NP-complete problems arise frequently in practice. In today’s lecture, we discuss various approaches for coping with NP-complete problems. When NP-completeness was discovered, algorithm designers could take it as an excuse. It is true, that I have not found an efficient algorithm. However, nobody will ever find an efficient algorithm (unless P = NP). Nowadays, NP-completeness is considered a ch...

We give a proof of SAT’s NP-completeness based upon a syntaxic characterization of NP given by Fagin at 1974. Then, we illustrate a part of our proof by giving examples of how two well-known problems, MAX INDEPENDENT SET and 3COLORING, can be expressed in terms of CNF. Finally, in the same spirit we demonstrate the min NPO-completeness of MIN WSAT under strict reductions.

For a given graph G, the Separator Problem asks whether a vertex or edge set of small cardinality (or weight) exists whose removal partitions G into two disjoint graphs of approximately equal sizes. Called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set, both problems are NP-complete for general unweighted graphs [6]. Desp...

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

A Skolem sequence is a sequence a1, a2, . . . , a2n (where ai ∈ A = {1, . . . , n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets of the form A = {1, . . . , n} are Skolem sets was solved by Thoralf Skolem [6] in 195...

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.

The optimization of large trusses often leads to a nearly optimal solution, rather than a truly optimal design. In fact, the problem space for truss optimization grows exponentially with the size of the truss. Using the method of problem reduction, this paper demonstrates that truss optimization is in the set of NP-complete problems. Hence, the only practical techniques for solving the truss pr...

Buttons & Scissors is a popular single-player puzzle. A level is played on an n-by-n grid, where each position is empty or has a single coloured button sewn onto it. The player’s goal is to remove all of the buttons using a sequence of horizontal, vertical, and diagonal scissor cuts. Each cut removes all buttons between two distinct buttons of the same colour, and is not valid if there is an in...

In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive different numbers and the sum of the numbers is minimized. Szkalicki [9] has shown that minimum sum coloring is NP-hard for interval graphs. Here we present a simpler proof of this result.