Circumventing d-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four

Combinatorial optimization include such tasks as finding the quickest route to work, scheduling jobs to specialists, and placing bus stops so as to minimize commuter times. We consider problems where one is given a collection of constraints with the objective of finding an assignment satisfying as many constraints as possible, also known as Constraint Satisfaction Problems (CSPs). Most CSPs are NP-hard to solve optimally and we turn to approximations - a solution is said to be a factor-c approximation if its satisfies at least c times the optimal number of constraints. This thesis presents new results on the approximation limits of CSPs in various settings.In ordering CSPs, one is given constraints which specify the relative order of items, and the objective is order the items so as to satisfy as many constraints as possible. We give improved approximation hardness results for two classical problems: it is NP-hard to approximate Maximum Acyclic Subgraph with a factor better than 14/15 and Maximum Betweenness with a factor better than 1/2. We present ordering problems which are NP-hard to approximate better than random assignments, and that there are ordering problems arbitrarily hard to approximate.Next, Gaussian elimination can efficiently find exact solutions for satisfiable collections of so-called parity constraints. We show that whenever constraints accept at least one assignment in addition to a parity, then the problem is NP-hard to approximate better than random assignments. Finally, we study the uselessness property which basically states that if one is given a collection where almost all constraints are simultaneously satisfiable and one is permitted to relax the constraints to accept or reject additional assignments, then it is still NP-hard to find solutions noticeably better than random assignments. We consider the setting where all variables appear unnegated and provide the first examples of non-trivially useless predicates assuming only P != NP.