduce any other np problem y to x in polynomial time intuitively this means that we can solve y quickly if we know how to solve x quickly. Precisely, y is reducible to x, if there is a polynomial time algorithm f to transform instances y of y to instances x fy of x. P, np, and npcompleteness weizmann institute of science. We can show that problems are np complete via the following steps. P, np, and np completeness siddhartha sen questions. In this lecture, professor demaine introduces npcompleteness. Example traveling salesperson problem 0n22n, knapsack problem 02n2 etc.
L b f computable in polynomial time a not harder than b. Example for the first group is ordered searching its time complexity is o log n time complexity of sorting is o n log n. There are algorithms for which there is no known solution, for example. What are the differences between np, npcomplete and nphard. The first part of an np completeness proof is showing the problem is in np. Tractability polynomial time ptime onk, where n is the input size and k is a constant. Np set of decision problems for which there exists a polytime certifier.
Informally, a search problem b is np hard if there exists some np complete problem a that turing reduces to b. Np complete the group of problems which are both in np and np hard are known as np complete problem. There are two classes of non polynomial time problems 1 np hard. Now suppose we have a np complete problem r and it is reducible to q then q is at least as hard as r and since r is an np hard problem.
Dont expect an efficient algorithm for this problem. A pushdown automata behaves like a turing machine when the number of auxiliary memory is 2 or more. Despite decades of research into computational complexity, the. A problem which is np complete will have the property that it can be solved in polynomial time iff all other np complete problems can also be. Problems basic concepts we are concerned with distinction between the problems that can be solved by polynomial time algorithm and problems for which no polynomial time algorithm is known. Nphard and npcomplete problems umsl mathematics and.
The second part is giving a reduction from a known np complete problem. Can find satisfiable assignment for 2cnf formula in. The problem in np hard cannot be solved in polynomial time, until p np. Sometimes, we can only show a problem np hard if the problem is in p, then p np, but the problem may not be in np. Class np contains all computational problems such that the corre sponding decision problem can be solved in a polynomial time by a.
33 1444 13 1219 621 470 1261 1386 1248 1123 243 814 1040 1342 1482 1328 288 575 726 538 1225 1329 1373 684 227 1028 249 1385 1315 734 212 832 439 1089 64 244 318