Hardness of Approximation

Cse 533: the Pcp Theorem and Hardness of Approximation Lecture 20: Course Summary and Open Problems

– 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 o...

3 Hardness of Approximation

Probabilistic Checking of Proofs and Hardness of Approximation

This paper introduces basic notions of classic computational complexity theory and probabilistically checkable proof. It sketches the proof of the PCP theorem, and discusses its applications to the hardness of approximation of various problems.

Coloring Graph Powers: Graph Product Bounds and Hardness of Approximation

We consider the question of computing the strong edge coloring, square graph coloring, and their generalization to coloring the k power of graphs. These problems have long been studied in discrete mathematics, and their “chaotic” behavior makes them interesting from an approximation algorithm perspective: For k = 1, it is well-known that vertex coloring is “hard” and edge coloring is “easy” in ...

The PCP Theorem and Hardness of Approximation

In this report, we give a brief overview of Probabilistically Chekable proofs, and present Irit Dinur’s proof of the celebrated PCP theorem. We also briefly deal with the importance of the PCP theorem in deciding hardness of approximation of various problems.

Hardness of approximation for orthogonal rectangle packing and covering problems

Bansal and Sviridenko [4] proved that there is no asymptotic PTAS for 2-dimensional Orthogonal Bin Packing (without rotations), unless P = NP. We show that similar approximation hardness results hold for several 2and 3-dimensional rectangle packing and covering problems even if rotations by ninety degrees are allowed. Moreover, for some of these problems we provide explicit lower bounds on asym...