I assume that most students have encountered Turing machines before. (Students who have not may want to look at Sipser’s book [3].) A Turing machine is defined by an integer k ≥ 1, a finite set of states Q, an alphabet Γ, and a transition function δ : Q×Γk → Q×Γk−1×{L, S,R}k where: • k is the number of (infinite, one-dimensional) tapes used by the machine. In the general case we have k ≥ 3 and ...

Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas West’s graph theory book, and also from wikipedia. There’s nothing remotely original about these notes. I just wanted all this material to filter through my brain and onto paper. I also wanted to collect everything together in a way I like. Here’s what thes...

For a fixed graph H, let H-CON denote the problem of determining whether a given graph is contractible to H. The complexity of H-CON is studied for H belonging to certain classes of graphs, together covering all connected graphs of order at most 4. In particular, H-CON is NP-complete if H is a connected triangle-free graph other than a star. For each connected graph H of order a t most 4 other ...

In this note, we answer two questions arising in broadcasting problems in networks. We rst describe a new family of minimum broadcast graphs. Then we give a proof due to Alon of the NP-completeness of nding disjoint spanning trees of minimum depth, rooted at a given vertex.

• INDEPENDENT -SET = {[G, k] : G is an undirected graph with an independent set of vertices I ⊆ V (G) of size ≥ k} where a subset I is independent iff it contains no edges of G. – Guess a set I of size k. Verify that it contains no edges of G. • CLIQUE = {[G, k] : G is an undirected graph with an independent set of vertices C ⊆ V (G) of size ≥ k} where a subset C is a clique iff every edge on t...

For a quarter of a century now NP completeness has been computer science s favorite paradigm fad punching bag buzzword alibi and intellectual export This paper is a fragmentary commentary on its origins its nature its impact and on the attributes that have made it so pervasive and contagious A keyword search in Melvyl the University of California s on line library reveals that about papers each...