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The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only (1-ε)-satisfiable (where the parameter is the number of variables) for some constant 0 < ε < 1. We consider a minimization version of CSPs (Min-CSP), where one may assign r values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the r × r pairs of values assigned to its variables (call such a CSP instance r-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every r ≥ 1, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even r-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well. 

Promise Constraint Satisfaction Problems (PCSPs) are a generalization ofConstraint Satisfaction Problems (CSPs) where each predicate has a strong and aweak form and given a CSP instance, the objective is to distinguish if thestrong form can be satisfied vs. even the weak form cannot be satisfied. Sincetheir formal introduction by Austrin, Guruswami, and H\aa stad, there has beena flurry of works on PCSPs [BBKO19,KO19,WZ20]. The key tool in studying PCSPsis the algebraic framework developed in the context of CSPs where the closureproperties of the satisfying solutions known as the polymorphisms are analyzed. The polymorphisms of PCSPs are much richer than CSPs. In the Boolean case, westill do not know if dichotomy for PCSPs exists analogous to Schaefer'sdichotomy result for CSPs. In this paper, we study a special case of BooleanPCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate$$x \leq y$$. In the algebraic framework, this is the special case of BooleanPCSPs when the polymorphisms are monotone functions. We prove that BooleanOrdered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1Conjecture [BKM21] which is a perfect completeness surrogate of the UniqueGames Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP canbe solved in polynomial time if for every $$\epsilon>0$$, it has polymorphismswhere each coordinate has Shapley value at most $$\epsilon$$, else it is NP-hard.The algorithmic part of our dichotomy is based on a structural lemma thatBoolean monotone functions with each coordinate having low Shapley value havearbitrarily large threshold functions as minors. The hardness part proceeds byshowing that the Shapley value is consistent under a uniformly random 2-to-1minor. Of independent interest, we show that the Shapley value can beinconsistent under an adversarial 2-to-1 minor. 

In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works [R. Saket, 2021; R. Saket, 2022] which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most 2 which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of [R. Saket, 2021] which gave a (2/5)-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs. Next, we prove the hardness of satisfying more than 1/2 + o(1) fraction of such bags using a t-DNF (i.e. DNF where each term has ≤ t literals) for any constant t. In usual PAC learning such a hardness was known [S. Khot and R. Saket, 2008] only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than (q/2^{q-1} + o(1))-fraction of q-sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a (1/2^{q-2})-approximation.