Universal Factor Graphs for Every NP-Hard Boolean CSP

An instance of a Boolean constraint satisfaction problem can be divided into two parts. One part, that we refer to as the factor graph of the instance, specifies for each clause the set of variables that are associated with the clause. The other part, specifies for each of the given clauses what is the constraint that is evaluated on the respective variables. Depending on the allowed choices of constraints, it is known that Boolean constraint satisfaction problems fall into one of two classes, being either NP-hard or in P. This paper shows that every NP-hard Boolean constraint satisfaction problem (except for an easy to characterize set of natural exceptions) has a universal factor graph. That is, for every NP-hard Boolean constraint satisfaction problem, there is a family of at most one factor graph of each size, such that the problem, restricted to instances that have a factor graph from this family, cannot be solved in polynomial time unless NP is contained in P/poly. Moreover, we extend this classification to one that establishes hardness of approximation

Max-3-Lin over Non-Abelian Groups with Universal Factor Graphs

Factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely defined by its factor graph and the list of predicates. We show inapproximability of Max-3-LIN over non-abelian groups (both in the perfect completeness case and in the imperfect completeness case), with the same inapproximability factor as in the general case, even when the factor graph is fixed. Along the way, we also show that these optimal hardness results hold even when we restrict the linear equations in the Max-3-LIN instances to the form x? y? z = g, where x,y,z are the variables and g is a group element. We use representation theory and Fourier analysis over non-abelian groups to analyze the reductions

Ranking with Fairness Constraints

Ranking algorithms are deployed widely to order a set of items in applications such as search engines, news feeds, and recommendation systems. Recent studies, however, have shown that, left unchecked, the output of ranking algorithms can result in decreased diversity in the type of content presented, promote stereotypes, and polarize opinions. In order to address such issues, we study the following variant of the traditional ranking problem when, in addition, there are fairness or diversity constraints. Given a collection of items along with 1) the value of placing an item in a particular position in the ranking, 2) the collection of sensitive attributes (such as gender, race, political opinion) of each item and 3) a collection of fairness constraints that, for each k, bound the number of items with each attribute that are allowed to appear in the top k positions of the ranking, the goal is to output a ranking that maximizes the value with respect to the original rank quality metric while respecting the constraints. This problem encapsulates various well-studied problems related to bipartite and hypergraph matching as special cases and turns out to be hard to approximate even with simple constraints. Our main technical contributions are fast exact and approximation algorithms along with complementary hardness results that, together, come close to settling the approximability of this constrained ranking maximization problem. Unlike prior work on the approximability of constrained matching problems, our algorithm runs in linear time, even when the number of constraints is (polynomially) large, its approximation ratio does not depend on the number of constraints, and it produces solutions with small constraint violations. Our results rely on insights about the constrained matching problem when the objective function satisfies certain properties that appear in common ranking metrics such as discounted cumulative gain (DCG), Spearman\u27s rho or Bradley-Terry, along with the nested structure of fairness constraints

Optimal Inapproximability with Universal Factor Graphs

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remains as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.Examples of results obtained for this restricted setting are:1. Optimal inapproximability for Max-3-Lin.2. Approximation resistance for predicates supporting pairwise independent subgroups.3. Hardness of the "(2+epsilon)-Sat" problem and other Promise CSPs.The main technical tool used to establish these results is a new way of folding the long code which we call "functional folding"