New Directions in Approximation Algorithms and Hardness of Approximation

Combinatorial optimization encompasses a wide range of important computational tasks such as UNIFORMSPARSESTCUT (also known as NORMALIZEDCUT), MAXCUT, TRAVELINGSALESMANPROBLEM, and VERTEXCOVER. Most combinatorial optimization problems are NP-hard to be solved optimally. On one hand, a natural way to cope with this computational intractability is via designing approximation algorithms to efficiently approximate the optimal solutions with provable guarantees. On the other hand, given an NPhard optimization problem, we are also interested in the best possible approximation guarantee that any polynomial-time algorithm could achieve, i.e. the hardness of approximation of the problem. Both approximation algorithms and hardness of approximation results contribute to understanding the approximability of combinatorial optimization problems. In the last two decades, the research frontier of approximation algorithm design has been greatly advanced thanks to the convex optimization techniques such as linear and semidefinite programming. However, the exact approximability for many problems remains mysterious but some common barriers for progress revolving around a problem called Unique Games has been identified. The limitations of convex relaxation techniques answer the question that what is the best possible approximation guarantee to be achieved by the state-of-the-art algorithmic design tools, and shed light on the real approximability. Therefore, the study of the power of convex relaxations becomes a valuable new research direction to get around the current barrier on hardness proofs. In this thesis, using constraint satisfaction problems, assignment problems, graph partitioning problems (BALANCEDSEPARATOR, UNIFORMSPARSESTCUT, DENSEkSUBGRAPH), and graph isomorphism as examples, we explore both the effectiveness and limitations of the most powerful convex relaxation techniques – convex relaxation hierarchies. We also use a proof complexity view of the convex relaxation hierarchies to analyze their performance on constraint satisfaction problems, and show that the so-called Parrilo–Lasserre semidefinite programming relaxation hierarchy succeeds on all hard instances constructed in literature for UNIQUEGAMES, MAXCUT, and BALANCEDSEPARATOR. This thesis also contains a collection of approximation algorithms for almost satisfiable constraint satisfaction problems and MAXBISECTION, detection of almost isomorphic trees, and estimation of the 2 → 4 operator norm of random linear operators. There are also a few (conditional) hardness of approximation results for almost satisfiable linear systems over integers, almost satisfiable MAXHORN3-SAT, and detection of almost isomorphic graphs.