Cse 533: the Pcp Theorem and Hardness of Approximation Lecture 20: Course Summary and Open Problems
نویسندگان
چکیده
– PCP theorem. The classic PCP theorem is most useful for proving that optimization problems have no PTAS. – E3LIN2. The 1− 2 vs 1/2− 2 hardness of this problem is most useful for proving that problems have no approximation beyond a fixed constant factor; i.e., for statements like “such-and-such maximization problem has no 77 78 -approximation unless P = NP”. – Raz’s Label Cover. The hardness of this problem is often useful for showing strong or super-constant hardness-of-approximation results. • Other areas on theoretical computer science that are related to PCPs and hardness of approximation:
منابع مشابه
CSE 533 : The PCP Theorem and Hardness of Approximation ( Autumn 2005 ) Lecture 17 : MAX - CUT : Approximation and Hardness Nov
As an example, we note that the bipartite graph has maxcut exactly equal to the sum of the weights of all its edges. A particular case of the MAX-CUT problem occurs when all the edge weights are reduced to unity. In that case it becomes equivalent to the MAX-2COLORING problem. In the MAX-2COLORING problem, we try to find the maximum number of edges in a given graph G which can be colored by usi...
متن کاملCSE 533 : The PCP Theorem and Hardness of Approximation ( Autumn 2005 ) Lecture 13 : “ Confuse / Match ” Games ( II )
If G is repeated in parallel poly(1/2) times, with overwhelming probability (as a function of 2), there will be poly(1/2) many “confuse rounds” and poly(1/2) many “match rounds”. These rounds will be randomly ordered. Further, it only helps the provers if we fix the questions in some rounds and tell them everything chosen in those rounds. For all of these reasons, it suffices (and indeed is equ...
متن کاملCSE 533 : The PCP Theorem and Hardness of Approximation ( Autumn 2005 ) Lecture 18 : Majority Is Stablest and Unique Games Conjecture
The table below shows the success probability of our 2-query test on various Boolean functions. Recall that another way to look at this probability is the following: Imagine the points of the discrete cube {−1, 1}m lying in m-dimensional Euclidean space on the unit sphere (after scaling). Imagine connecting pairs of points by edges if their inner product is roughly ρ. Then a Boolean function is...
متن کاملCse 533: the Pcp Theorem and Hardness of Approximation Lecture 4: Pcp Theorem Proof: Degree Reduction, Expanderization
1 Constraint graphs and the PCP Theorem Definition 1.1. A constraint graph (G, C) over alphabet Σ is defined to be an undirected graph G = (V, E) together with a set of " constraints " C, one per edge. The constraint specified for edge e ∈ E, call it c e ∈ C, is just a subset of Σ × Σ. Note that the graph is allowed to have multi-edges (representing multiple constraints over the same pair of ve...
متن کاملLecture 1 : Course Overview and Turing machine complexity
1. Basic properties of Turing Machines (TMs), Circuits & Complexity 2. P, NP, NP-completeness, Cook-Levin Theorem. 3. Hierarchy theorems, Circuit lower bounds. 4. Space complexity: PSPACE, PSPACE-completeness, L, NL, Closure properties 5. Polynomial-time hierarchy 6. #P and counting problems 7. Randomized complexity 8. Circuit lower bounds 9. Interactive proofs & PCP Theorem Hardness of approxi...
متن کامل