By Chen J.

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Planar Graph Vertex-Cover (D) Given a planar graph G and an integer k, is there a subset S of at most k vertices of G such that every edge in G has at least one end in S? 28 INTRODUCTION Hamiltonian Circuit Given a graph G of n vertices, is there a simple cycle in G that contains all vertices? Euclidean Traveling Salesman (D) Given a set S of n points in the plane and an integer k, is there a tour of length bounded by k that visits all points in S? Maximum Cut (D) Given a graph G and an integer k, is there a partition of the vertices of G into two sets V1 and V2 such that the number of edges with one end in V1 and the other end in V2 is at least k?

This hints a lower bound on the computational complexity for the Satisfiability problem. Motivated by this theorem, we introduce the following definition. 5 A decision problem Q is NP-hard if every problem in the class NP is polynomial-time many-one reducible to Q. A decision problem Q is NP-complete if Q is in the class NP and Q is NP-hard. In particular, the Satisfiability problem is NP-hard and NP-complete (it is easy to see that the Satisfiability problem is in the class NP). 1, if an NP-hard problem can be solved in polynomial time, then so can all problem in NP.

Remark. There is no special restriction on the directed graph G that models a flow network. In particular, we allow edges in G to be directed into the source and out of the sink. Intuitively, a flow in a flow network should satisfy the following three conditions: (1) the amount of flow along an edge should not exceed the capacity of the edge (capacity constraint); (2) a flow from a vertex u to a vertex v can be regarded as a “negative” flow of the same amount from vertex v to vertex u (skew symmetry); and (3) except for the source s and the sink t, the amount of flow getting into a vertex v should be equal to the amount of flow coming out of the vertex (flow conservation).