A characterization of approximability for biased CSPs

A µ-biased MAX-CSP instance with predicate ψ : 0, 1 r → 0, 1 is an instance of Constraint Satisfaction Problem (CSP) where the objective is to find a labeling of relative weight at most µ which satisfies the maximum fraction of constraints. Biased CSPs are versatile and express several well studied problems such as DENSEST-k-SUB(HYPER)GRAPH and SMALLSETEXPANSION. In this work, we explore the role played by the bias parameter µ on the approximability of biased CSPs. We show that the approximability of such CSPs can be characterized (up to loss of factors of arity r) using the bias-approximation curve of DENSEST-k-SUBHYPERGRAPH (DkSH). In particular, this gives a tight characterization of predicates which admit approximation guarantees that are independent of the bias parameter µ. Motivated by the above, we give new approximation and hardness results for DkSH. In particular, assuming the Small Set Expansion Hypothesis (SSEH), we show that DkSH with arity r and k = µn is NP-hard to approximate to a factor of Ω(r 3 µ r-1 log(1/µ)) for every r ≥ 2 and µ < 2 -r . We also give a O(µ r-1 log(1/µ))-approximation algorithm for the same setting. Our upper and lower bounds are tight up to constant factors, when the arity r is a constant, and in particular, imply the first tight approximation bounds for the DENSEST-k-SUBGRAPH problem in the linear bias regime. Furthermore, using the above characterization, our results also imply matching algorithms and hardness for every biased CSP of constant arity. 1 Here, we say that a α-factor approximation is efficiently achievable if the problem of finding an α-approximate solution to the biased MAX-CSP problem is in P. 2 Here the containment relationship refers to the containment relationship induced by interpreting the Boolean strings as indicators of subsets. 3 Given a labeling σ : V → 0, 1, its relative weight with respect to vertex weight function w : V → 0, 1 is defined as w(σ) := ∑ i:σ(i)=1 w(i)/w(V), where w(V) denotes the total vertex weight.