NP-complete problems fall into a range of difficulty and it is of considerable interest to identify the general characteristics of these problems that account for this difference. A method for measuring problem difficulty is developed based on the amount of time it takes to find an optimal solution. This reveals that there are layers of difficulty and that any NP-complete problem can be separat...

Two subproblems that arise when routing channels with interchangeable terminals are shown to be NP-hard. These problems are: P1: Is there a net to terminal assignment that results in an acyclic vertical constraint graph? P2: For instances with acyclic vertical constraint graphs, obtain net to terminal assignments for which the length of the longest path in the vertical constraint graph is minimum.

A graph is a tree of paths (cycles), if its vertex set can be partitioned into clusters, such that each cluster induces a simple path (cycle), and the clusters form a tree. Our main result states that the problem whether or not a given graph is a tree of paths (cycles) is NP-complete. Moreover, if the length of the paths (cycles) is bounded by a constant, the problem is in P.

We prove that all of Karp’s 21 original NP-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate log(k) of the magnitude of these problems to within any co...

The discrete gate sizing problem has been studied by several researchers recently. Some Complexity results have been obtained, ana' a number of heuristic algorithms have been proposed. For circuit networks that are restricted to the set of trees, or series-parallel graphs pseudo-polynomial time algorithms to obtain the exact solution have also been proposed, though none can be extended to arbit...

We present a new optical method for solving bounded (input-length-restricted) NP-complete combinatorial problems. We have chosen to demonstrate the method with an NP-complete problem called the traveling salesman problem (TSP). The power of optics in this method is realized by using a fast matrix-vector multiplication between a binary matrix, representing all feasible TSP tours, and a gray-scal...

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were assumed to be integers. We re-examine here the computational complexity of all these problems in the setting where the weights and profits are allowed to be...

A polynomial algorithm for solving the ”Hamiltonian circuit” problem is presented in the paper. Computational complexity of the algorithm is equal to O ( n log 2 n ) where n is the cardinality of the observed graph vertex set. Thus the polynomial solvability for NP -complete problems is proved.

We investigate here NP optimization problems from a logical deenability standpoint. We show that the class of optimization problems whose optimum is deenable using rst-order formulae coincides with the class of polynomially bounded NP optimization problems on nite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this fram...