Approximation Algorithms and Hardness of Approximation

S (in alphabetic order by speaker surname) Speaker: Hyung-Chan An (EPFL, Lausanne) Title: LP-Based Algorithms for Capacitated Facility Location Abstract: Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that fundamental concepts from the matching theory, including alternating paths and residual networks, provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding. Our results resolve one of the ten open problems selected by the textbook on approximation algorithms of Williamson and Shmoys. This is joint work with Mohit Singh and Ola Svensson. Speaker: Per Austrin (KTH Stockholm) Title: (2 + )-SAT is NP-hard Abstract: We prove the following hardness result for a natural promise variant of the classical CNFsatisfiability problem: given a CNF-formula where each clause has width w and the guarantee that there exists an assignment satisfying at least g = w/2−1 literals in each clause, it is NP-hard to find a satisfying assignment to the formula (that sets at least one literal to true in each clause). On the other hand, when g = w/2, it is easy to find a satisfying assignment via simple generalizations of the algorithms for 2-SAT. We also prove that given a (2k+ 1)-uniform hypergraph that can be 2-colored such that each edge has perfect balance (at most k + 1 vertices of either color), it is NP-hard to find a 2-coloring that avoids a monochromatic edge. In other words, a set system with discrepancy 1 is hard to distinguish from a set system with worst possible discrepancy. Joint work with Venkatesan Guruswami and Johan H̊astad. Speaker: Moses Charikar (Princeton) Title: Smoothed Analysis of Tensor Decompositions Abstract: Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because of hardness results. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error – a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, these are not known to be stable to noise.