The proof that the Ellipsoid algorithm can solve linear programming in polynomial time is due to Leonid Khachiyan (1979). This discovery took place a full 20 years after the invention of the simplex algorithm. Today we sketched its main idea. In the next lecture we will be looking at some of the details.

I mentioned today some examples of problems in different complexity classes. Let me recap here:

Q10. Think about the dual to "maximize $0^T x$ s.t. $Ax\leq b$.

Q11. Here you are being asked to show SAT $\leq_p$ 3SAT. So you need to find a way to make a polynomial time reduction from a SAT instance to a 3SAT instance.

Q12(b). Here are you being given some hints as to how to make a polynomial time reduction from SAT by which to show SAT $\leq_p$ Hamiltonian Path Problem.

I mentioned today some examples of problems in different complexity classes. Let me recap here:

**P**includes "Is there a path from $i$ to $j$ in this graph with cost $\leq k$?", 2SAT and "Is the set $P=\{x : Ax\leq b\}$ nonempty?"**NP-complete**includes SAT, 3SAT, Subset Sum, Hamiltonian Path, Travelling Salesman Decision Problem ("Does there exist a tour of length $\leq k$?"), and Subgraph Isomorphism.**NP-hard, but not in NP**includes TSP optimization problem ("What is the shortest TSP tour?") and the Halting Problem. Note that is it wrong to say, as some do, that the travelling salesman problem of finding the best tour is NP-complete. It is not NP-complete. It is NP-hard. In general, optimization forms of NP-complete decision problems should be spoken of as being NP-hard.**NP, but not in P or NP-complete**, is conjectured to include Graph Isomorphism. If this could be proved then it would follow that P$\neq $NP.**EXAMPLE SHEET 1.**Let me make some remarks on the examples sheet which you should now be attempting.Q10. Think about the dual to "maximize $0^T x$ s.t. $Ax\leq b$.

Q11. Here you are being asked to show SAT $\leq_p$ 3SAT. So you need to find a way to make a polynomial time reduction from a SAT instance to a 3SAT instance.

Q12(b). Here are you being given some hints as to how to make a polynomial time reduction from SAT by which to show SAT $\leq_p$ Hamiltonian Path Problem.