References
[CLR] Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest,
Introduction to Algorithms, The MIT Press, 1990.
[Sip] Michael Sipser, Introduction to the Theory of Computation,
PWS Publishing Company, 1997.
[PAP] Christos H. Papadimitriou, Computational Complexity,
[MR] Rajeev Motwani, Prabhakar Raghavan, Randomized Algorithms
Addison-Wesley Publishing Company, 1995.
[V]
Vijay V. Vazirani, Approximation Algorithms. Springer.
1
1
2
1.03.04
Oded
4.03.04
Amnon
8.03.04
Oded
2
11.03.04
Amnon
3
15.04.04
Oded
3
18.03.04
Amnon
22.03.04
Oded
25.03.04
Amnon
4
4
5
5
6
29.03.04
Oded
15.04.04
Amnon
19.04.04
Oded
Complexity of functions. [CLR 2-4], [Sip 7.1]
Turing Machines (example – palindromes). [PAP 2.1-2.3],
Multiple tape T.M. vs. Single tape ones. [PAP 2.3],[Sip 3.2]
(doesn’t include time analysis).
Time and Space complexity. [PAP 7.1]
The class P. [PAP 2.6]
Multiple tape – cont.
s-t-connectivity –in SPACE(log2n) – an explicit machine.
[PAP 7.3], [Sip 8.1]
Composition of space bounded machines.
s-t-connectivity –in SPACE(log2n) – the more general framework.
[PAP 7.3]
Diagonalization and universal TM for Halt [PAP 3.1-2] and for
complexity class separation – The Time and space hierarchies. [PAP 7.2]
Non-deterministic TM – two possible definitions. NTIME.
Co-classes. [PAP 2.7,7.1] Diagonalization and universal TM for Halt
and for complexity class separation. [PAP 7.2], [Sip 9.1]
Circuits [PAP 11.4]
Reductions, [PAP 8.1] STCONN is NL-complete [PAP 7.3,16.1],
CVAL is P-Complete [PAP 8.2] .
CVAL is P-Complete [PAP 8.2]
NP. All guesses can be made at beginning.
NP as a game between an all powerful prover and a polynomial time
verifier.
CSAT is NP-Complete. SAT, 3SAT are NP-Complete. [PAP 8.2,9.2]
2SAT is in P, in fact in co-NL. [PAP 9.2]
2-SAT is in co-NL. [PAP 9.2]
3-Col is NPC assuming 3-NAE is NPC. [PAP 9.3]
Clique, IS, VC, SetCover are NP-complete. [PAP 9.2,9.3]
Approximation algorithms [V 2.1]
Approximating VC with a factor of 2 [PAP 13.1]
Approximating Set-Cover with a factor ln(n).
Self reducibility of 3-SAT: reducing the search to the decision problem
by a polynomial Turing/Cook reduction. [PAP 10.3]
6
7
7
22.04.04
Amnon
Tirgul (taught over
several weeks)
D-HP, D-HC, U-HC, TSP and TSP with triangle inequality. [PAP 9.3]
2-approximation and 1.5 approximation for TSP with triangle
inequality. [V 3.2]
3-NAE < MAX-CUT
0.5 approximation for max-cut.
29.04.04
Amnon
3.05.04
Oded
TQBF-k is Sigma_k complete. [PAP 17.2]
NL=coNL [PAP 7.3].
Separating AvP from EXP
Separating P from Time(2n)
Separating AvP from Time(2n) [ans2]
Parallel computation, [PAP 15.1] . sorting in parallel.
NC [PAP 15.3]. NC in DSPACe(polylog(n)) .
Perfect Matching in a bipartite graph. Solving by determinants.
A randomized algorithm.
4-NAE is NPC, 3-NAE is NPC.
PSPACE = NPSPACE revisited.
Pratt's Theorem.
Probabilistic computation [PAP 11.1,11.2]. Amplification. Chernoof.
Schwartz's lemma, Perfect Matching in bipartite graph in RNC.
matrix-multiplication-verification.
Checking x=y with low communication complexity. [MR 7.4]
The min-cut algorithm. [MR 1.1, 10.2]
BPP can be solved by polynomial-size circuits. [PAP 11.4]
BPP in Sigma_2 . [PAP 17.2]
GNISO has an interactive proof!
Interactive proofs with public and private coins.
A detailed proof of coNP is in IP. Arithmization. Low degree extension. .
[PAP 19.2]
A short discussion of Multi-prover proofs and the MIP=NEXP result.
Oded: A randomized algorithm for pattern matching, IP IN PSPACE.
[Oded Goldreich’s lecture notes] [Mathematica file]
PCP as an interactive proof system, one round, few bits read from
the witness. Encoding the witness so that a single error
propagates everywhere. PCP(log(n),1) reduces to sat,
and is in NP. An equivalent formulation to the PCP theorem:
Gap-Max3sat is NP-hard. Hardness of approximating Max-3SAT.
Clique is hard to approximate to within some constant factor. .
[VAZ 29.1-3]
8
6.04.04
Amnon
8
10.05.04
Oded
9
13.05.04
Amnon
10
08.05.2003
11
15.05.2003
Oded
12
Padding argument [PAP 20.1] applied to ex2 q6 [ans2] and to
Savitch theorem.
TQBF is PSPACE-complete. [PAP 19.1] Oracle machines[PAP 14.3] ;
PH in terms of oracles, Sigma_k=co-Pi_k.
22.05.2003
13
29.05.2003
Reducing the error with few random bits by a random walk on
Exapnders (informal).
NP in PCP(O(log n),O(log n)) with 1/n error.
Clique is hard to approximate to within some n^a, for some
constant a [VAZ 29.6].
Max 3SAT[29] is hard to approximate to within
some constant [VAZ 29.4].