Conditional Dichotomy of Boolean Ordered Promise CSPs
aa r X i v : . [ c s . CC ] F e b Conditional Dichotomy of Boolean Ordered Promise CSPs
Joshua Brakensiek * Venkatesan Guruswami † Sai Sandeep ‡ February 2021
Abstract
Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Sat-isfaction Problems (CSPs) where each predicate has a strong and a weak form and given aCSP instance, the objective is to distinguish if the strong form can be satisfied vs. even theweak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, andHåstad [AGH17], there has been a flurry of works on PCSPs, including recent breakthroughsin approximate graph coloring [BBKO19, KO19, WZ20]. The key tool in studying PCSPs is thealgebraic framework developed in the context of CSPs where the closure properties of thesatisfying solutions known as polymorphisms are analyzed.The polymorphisms of PCSPs are significantly richer than CSPs—this is illustrated by thefact that even in the Boolean case, we still do not know if there exists a dichotomy result forPCSPs analogous to Schaefer’s dichotomy result [Sch78] for CSPs. In this paper, we studya special case of Boolean PCSPs, namely Boolean
Ordered
PCSPs where the Boolean PCSPshave the predicate x ≤ y . In the algebraic framework, this is the special case of Boolean PCSPswhen the polymorphisms are monotone functions . We prove that Boolean Ordered PCSPsexhibit a computational dichotomy assuming the Rich -to- Conjecture [BKM21] which is aperfect completeness surrogate of the Unique Games Conjecture.In particular, assuming the Rich -to- Conjecture, we prove that a Boolean Ordered PCSPcan be solved in polynomial time if for every ǫ > , it has polymorphisms where each coor-dinate has Shapley value at most ǫ , else it is NP-hard. The algorithmic part of our dichotomyresult is based on a structural lemma showing that Boolean monotone functions with eachcoordinate having low Shapley value have arbitrarily large threshold functions as minors. Thehardness part proceeds by showing that the Shapley value is consistent under a uniformlyrandom -to- minor. As a structural result of independent interest, we construct an exam-ple to show that the Shapley value can be inconsistent under an adversarial -to- minor. Constraint satisfaction problems (CSP) have played a very influential role in the theory of com-putation, providing an excellent testbed for the development of both algorithmic and hardnesstechniques, which then extend to more general settings. A CSP over domain D is specified by afinite collection A of predicates over D , and is denoted as CSP( A ). Given an input containing n * Stanford University, [email protected] . Research supported in part by an NSF Graduate Research Fel-lowship. † Carnegie Mellon University, [email protected] . Research supported in part by NSF grant CCF-1908125 and aSimons Investigator award. ‡ Carnegie Mellon University, , [email protected] . Research supported in part by NSF grants CCF-1563742and CCF-1908125. D to the variables that satisfies all the constraints. Examples of CSPsinclude classical problems such as -SAT and -Coloring of graphs.When the domain is Boolean, Schaefer [Sch78] proved that every CSP is either in P or isNP-Complete. Feder and Vardi [FV98] conjectured that the same should hold over arbitrarydomains as well. They also showed that the then known algorithmic results all follow by thealgebraic closure properties of the CSPs. This notion was formalized by Jeavons, Cohen, andGyssens [JCG97, Jea98] and other works [BJK05] that crystallized the (universal) algebraic ap-proach to CSPs. In the algebraic approach, the higher-order closure properties obeyed by thepredicates, namely their polymorphisms , are studied. A polymorphism is a function that whenapplied coordinate-wise to arbitrary satisfying assignments to the predicate, is guaranteed toproduce an output that satisfies the predicate. For example, consider an arbitrary instance I of the -SAT problem over n variables, and suppose that x , y , z ∈ { , } n are three assignmentsthat satisfy all the constraints in I . Now, if we compute u ∈ { , } n that is obtained by setting u i = MAJ ( x i , y i , z i ) for all i ∈ [ n ] , the assignment u also satisfies all the constraints of I . Thus,the majority function on bits is a polymorphism of the -SAT CSP. On the other hand, for the -SAT problem, it is not hard to prove that the only polymorphisms are the dictator functions.The algebraic approach has been immensely successful and culminated in the recent resolutionof Feder-Vardi conjecture by Bulatov [Bul17] and Zhuk [Zhu20]. Further, these proofs yield a pre-cise understanding of the mathematical structure underlying efficient algorithms: if the CSP hasa “non-trivial” polymorphisms, the CSP is polytime solvable, and otherwise, it is NP-complete.In this paper, we study Promise Constraint Satisfaction Problems (PCSPs) that vastly gener-alize the CSPs. In the PCSPs, each predicate has a weak and a strong form–given an instanceof PCSP containing n variables with the constraints, the goal is to distinguish between the casethat the stronger form can be satisfied vs. even the weaker one cannot be satisfied. A classi-cal example of PCSP is the approximate graph coloring, where given a graph G , the goal is todistinguish between the cases that G can be colored with c colors vs. it cannot be colored with s colors for some c ≤ s . Another example is the ( -in- SAT, NAE- -SAT), wherein given a -in- -SAT instance that is promised to be satisfiable, the objective is to assign , values to thevariables such that each constraint is satisfied as a NAE- -SAT instance, i.e., both and occurin every constraint. While the individual CSPs, namely -in- -SAT and NAE- -SAT are both NP-hard, the above PCSP is in P. The study of PCSPs was formally initiated by Austrin, Guruswami,and Håstad [AGH17]. and since then, there has been a lot of recent interest in PCSPs, includ-ing the development of a systematic theory in [BG18, BBKO19] and leading to breakthroughs inapproximate graph coloring [BBKO19, KO19, WZ20].The central question in the study of PCSPs is whether there exists a complexity dichotomyfor PCSPs i.e. if every PCSP is either in P or is NP-complete. As is the case with CSPs, the key tooltowards establishing such a dichotomy result is the algebraic approach. The Galois correspon-dence from the CSP world extends to PCSPs, i.e., the polymorphisms fully capture the computa-tional complexity of the underlying PCSP [Pip02,BG18]. This has been extended to show that justthe identities satisfied by the polymorphisms suffice to capture the computational complexityof the underlying PCSP [BBKO19]. However, the polymorphisms of PCSPs are much richer, andcharacterizing which polymorphisms lead to algorithms and which ones lead to hardness hasbeen a challenging problem. Conceptually, the principal difficulty is that the polymorphismsfor CSPs are closed under composition (hence referred to as clones ), whereas for PCSPs, this isno longer the case.As a result, even in the Boolean case, we do not have a dichotomy theorem for PCSPs. To-2ards establishing a potential Boolean PCSP dichotomy, progress has been made by Ficak, Kozik,Olsák and Stankiewicz [FKOS19], who obtained a dichotomy result when each predicate is sym-metric. In this paper, we study Boolean PCSPs that contain the simplest non-symmetric predi-cate , x → y . We call such Boolean PCSPs Ordered as we can also view the implication constraintas an ordering requirement x ≤ y .Ordered Boolean PCSPs have come under recent study. The work of Petr [Pet20] (inspired bywork of Barto [Bar18, Bar20]) considered a special class of Ordered Boolean PCSPs which havean additional predicate x = y (this corresponds to allowing negations in the constraints) as wellas the requirement that the majority on three bits is not a polymorphism. In this setting Petr wasable to show that such Ordered Boolean PCSPs are NP-hard. However, the approach considereddoes not seem immediately extendable to analyzing general Ordered Boolean PCSPs [Bar20].The main motivation for studying these PCSPs comes from the fact that adding the additional x ≤ y predicate is equivalent to restricting the polymorphisms of the PCSPs to be monotonefunctions . Monotonicity is an influential theme in the study of Boolean functions and complex-ity theory, and understanding the structure of polymorphisms in the monotone case is an im-portant (and certainly necessary) subcase towards a general characterization of polymorphismsvs. tractability for arbitrary Boolean PCSPs. For the special case of Boolean Ordered PCSPs whichinclude negation constraints, it was conjectured in [Bar20] that polynomial time tractability ischaracterized by the existence of majority polymorphisms of arbitrarily large arity.Our main result is that Boolean Ordered PCSPs exhibit a dichotomy, under the recently in-troduced Rich -to- Conjecture of Braverman, Khot, and Minzer [BKM21].
Theorem 1.
Assuming the Rich -to- Conjecture, every Ordered Boolean PCSP is either in P oris NP-Complete. Furthermore, an Ordered PCSP Γ is in P if and only if for every ǫ > , there arepolymorphisms of Γ with every coordinate having Shapley value at most ǫ . Equivalently, Γ is in Pif and only if it has threshold polymorphisms of arbitrarily large arity. As a concrete example, recall the earlier mentioned example of ( -in- -SAT, NAE- -SAT). Asit has threshold polymorphisms of arbitrarily large arity, it remains polynomial time solvableeven after adding the predicate x → y . However, if we also add another two-variable predicate x = y , the PCSP no longer has threshold polymorphisms, and by our above result, it becomesNP-Complete.We obtain the conditional dichotomy result by analyzing the polymorphisms of the OrderedPCSPs. The key idea in the algebraic approach to PCSPs is that the PCSP is tractable if thepolymorphisms are close to symmetric, and the PCSP is hard if all the polymorphisms havea small number of “important” coordinates. More concretely, on the algorithmic front, it hasbeen proved that symmetric polymorphisms of arbitrarily large arities lead to polynomial timealgorithms for PCSPs [BGWZ20]. On the hardness side, if all the polymorphisms depend on abounded number of coordinates, then the underlying PCSP is NP-hard [AGH17]. This has beenextended to various other notions, including combinatorial ones such as C -fixing [BG16], andtopological ones such as having a bounded number of coordinates with non-zero winding num-ber [KO19]. In this paper, we study the monotone polymorphisms using analytical techniques.In particular, we use Shapley value to analyze the monotone polymorphisms. For a mono-tone function f : { , } n → { , } , Shapley value of a coordinate i is the probability that ona random path from { , , . . . , } to { , , . . . , } , the function value turns from to when weswitch the i th coordinate to . Initially studied to understand the power of an individual in vot-ing systems [SS54], Shapley value has now found applications in various settings, especially ingame theory [MKS +
13, NN11]. In our setting, there are two advantages of using Shapley value3o study the polymorphisms. First, it is a relative measure of the importance of a coordinate, asopposed to other notions of Influence which are absolute. This helps in bounding the numberof coordinates with Shapley value above a certain threshold. Second, it is a versatile measurewith combinatorial and analytical interpretations [DDS17] which helps in proving that Shapleyvalue stays consistent under function minors , a key property necessary in both the algorithmand the hardness. Algorithm Overview.
We obtain our algorithmic result by using the Basic Linear Programmingwith Affine relaxation (BLP+Affine relaxation), combined with a structural result regarding themonotone functions with bounded Shapley value. As mentioned earlier, PCSPs with symmetricpolymorphisms of arbitrarily large arities can be solved in polynomial time using the BLP+Affinerelaxation algorithm [BGWZ20]. Our main structural result is that Boolean functions with boundedShapley value have arbitrarily large threshold functions as minors. Since the set of polymor-phisms of a PCSP are closed under taking minors, this proves that the underlying PCSP Γ hasarbitrarily large threshold functions as polymorphisms, which then implies that Γ is in P. Thekey tool underlying our structural result is a result of Kalai [Kal04] that states that under cer-tain conditions, monotone Boolean functions with arbitrarily small Shapley value have a sharpthreshold. Hardness Overview.
We obtain our hardness result assuming the Rich -to- Conjecture. Braver-man, Khot, and Minzer [BKM21] introduced the conjecture as a perfect completeness surrogateof the well known Unique Games Conjecture [Kho02b]. They also proved that the conjecture isequivalent to Unique Games Conjecture when we relax the perfect completeness requirement.The reduction from the Rich -to- Conjecture to PCSPs follows using the standard Label Cover-Long Code paradigm. The key ingredient in this reduction is a decoding of the Long Codes toa bounded number of coordinates that is consistent under function minors. We decode eachLong Code function to the coordinates with
Ω(1)
Shapley value—as the sum of Shapley values ofall the coordinates of any monotone function is equal to , there is a bounded number of suchcoordinates. We argue about the consistency of this decoding using a structural result that statesthat under a uniformly random minor, Shapley value is roughly preserved. On the necessity of “richness” in -to- Conjecture.
A natural question is whether our hardnessresult can be obtained using a weaker assumption such as the -to- conjecture (whose im-perfect completeness version was recently established [KMS17, DKK + + f : { , } n → { , } and g : { , } n → { , } such that g is a minor of f with respect to the -to- function π , both the functions f and g have exactly one coordinate i , i respectively, with Ω(1)
Shapley value, and yet π ( i ) = i . Such an adversarial example is interesting from two an-gles: first, it shows that even using the -to- conjecture, the Shapley value based decoding isnot consistent. Second, it gives an example of agents pairing up maliciously to completely alterthe Shapley value. The underlying phenomenon is that the rich -to- games have “subcode-covering” property, which is absent in the standard -to- games, helping in preserving the con-sistency of any biased influence measure such as the Shapley value. Organization.
In Section 2, we formally define PCSPs, polymorphisms, and Shapley value. Wepresent the algorithmic and hardness parts of our dichotomy result in Section 3 and Section 4respectively. We present the adversarial example of a -to- minor that alters the Shapley valuein Section 5. A minor(formally defined in Section 2) of a function f : { , } m → { , } is a function g : { , } n → { , } ofsmaller arity n ≤ m obtained from f by identifying sets of variables together. Preliminaries
Notations.
We use [ n ] to denote the set { , , . . . , n } . For a k -ary relation A ⊆ [ q ] k , we abuse thenotation and use A both as a subset of [ q ] k , and also as a predicate A : [ q ] k → { , } . For a vector x = ( x , x , . . . , x n ) ∈ { , } n , we use hw ( x ) to denote P ni =1 x i . For two vectors x , y ∈ { , } n , wesay that x ≤ y if x i ≤ y i for all i ∈ [ n ] . A Boolean function f : { , } n → { , } is called monotoneif f ( x ) ≤ f ( y ) for all x ≤ y . PCSPs and Polymorphisms.
We first define Constraint Satisfaction Problems(CSP).
Definition 2. (CSP) Given a k -ary relation A : D k → { , } over a domain D , the Constraint Satis-faction Problem(CSP) associated with the predicate A takes a set of variables V = { v , v , . . . , v n } as input which are to be assigned values from D . There are m constraints ( e , e , . . . , e m ) eachconsisting of e i = (( e i ) , ( e i ) , . . . , ( e i ) k ) ⊆ V k that indicate that the corresponding assignmentshould belong to A . The objective is to identify if there is an assignment V → D that satisfies allthe constraints. In general, we can have multiple relations A , A , . . . , A l , and different constraints can usedifferent relations. We denote such a CSP by CSP ( A , A , . . . , A l ) .We formally define Promise Constraint Satisfaction Problems(PCSP). Definition 3. (PCSP) In a Promise Constraint Satisfaction Problem
P CSP (Γ) over a pair of do-mains D , D , we have a set of pairs of relations Γ = { ( A , B ) , ( A , B ) , . . . , ( A l , B l ) } such thatfor every i ∈ [ l ] , A i is a subset of D k i and B i is a subset of D k i . Furthermore, there is a homomor-phism h : D → D such that for all i ∈ [ l ] and x ∈ D k i , x ∈ A i implies h ( x ) ∈ B i . Given a CSP ( A , A , . . . , A l ) instance, the objective is to distinguish between the two cases:1. There is an assignment to the variables from D that satisfies every constraint when viewedas CSP ( A , A , . . . , A l ) .2. There is no assignment to the variables from D that satisfies every constraint when viewedas CSP ( B , B , . . . , B l ) . We now define Boolean Ordered PCSPs.
Definition 4. (Boolean Ordered PCSP) A PCSP
P CSP (Γ) over a pair of domains D , D with theset of pairs of relations Γ = { ( A , B ) , ( A , B ) , . . . , ( A l , B l ) } is said to be Boolean Ordered if thefollowing hold.1. The domains are both Boolean i.e., D = D = { , } .2. There exists i ∈ [ l ] such that A i = B i = { (0 , , (0 , , (1 , } . Associated with every PCSP, there are polymorphisms that capture the closure properties ofthe satisfying solutions to the PCSP. More formally, we can define polymorphisms of a PCSP asfollows.
Definition 5. (Polymorphisms) For
P CSP (Γ) with
Γ = { (( A , B ) , ( A , B ) , . . . , ( A l , B l )) } wherefor every i ∈ [ l ] , A i : [ q ] k i → { , } , B i : [ q ] k i → { , } , a polymorphism of arity n is a function f : [ q ] n → [ q ] that satisfies the below property for all i ∈ [ l ] . For all ( v , v , . . . , v k i ) such that forall j ∈ [ n ] , (( v ) j , ( v ) j , . . . , ( v k i ) j ) ∈ A i , we have ( f ( v ) , f ( v ) , . . . , f ( v k i )) ∈ B i We use
Pol (Γ) to denote the family of all the polymorphisms of
P CSP (Γ) . (Γ) is that the family of functions is closed under takingminors. We first define the minor of a function formally. Definition 6. (Minor of a function) For a Boolean function f : [ q ] n → [ q ′ ] and m ≤ n , the function g : [ q ] m → [ q ′ ] is said to be a minor of f with respect to the function π : [ n ] → [ m ] if g ( x , x , . . . , x m ) = f ( x π (1) , x π (2) , . . . , x π ( n ) ) ∀ x , x , . . . , x m ∈ [ q ] We say that a function g is a minor of f if there exists some π such that g is a minor of f with respectto π . We are often interested in -to- minors. A function g is said to be a -to- minor of f if thereexists a -to- function π such that g is a minor of f with respect to π , where -to- function isdefined below. Definition 7. ( -to- function) A function π : [2 n ] → [ n ] is said to be a -to- function if | π − ( i ) | = 2 ∀ i ∈ [ n ] We use F → ( n ) to denote the set of all the -to- functions from [2 n ] to [ n ] . By the definition of the polymorphisms, we can infer that if f ∈ Pol (Γ) for a PCSP Γ , then forall functions g such that g is a minor of f , we have g ∈ Pol (Γ) . Such a family of functions that isclosed under taking minors is called as a minion . We often refer to the family of polymorphismsof a PCSP as the polymorphism minion.We refer the reader to [BBKO19] for an extensive introduction to PCSPs and polymorphisms.
Shapley value.
Let f : { , } n → { , } be a monotone Boolean function. We can view themonotone Boolean function f as a voting scheme between two parties, and n agents: the winnerof the voting scheme when the i th agent votes for x i ∈ { , } is f ( x ) . The relative power of anagent in a voting scheme is typically measured using the Shapley-Shubix Index, also known asShapley Value.Informally speaking, the Shapley Value of a coordinate i is the probability that the i th agentis the altering vote when we start with all zeroes and flip the votes in a uniformly random order.More formally, Definition 8. (Shapley value) Let f : { , } n → { , } be a monotone Boolean function. Let σ ∈ S n be a uniformly random permutation of [ n ] . For an integer j ∈ [ n ] , let P j denote the the set of first j elements of σ i.e., P j := { σ (1) , σ (2) , . . . , σ ( j ) } . The Shapley value Φ f ( i ) of the coordinate i ∈ [ n ] isdefined as Φ f ( i ) := Pr σ {∃ j ∈ [ n ] : σ ( j ) = i, f ( P j − ) = 0 , f ( P j ) = 1 } We also give an alternate definition of Shapley value using the notion of boundary of a coor-dinate. For a monotone Boolean function f : { , } n → { , } and coordinate i ∈ [ n ] , let B f ( i ) denote the boundary of the coordinate i i.e. B f ( i ) := { S ⊆ [ n ] \ { i } : f ( { i } ∪ S ) = 1 , f ( S ) = 0 } By the monotonicity of f , we can infer that B f ( i ) satisfies the following sandwich property thatwill be useful later. Proposition 9.
Let f : { , } n → { , } be a monotone Boolean function and let i ∈ [ n ] . Then, forevery pair of sets S , S ∈ B f ( i ) with S ⊆ S , we have S ∈ B f ( i ) for all S such that S ⊆ S ⊆ S . roof. By the monotonicity of f , we have f ( S ∪ { i } ) ≥ f ( S ∪ { i } ) = 1 , and thus, f ( S ∪ { i } ) = 1 .Similarly, we have f ( S ) ≤ f ( S ) = 0 , and thus, f ( S ) = 0 .For an index j ∈ { , , . . . , n − } , let µ f ( j ) ( i ) denote the fraction of subsets of [ n ] of size j thatare in B f ( i ) i.e. µ f ( j ) ( i ) := (cid:12)(cid:12)(cid:12) B f ( i ) ∩ (cid:0) [ n ] j (cid:1)(cid:12)(cid:12)(cid:12) .(cid:0) nj (cid:1) . We can rewrite the definition of Shapley value of the i th coordinate as the following [Web77]: Φ f ( i ) = P n − j =0 µ f ( j ) ( i ) n . (1) In this section, we show that monotone Boolean functions where each coordinate has boundedShapley value has arbitrarily large threshold functions as minors, thereby proving the algorith-mic part of our dichotomy result.Let L be a positive integer and ≤ τ ≤ L be a non-negative integer. We let THR
L,τ : { , } L →{ , } be the threshold function on L variables with threshold τ . More formally, THR
L,τ ( x ) := ( if hw ( x ) ≥ τ otherwise.For a monotone Boolean function f : { , } n → { , } and real number p ∈ [0 , , let P p ( f ) denote the expected value of f ( x ) where each element x i , i ∈ [ n ] is independently set to be withprobability p and with probability − p . For every monotone function f , the function P p ( f ) isa strictly monotone continuous function in p on the interval [0 , . The value p c = p c ( f ) at which P p c ( f ) = is called the critical probability of f .Using the Russo-Margulis Lemma [Rus82, Mar74] and Poincaré Inequality, we can show thefollowing lemma that we need later. Lemma 10 (Exercise 8.29(e) in [O’D14]) . Let f be a non-constant monotone Boolean functionwith critical probability p c ≤ . Let p := ν ) p c for ν > . If p ≤ , then P p ( f ) ≥ − ν . We now define the threshold interval of f . Definition 11.
For a monotone function f and < ǫ < , we define T ǫ ( f ) := p − p , where p and p are such that P p ( f ) = ǫ, P p ( f ) = 1 − ǫ . Kalai [Kal04] proved the following result regarding monotone Boolean functions.
Theorem 12.
For every a, ǫ, γ > , there exists δ := δ ( a, ǫ, γ ) > such that for every monotoneBoolean function f : { , } n → { , } with Φ f ( i ) ≤ δ for all i ∈ [ n ] and a ≤ p c ( f ) ≤ − a , then T ǫ ( f ) ≤ γ . We will use this result to show that for every monotone function where each coordinate hasbounded Shapley value has arbitrarily large threshold functions as minor.7 emma 13.
For every L ≥ , there exists a δ := δ ( L ) > such that the following holds. For anymonotone Boolean function f : { , } n → { , } with Φ f ( i ) ≤ δ ∀ i ∈ [ n ] there exists a positive integer L ′ ∈ { L, L + 1 } and a non-negative integer τ such that THR L ′ ,τ is aminor of f .Proof. We obtain δ := δ ( L ) > from Theorem 12 by setting ǫ = L +1 , γ = a = L . Our goal isto show that for this parameter δ , for every monotone Boolean function f with each coordinatehaving Shapley value at most δ , there exists L ′ ∈ { L, L + 1 } and τ such that THR L ′ ,τ is a minor of f . We assume that f is a non-constant function, else we have a trivial minor by setting τ = 0 or τ = L ′ . Let p c be the critical probability of f . Case 1: p c < a = L .Let p = L p c < L . Using Lemma 10, we can conclude that P p ( f ) ≥ − L . As P p ( f ) ismonotone, we get that P L ( f ) > − L . We let g : { , } L → { , } be a uniformly randomminor of f i.e. we choose the function π : [ n ] → [ L ] by choosing each value π ( i ) uniformly andindependently at random from [ L ] , and we let g to be the minor of f with respect to π .Note that for every i ∈ [ L ] , the distribution of g ( { i } ) over the random minor g is the same assampling a random input to f where we set each bit to with probability L . As P L ( f ) ≥ − L ,we get that for each i ∈ [ L ] , g ( { i } ) = 1 with probability at least − L . By union bound, withprobability at least , g ( { i } ) = 1 for all i ∈ [ L ] . As f (0 , , . . . ,
0) = 0 , g ( φ ) = 0 as well. Thus, withprobability at least , g = THR L, . Hence, THR L, is a minor of f . Case 2: p c > − a = 1 − L .Let f † be the Boolean dual of f defined as f † ( x ) = 1 − f ( x ) . Note that P p ( f † ) = 1 − P − p ( f ) for all p ∈ [0 , . Thus, p c ( f † ) = 1 − p c < a . Using the previous case, we can infer that THR L, isa minor of f † with respect to a funtion π : [ n ] → [ L ] . The same function π proves that THR † L, = THR
L,L is a minor of f . Case 3: a ≤ p c ≤ − a .Using Theorem 12, we obtain p such that P p ( f ) ≤ ǫ , and P p + γ ≥ − ǫ , where ǫ = L +1 , γ = L . As γ < L ( L +1) , there exists L ′ ∈ { L, L + 1 } and τ ∈ [ L ′ ] such that p + γ < τL ′ and p > τ − L ′ .Thus, we get that P τL ′ ( f ) > − ǫ and P τ − L ′ < ǫ . Let g : { , } L ′ → { , } be a uniformly randomminor of f i.e. we choose π : [ n ] → [ L ′ ] by setting each value uniformly and independentlyat random from [ L ′ ] and set g to be the minor of f with respect to π . For a vector x ∈ { , } L ′ with hw ( x ) = τ , with probability greater than − L +1 , g ( x ) = 1 . Similarly, for x ∈ { , } L ′ with hw ( x ) = τ − , with probability greater than − L +1 , g ( x ) = 0 . Thus, with non-zero probability, g ( x ) = 1 for all x ∈ { , } L ′ with hw ( x ) = τ and g ( x ) = 0 for all x ∈ { , } L ′ with hw ( x ) = τ − . Inother words, with non-zero probability, g is equal to THR L ′ ,τ . Thus, THR L ′ ,τ is a minor of f .Using the existence of arbitrarily large arity threshold minors, the algorithmic part of ourDichotomy result follows immediately. Theorem 14.
Let Γ be a Promise CSP template. Suppose that for every ǫ > , there exists a function f ∈ Pol(Γ) , f : { , } n → { , } such that Φ i ( f ) ≤ ǫ for all i ∈ [ n ] . Then, PCSP(Γ) ∈ P . f f ′ g Figure 1: An illustration of the two step minor approach: Here f : { , } → { , } is a Booleanfunction, f ′ : { , } → { , } is a minor of f with respect to the function π : [6] → [5] with π ( i ) = max( i − , , and g is a minor of f ′ with respect to the function π : [5] → [3] with π ( i ) = ⌈ i +12 ⌉ . Proof.
Using Lemma 13, we can conclude that there are infinitely many positive integers L suchthat there exists τ ∈ { , , . . . , L } with THR
L,τ ∈ Pol(Γ) . As the threshold functions are symmet-ric , Pol(Γ) has symmetric polymorphisms of infinitely many arities. Thus, using the BLP+Affinealgorithm of [BGWZ20],
PCSP(Γ) can be solved in polynomial time.We remark that the above result is inspired by a special case shown by Barto [Bar18] that aBoolean Ordered PCSP is polytime tractable if it has cyclic polymorphisms of arbitrarily largearities.
In this section, we prove the hardness part of our dichotomy result. First, we prove that Shapleyvalue is preserved under uniformly random -to- minors, and then we use this to show thehardness assuming the Rich -to- Conjecture.
Let f : { , } n → { , } be a monotone Boolean function with Φ f (1) ≥ λ for some absoluteconstant λ > . Let g : { , } n → { , } be a minor of f with respect to the uniformly random -to- function π : [2 n ] → [ n ] . Our goal in this subsection is to show that E π [Φ g ( π (1))] ≥ γ forsome function γ := γ ( λ ) > . We prove this in two steps. (See Figure 1)1. First, we consider the minor of f , f ′ : { , } n − → { , } obtained with respect to π :[2 n ] → [2 n − where π (1) = π (2) = 1 , π ( i ) = i − ∀ i ∈ { , , . . . , n } . We show that Φ f ′ (1) ≥ λ .2. Next, we consider a minor g of f ′ obtained with respect to the function π : [2 n − → [ n ] which has π (1) = 1 while the rest n − values are chosen using a uniformly randompartition of [2 n − into n − pairs. We show that E π [Φ g (1)] ≥ γ for some function γ := γ ( λ ) > .Note that the process of first taking the f ′ minor and then obtaining g by partitioning [2 n − into n − uniformly random pairs is equivalent to taking a uniformly random -to- minor of f . A function f : { , } n → { , } is said to be symmetric if it is unchanged by any permutation of the input variables. -to- minor.The first step is captured by the following lemma. Lemma 15.
Let f : { , } n → { , } and f ′ : { , } n − → { , } be monotone Boolean functionssuch that f ′ is a minor of f with respect to the function π : [2 n ] → [2 n − defined as π ( i ) =max( i − , . If Φ f (1) ≥ λ , then Φ f ′ (1) ≥ λ .Proof. We recall a bit of notation: let B f (1) denote the boundary of the coordinate in the func-tion f i.e. the family of all the sets S ⊆ [2 n ] \ { } such that f ( S ) = 0 , f ( S ∪ { } ) = 1 . For aninteger j ∈ { , , . . . , n − } , let µ f ( j ) (1) denote the fraction of subsets of [2 n ] \ { } of size j thatare in B f (1) . For ease of notation, we let µ ( j ) = µ f ( j ) (1) , and µ ′ ( j ) = µ f ′ ( j ) (1) . Consider a set S ⊆ [2 n ] \ { } such that S ∈ B f (1) . Note that S ′ = { i − i > , i ∈ S } satisfies S ′ ∈ B f ′ (1) . Suppose that S , S ∈ B f (1) such that | S | = | S | = j , S = S and / ∈ S ∪ S .Then, the above definition satisfies S ′ = S ′ , S ′ , S ′ ∈ B f ′ (1) and | S ′ | = | S ′ | = j . This implies that (cid:12)(cid:12)(cid:12) { S : S ∈ B f (1) , | S | = j, / ∈ S } (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) B f ′ (1) ∩ (cid:18) [2 n − \ { } j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Similarly, (cid:12)(cid:12)(cid:12) { S : S ∈ B f (1) , | S | = j, ∈ S } (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) B f ′ (1) ∩ (cid:18) [2 n − \ { } j − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Summing the two, we obtain that (cid:12)(cid:12)(cid:12) { S : S ∈ B f (1) , | S | = j } (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) B f ′ (1) ∩ (cid:18) [2 n − \ { } j − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) B f ′ (1) ∩ (cid:18) [2 n − \ { } j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) We can rewrite it as (cid:18) n − j (cid:19) µ ( j ) ≤ (cid:18) n − j (cid:19) µ ′ ( j ) + (cid:18) n − j − (cid:19) µ ′ ( j − ∀ j ∈ [2 n − As (cid:0) n − j (cid:1) = (cid:0) n − j (cid:1) + (cid:0) n − j − (cid:1) for every j ∈ [2 n − , we get that µ ( j ) ≤ µ ′ ( j ) + µ ′ ( j − for all j ∈ [2 n − . Also note that µ (0) ≤ µ ′ (0) , and µ (2 n − ≤ µ ′ (2 n − . Summing over allthese inequalities, we get that X j ∈{ , ,..., n − } µ ′ ( j ) ≥ X j ∈{ , ,..., n − } µ ( j ) ≥ λ (2 n )2 = nλ Thus, Φ f ′ (1) = P j ∈{ , ,..., n − } µ ′ ( j )2 n − ≥ λ . Before proving the second step, we prove the following key lemma regarding the distributionof the boundary subsets. 10 emma 16.
Let f ′ : { , } n − → { , } be a monotone Boolean function such that Φ f ′ (1) = λ with λ ≥ n . For an integer j ∈ { , , . . . , n − } , let µ ′ ( j ) = µ f ′ ( j ) (1) . Then, there exists an absoluteconstant γ := γ ( λ ) > such that P n − j =0 µ ′ (2 j ) n ≥ γ Proof.
We prove that there exist constants (depending on λ ) c < c , c > such that for all j suchthat c n ≤ j ≤ c n , we have µ ′ ( j ) ≥ c , and c − c ≥ λ . This directly implies the required claimwith γ = Ω( cλ ) .For a pair of integers ≤ i < j ≤ n − , we define the following parameter µ ′ ( i, j ) as thefraction of the pair of subsets ( S, T ) where S, T ⊆ { , , . . . , n − } , | S | = i, | T | = j, S ⊆ T thatsatisfy S ∈ B f ′ (1) , T ∈ B f ′ (1) . µ ′ ( i, j ) = (cid:12)(cid:12) { ( S, T ) : | S | = i, | T | = j, S ⊆ T, S ∈ B f ′ (1) , T ∈ B f ′ (1) } (cid:12)(cid:12)(cid:0) n − i (cid:1)(cid:0) n − − ij (cid:1) We first claim that there exist constants (depending on λ ) c < c , c > such that µ ′ ( c n, c n ) ≥ c , and c − c ≥ λ . Consider a uniformly random permutation of [2 n − \ { } denoted by σ = ( σ (1) , σ (2) , . . . , σ (2 n − . For an integer j ∈ { , , . . . , n − } , let S j be the random variablethat is the union of the prefix of σ containing the first j elements. S j := { σ (1) , σ (2) , . . . , σ ( j ) } , ∀ j ∈ { , , . . . , n − } . For each j ∈ { , , . . . , n − } , the subset S j is uniformly distributed in (cid:0) [2 n − \{ } j (cid:1) . For j ∈{ , , . . . , n − } , let X j be the indicator random variable for the event that S j ∈ B f ′ (1) . By thedefinition of µ ′ ( j ) , we get E [ X j ] = µ ′ ( j ) ∀ j ∈ { , , . . . , n − } . Let X = X + X + . . . + X n − be the number of subsets in the set family ( φ = S ⊂ S ⊂ S . . . ⊂ S n − = [2 n − \ { } ) that are in B f ′ (1) . Using Equation (1), we get E [ X ] = λ (2 n − . Using Jensen’s inequality, we get that E (cid:20)(cid:18) X (cid:19)(cid:21) ≥ (cid:18) λ (2 n − (cid:19) = 12 · λ (2 n − (cid:18) λ n −
2) + nλ − (cid:19) ≥ λ (cid:18) n − (cid:19) wherein the final inequality, we used the fact that λn ≥ . Note that for every i < j , the marginaldistribution of ( S i , S j ) is the uniform distribution over all the pairs of subsets ( S, T ) where S, T ⊆{ , , . . . , n − } , | S | = i, | T | = j, S ⊆ T . Thus, by the definition of µ ′ ( i, j ) , we get that µ ′ ( i, j ) = E [ X i X j ] , for ≤ i < j ≤ n − . Therefore we have E (cid:20)(cid:18) X (cid:19)(cid:21) = E X ≤ i Suppose that f ′ : { , } n − is a monotone Boolean function such that Φ f ′ (1) ≥ λ with λ ≥ n . Let g be a random minor of f ′ with respect to π : [2 n − → [ n ] which is obtained by setting π (1) = 1 , and for every i > , we randomly choose j , j ∈ [2 n − \ { } (without replacements)and set π ( j ) = π ( j ) = i . In other words, we choose a uniformly random partition of [2 n − \{ } into n − pairs P , P , . . . , P n and set π ( j ) = i ∀ j ∈ P i . Then, there exists γ := γ ( λ ) > such that E π [Φ g (1)] ≥ γ . Proof. For ease of notation, we let µ ′ ( j ) = µ f ′ ( j ) (1) and µ g ( j ) = µ g ( j ) (1) . For a set S ⊆ [ n ] \ { } and a function π : [2 n − → [ n ] with π (1) = 1 , and | π − ( i ) | = 2 for all i ∈ { , , . . . , n } , let π − ( S ) be the | S | sized subset of { , , . . . , n − } defined as follows: π − ( S ) := { π − ( i ) : i ∈ S } For every set S ⊆ { , , . . . , n } , when π : [2 n − → [ n ] is a uniformly random -to- minor with π (1) = 1 , and the rest n − elements are partitioned into n − pairs uniformly at random,the set π − ( S ) is distributed uniformly in (cid:0) [2 n − \{ } | S | (cid:1) . Also note that S ∈ B g (1) if and only if π − ( S ) ∈ B f ′ (1) . Thus, for every set S ⊆ { , , . . . , n } , the probability that S ∈ B g (1) (over thechoice of π ) is equal to µ ′ (2 | S | ) . Summing over all such sets of size j , we get that for every j ∈ { , , . . . , n − } , the expected value of µ g ( j ) is equal to µ ′ (2 j ) . E π [ µ g ( j )] = µ ′ (2 j ) ∀ j ∈ { , , . . . , n − } By using Lemma 16, we can infer that there exists γ = γ ( λ ) > such that P n − j =0 E π [ µ g ( j )] = P n − j =0 µ ′ (2 j ) ≥ γn . Using Equation (1), we get E π [Φ g (1)] = E π " P n − j =0 µ g ( j ) n = P n − j =0 E π [ µ g ( j )] n ≥ γ. Lemma 15 and Lemma 17 together prove that Shapley value behaves well under uniformlyrandom -to- minors for monotone Boolean functions. Lemma 18. Suppose that f : { , } n → { , } is a monotone Boolean function such that Φ f (1) ≥ λ for some absolute constant λ > with λ ≥ n . Then, there exists γ := γ ( λ ) > such that E π [Φ g ( π (1))] ≥ γ where g is a minor of f with respect to the uniformly random -to- function π . roof. Combining Lemma 15 and Lemma 17, we can conclude that for every i ∈ [2 n ] , i > ,when π : [2 n ] → [ n ] is a uniformly random -to- minor conditioned on the fact that π (1) = π ( i ) ,we have E π [Φ g ( π (1))] ≥ γ . Taking average over all the i ∈ [2 n ] , i > , we get a proof that the sameinequality holds when π is a uniformly random -to- minor. We first formally define the Label Cover problem and state the Rich -to- Conjecture. Definition 19. (Label Cover) In the Label Cover problem G = ( G, Σ L , Σ R , Π) , the input is a bi-partite graph G = ( L ∪ R, E ) with projection constraint Π e : Σ L → Σ R on every edge e ∈ E . Alabeling σ which assigns values from Σ L to L and from Σ R to R satisfies the constraint Π e on theedge e = ( u, v ) if Π e ( σ ( u )) = σ ( v ) . The objective is to identify if there is a labeling that satisfies allthe constraints. For every constant ǫ > , it is NP-hard to distinguish between the case that a given LabelCover instance has a labeling that satisfies all the constraints vs. no labeling can satisfy morethan ǫ fraction of the constraints. This hardness result for Label Cover has been instrumen-tal in showing numerous strong, and sometimes optimal, inapproximability results for variouscomputational problems. However, standard Label Cover seems insufficient as a starting pointtowards proving hardness results for approximate graph coloring and other -variable PCSPs.To circumvent this, the hardness of Label Cover on structured instances such as Unique Games,smooth Label Cover, etc. has been studied.In his celebrated work proposing the Unique Games Conjecture [Kho02a], Khot also pro-posed the “ -to- conjecture” that the strong hardness of Label Cover holds when all the con-straints of the Label Cover are -to- functions. The imperfect completeness version of this con-jecture was recently established in a striking sequence of works [KMS17, DKK + + -to- functions on edges incidenton every vertex u ∈ L is uniform over F → . Definition 20. (Rich -to- Label Cover instances) We all a Label Cover instance G = ( G, Σ L , Σ R , Π) with G = ( L ∪ R, E ) a rich -to- instance if the following hold.1. There exists an integer Σ such that Σ L = [2Σ] , Σ R = [Σ] , and every projection constraint Π e , e ∈ E is a -to- function.2. For every vertex u ∈ L , the distribution of -to- functions P u obtained by first sampling auniformly random neighbor v of u , and then picking Π e , e = ( u, v ) , is uniform over F → (Σ) . Conjecture 21. (Rich -to- Conjecture) [BKM21] For every ǫ > , there exists an integer Σ = Σ( ǫ ) such that given a rich -to- Label Cover instance G , it is NP-Hard to distinguish between thefollowing.1. There is a labeling that satisfies all the constraints of G .2. No labeling can satisfy more than ǫ fraction of the constraints of G . We are now ready to state the hardness part of our dichotomy. It is proved using the La-bel Cover-Long Code framework. This reduction is standard in the PCSP literature, see e.g.,[BBKO19]. 13 heorem 22. Assume the Rich -to- Conjecture. Let PCSP (Γ) be a Boolean Ordered PCSP suchthat there exists an absolute constant λ > with max i ∈ [ n ] Φ f ( i ) ≥ λ for all functions f : { , } n →{ , } , f ∈ Pol (Γ) . Then PCSP (Γ) is NP-Hard.Proof. Let Γ = { ( A , B ) , ( A , B ) , . . . , ( A l , B l ) } be the PCSP under consideration, where each A i is a subset of { , } k i for all i ∈ [ l ] , and similarly, each B i is a subset of { , } k i for all i ∈ [ l ] . Westart from a rich -to- Label Cover instance G = ( G, [2Σ] , [Σ] , Π) with G = ( L ∪ R, E ) . For ease ofnotation, we use Σ w to denote if w ∈ L , and Σ if w ∈ R . For every vertex w ∈ L ∪ R , we have aset of Σ w nodes denoted by L w = { w } × { , } Σ w referred to as the long code corresponding to w . The elements of our output PCSP instance V is the union of all the long code nodes. V = [ w ∈ L ∪ R L w We add two types of constraints.1. Polymorphism Constraints. For every i ∈ [ l ] , we add the following constraints using thepair of predicates ( A i , B i ) . For every w ∈ L ∪ R , and multiset of vectors x , x , . . . , x k i ∈{ , } Σ w satisfying ( x j , x j , . . . , x k i j ) ∈ A i ∀ j ∈ [Σ w ] , we add the constraint on the k i nodes { w, x } , { w, x } , . . . , { w, x k i } .2. Equality Constraints. For every edge e = ( u, v ) of the Label Cover instance with the con-straint Π e : [2Σ] → [Σ] , we add the following set of equality constraints. For every x ∈{ , } and y ∈ { , } Σ such that for all j ∈ [2Σ] , x j = y Π e ( j ) , we add an equality con-straint between { u, x } and { v, y } ensuring that the two nodes are assigned the same value.The fact that we can add the equality constraints follows either by identifying the variablestogether, or by observing that the polymorphism minion of any PCSP remains the samewhen we add the equality predicate (see e.g., [BBKO19, GS20]). Completeness. Suppose that there exists a labeling σ that satisfies all the constraints of the LabelCover instance. For every node { w, x } ∈ V , we assign the dictator function x σ ( w ) ∈ { , } . By theway we have added the polymorphism constraints, any dictator assignment satisfies them. Theequality constraints are also satisfied as the labeling satisfies all the constraints of G . Soundness. Suppose that there exists an assignment f : V → { , } that satisfies all the poly-morphism constraints and the equality constraints. Then, we claim that there exists a labeling σ that satisfies ǫ := ǫ ( λ ) > fraction of the constraints of the Label Cover instance G .For a vertex w ∈ L ∪ R , let f w : { , } Σ w → { , } denote the function f restricted to L w . Notethat f w is a polymorphism of the PCSP Γ for all w ∈ L ∪ R . As every polymorphism of Γ has acoordinate with Shapley value at least λ , for every u ∈ L , we define the set S ( u ) that is non-emptyas follows: S ( u ) = { i ∈ [2Σ] : Φ f u ( i ) ≥ λ } As P i ∈ [ n ] Φ f ( i ) = 1 for all functions f : { , } n → { , } , we have | S ( u ) | ≤ λ for all u ∈ L .As a corollary of Lemma 18, we can conclude that there exists γ = γ ( λ ) > such that forevery monotone Boolean function f : { , } → { , } with Φ f ( i ) ≥ λ , when g is a minor of f with respect to a uniformly random -to- function π : [2Σ] → [Σ] , Φ g ( π (1)) ≥ γ with probabilityat least γ . Note that applying Lemma 18 requires that λ ≥ . However, even when λ < , by14icking the coordinate with the largest Shapley value, we can still assume that in every long codefunction, there is a coordinate with Shapley value at least = Θ( λ ) , and then apply Lemma 18.Using this γ , for every v ∈ R , we define the set S ( v ) as S ( v ) = n i ∈ [Σ] : Φ f v ( i ) ≥ γ o By definition, we have | S ( v ) | ≤ γ for all v ∈ R . As the Label Cover instance is rich -to- , forevery u ∈ L , when we pick a uniformly random edge e = ( u, v ) adjacent to u with constraint Π e : [2Σ] → [Σ] , with probability at least γ , there exist i ∈ [2Σ] , i ∈ [Σ] such that Φ f u ( i ) ≥ λ , Φ f v ( i ) ≥ γ , and Π e ( i ) = i .We now pick a labeling σ of G by picking uniformly random label from S ( w ) for all w ∈ L ∪ R . By the above argument, for every u ∈ L , the expected number of constraints of G that areadjacent to u that the labeling σ satisfies is at least γ · λ γ . Summing over all u ∈ L , σ satisfies atleast Ω( λγ ) fraction of the constraints of G in expectation. Thus, there exists a labeling to G thatsatisfies ǫ = Ω( λγ ) > fraction of the constraints, which completes the proof. We construct an example of a -to- minor where the Shapley value alters completely after takingthe minor. Theorem 23. Let n ≥ be a positive integer. There exist two monotone Boolean functions f : { , } n → { , } and g : { , } n → { , } such that g is a -to- minor of f with respect to the -to- function π : [2 n ] → [ n ] defined as π ( i ) = ⌈ i ⌉ . Furthermore,1. Φ g (1) = Ω(1) , and Φ g ( j ) = o (1) for all j > .2. Φ f (3) = Ω(1) , and Φ f ( i ) = o (1) for all i ∈ [2 n ] , i = 3 .Proof. Similar to the proof of Theorem 22, we construct the minor function pair in two steps.1. First, we construct Boolean monotone functions f : { , } n − → { , } and g : { , } n →{ , } such that g is a minor of f with respect to the function π : [2 n − → [ n ] defined as π (1) = 1 , π ( i ) = ⌈ i +12 ⌉ for all i > . Furthermore, Φ g (1) = Ω(1) , and Φ g ( j ) = o (1) for all j > . We also have Φ f (2) = Ω(1) , and Φ f ( i ) = o (1) for all i ∈ [2 n − , i = 2 .2. We define the function f ′ : { , } n → { , } as f ′ ( y , y , . . . , y n ) = f ( y , y , . . . , y n ) Note that g is a minor of f ′ with respect to the -to- function π : [2 n ] → [ n ] defined as π ( i ) = ⌈ i ⌉ . Furthermore, by definition, we have Φ f ′ (3) = Ω(1) , and Φ f ′ ( i ) = o (1) for all i ∈ [2 n ] , i = 3 .Henceforth, our goal is to construct a pair of functions as in the first step above.We define a partial Boolean function to be a function from { , } n → { , , ? } . A partialBoolean function on n variables is monotone if for all p ∈ { , } n and q ∈ { , } n such that p ≤ q , if f ( p ) = 1 , then f ( q ) = 1 , and if f ( q ) = 0 , then f ( p ) = 0 .15ow, consider g : { , } n → { , } to be g ( x ) = if P nj =2 x j ≥ n if P nj =2 x j ≤ n x if n < P nj =2 x j < n By definition, g is a monotone function, and using Equation (1), we can infer that Φ g (1) = ,and Φ g ( j ) < n for all j > .We now construct f in three steps. Start with f = ′ ? ′ .1. (Preserving the minor) First, set the value of entries of f that are of the form ( x , x , x , · · · , x n , x n ) as f ( x , x , x , . . . , x n , x n ) = g ( x , x , . . . , x n ) ∀ x ∈ { , } n We then extend it both upwards and downwards i.e. if f ( p ) is set to and p ≤ q , then set f ( q ) = 1 as well, and similarly, if f ( q ) is set to , and p ≤ q , then we set f ( p ) = 0 . Thisensures that g is a minor of f and that the partial function f is monotone.2. (Destroying the influence of ) Next, we ensure that the Shapley value of the coordinate is low by the following operation: consider all y such that f ( y ) has not been set in the firststep, y = 0 and f (1 , y , · · · , y n − ) is already set to in the first step. Then set f ( y ) to be .Similarly, if y satisfies y = 1 and f (0 , y , · · · , y n − ) is already set to in the first step, set f ( y ) to be if it has not been set in the first step.We claim that the updated partial function f is still a monotone partial function. Consider p , q ∈ { , } n − such that p ≤ q . Suppose that f ( p ) is set to be . If it is set in the first step,as we extended the partial function upwards in the first step, f ( q ) = 1 as well. If f ( p ) is setto be in the second step, it implies that f ( p ′ ) has been set to in the first step, where p ′ is obtained from p by setting p to be . Let q ′ ∈ { , } n − be obtained from q by setting q = 1 . As p ′ ≤ q ′ , f ( q ′ ) has been set to in the first step as well. Thus, f ( q ) is set to be inthe second step. The same argument can be used to show that if f ( q ) = 0 , then f ( p ) = 0 aswell.3. (Adding influence to ) For all y for which f ( y ) = ′ ? ′ set f ( y ) = y . The fact that the finalfunction f is monotone follows from observing that any completion of a partial monotonefunction using a monotone function results in a monotone function.Finally, our goal is to argue about the Shapley value of the coordinates of the function f .First, we show that the Shapley value of the coordinate in f is o (1) . Suppose there exists p =(0 , y , y , · · · , y n − ) and q = (1 , y , y , · · · , y n − ) such that f ( p ) = 0 and f ( q ) = 1 . We claimthat both the values f ( p ) and f ( q ) are set in the first step of the above procedure. Suppose forcontradiction that this is not the case. If neither of them is set in the first step, then they will notbe set in the second step either, and in the third step, both of them will be assigned the samevalue, a contradiction. If exactly one of them is set in the first step, then in the second step, theother value would be set to be equal to it, a contradiction as well. Thus, both the values f ( p ) and f ( q ) are set in the first step.Let B = B g (1) ⊆ { , } n − be the boundary of the coordinate in g . As f ( q ) is set to be inthe first step, there exists x ∈ { , } n such that g ( x ) = 1 and ( x , x , x , · · · , x n , x n ) ≤ q . As x is not16ess than or equal to p , we can conclude that x = 1 and g (0 , x , x , · · · , x n ) = 0 . In other words, ( x , x , · · · , x n ) ∈ B . Similarly, there exists x ′ such that g ( x ′ ) = 0 and ( x ′ , x ′ , x ′ , · · · , x ′ n , x ′ n ) ≥ p .By the same argument as above, we can conclude that ( x ′ , x ′ , · · · , x ′ n ) ∈ B . Combining the both,we can conclude that there exist x , x ′ ∈ B such that ( x , x , x , x , . . . , x n , x n ) ≤ ( y , y , · · · , y n − ) ≤ ( x ′ , x ′ , x ′ , x ′ , . . . , x ′ n , x ′ n ) . Note that if the above inequality is true for a ( y , y , · · · , y n − ) , we di-rectly get that ( y , y , · · · , y n − ) is in the boundary of the coordinate in f .Observe that the boundary of coordinate in g is the set of vectors ( x , x , · · · , x n ) such that n ≤ P nj =2 x j ≤ n . By the previous argument, we can deduce that the boundary B ′ = B f (1) of the coordinate in f is the set of vectors y = ( y , y , · · · , y n − ) that satisfy the followingproperty: The number of i ∈ [ n − such that both y i = y i +1 = 1 is at least n . Similarly, thenumber of i ∈ [ n − such that y i = y i +1 = 0 is at least n . Observe that this implies that werequire that n ≤ P n − j =2 y j ≤ n . However, for every integer l such that n ≤ l ≤ n , whenwe sample a uniformly random vector y = ( y , y , . . . , y n − ) with P n − j =2 y j = l , the probabilitythat the number of i ∈ [ n − such that both y i = y i +1 = 1 is at least n is o ( n ) . Thus,using Equation (1), we can infer that the Shapley value of the coordinate in f is o (1) .We now show that the coordinate has Ω(1) Shapley value in f . Consider y = ( y , y , . . . , y n − ) such that n < hw ( y ) ≤ n . If the number of i such that both y i = y i +1 = 1 is less than n ,we have ( y , y , . . . , y n − ) ∈ B f (2) . However, for every integer l such that n ≤ l ≤ n , whenwe sample a uniformly random y with hw ( y ) = l , with probability − o (1) , the number of i suchthat both y i = y i +1 = 1 is less than n . Thus, using Equation (1), we can infer that the Shap-ley value of the coordinate is Ω(1) in the function f . Finally, by symmetry, we can observe that Φ f ( i ) = Φ f (3) for all i ≥ , and thus, Φ f ( i ) = o (1) for all i ≥ . Acknowledgments We thank Libor Barto, whose talk [Bar18] and insightful discussions inspired our work. References [AGH17] Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2+ ǫ )-SAT is NP-hard. SIAMJ. Comput. , 46(5):1554–1573, 2017.[Bar18] Libor Barto. Cyclic operations in promise constraint satisfaction prob-lems. Dagstuhl workshop The Constraint Satisfaction Problem: Complex-ity and Approximability, Schloss Dagstuhl, Germany , 2018. Available at .[Bar20] Libor Barto. Personal communication, 2018-20.[BBKO19] Libor Barto, Jakub Bulín, Andrei Krokhin, and Jakub Opršal. Algebraic approach topromise constraint satisfaction. 2019. arXiv-cs.CC:1811.00970.[BG16] Joshua Brakensiek and Venkatesan Guruswami. New hardness results for graph andhypergraph colorings. In ,pages 14:1–14:27, 2016. 17BG18] Joshua Brakensiek and Venkatesan Guruswami. Promise constraint satisfaction:Structure theory and a symmetric Boolean dichotomy. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 , pages1782–1801, 2018.[BGWZ20] Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, and Stanislav Zivný.The power of the combined basic linear programming and affine relaxation forpromise constraint satisfaction problems. SIAM J. Comput. , 49(6):1232–1248, 2020.[BJK05] Andrei A. Bulatov, Peter Jeavons, and Andrei A. Krokhin. Classifying the complexityof constraints using finite algebras. SIAM J. Comput. , 34(3):720–742, 2005.[BKM21] Mark Braverman, Subhash Khot, and Dor Minzer. On rich 2-to-1 games. In , volume 185 of LIPIcs , pages 27:1–27:20, 2021.[Bul17] Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In , pages 319–330. IEEEComputer Society, 2017.[DDS17] Anindya De, Ilias Diakonikolas, and Rocco A. Servedio. The inverse Shapley valueproblem. Games Econ. Behav. , 105:122–147, 2017.[DKK + Proceedings of the 50th An-nual ACM SIGACT Symposium on Theory of Computing, STOC 2018, , pages 940–951.ACM, 2018.[DKK + Proceedings of the 50th Annual ACM SIGACTSymposium on Theory of Computing, STOC 2018, , pages 376–389. ACM, 2018.[FKOS19] Miron Ficak, Marcin Kozik, Miroslav Olsák, and Szymon Stankiewicz. Dichotomyfor symmetric boolean PCSPs. In , volume 132 of LIPIcs , pages 57:1–57:12,2019.[FV98] Tomás Feder and Moshe Y. Vardi. The computational structure of monotonemonadic SNP and constraint satisfaction: A study through datalog and group the-ory. SIAM J. Comput. , 28(1):57–104, 1998.[GS20] Venkatesan Guruswami and Sai Sandeep. Rainbow coloring hardness via low sensi-tivity polymorphisms. SIAM J. Discret. Math. , 34(1):520–537, 2020.[JCG97] Peter Jeavons, David A. Cohen, and Marc Gyssens. Closure properties of constraints. J. ACM , 44(4):527–548, 1997.[Jea98] Peter Jeavons. On the algebraic structure of combinatorial problems. Theor. Com-put. Sci. , 200(1-2):185–204, 1998.[Kal04] Gil Kalai. Social indeterminacy. Econometrica , 72(5):1565–1581, 2004.18Kho02a] Subhash Khot. Hardness results for coloring 3-colorable 3-uniform hypergraphs. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2002 ,pages 23–32. IEEE, 2002.[Kho02b] Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings on34th Annual ACM Symposium on Theory of Computing, STOC 2002 , pages 767–775.ACM, 2002.[KMS17] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, andgrassmann graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium onTheory of Computing, STOC 2017, , pages 576–589. ACM, 2017.[KMS18] Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in Grassmann graphhave near-perfect expansion. In , pages 592–601. IEEE Computer Society, 2018.[KO19] Andrei A. Krokhin and Jakub Opršal. The complexity of 3-colouring H-colourablegraphs. In , pages 1227–1239. IEEE Computer Society, 2019.[Mar74] G. A. Margulis. Probabilistic characteristics of graphs with large connectivity (inRussian). Probl. Pered. Inform. , 10:101–108, 1974.[MKS + 13] Tomasz P. Michalak, Aadithya V. Karthik, Piotr L. Szczepanski, Balaraman Ravin-dran, and Nicholas R. Jennings. Efficient computation of the Shapley value forgame-theoretic network centrality. J. Artif. Intell. Res. , 46:607–650, 2013.[NN11] Ramasuri Narayanam and Yadati Narahari. A Shapley value-based approach to dis-cover influential nodes in social networks. IEEE Trans Autom. Sci. Eng. , 8(1):130–147, 2011.[O’D14] Ryan O’Donnell. Analysis of Boolean Functions . Cambridge University Press, 2014.[Pet20] Jan Petr. Monotone functions avoiding majorities, 2020. Undergraduate Thesis.Univerzita Karlova, Matematicko-fyzikální fakulta.[Pip02] Nicholas Pippenger. Galois theory for minors of finite functions. Discrete Mathe-matics , 254(1-3):405–419, 2002.[Rus82] Lucio Russo. An approximate zero-one law. Z. Wahrscheinlichkeitstheorie und Ver-wandte Gebiete , 61(1):129–139, 1982.[Sch78] Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the10th Annual ACM Symposium on Theory of Computing, STOC 1978 , pages 216–226.ACM, 1978.[SS54] L. S. Shapley and Martin Shubik. A method for evaluating the distribution of powerin a committee system. American Political Science Review , 48(3):787–792, 1954.[Web77] Robert Weber. Probabilistic values for games. Cowles Foundation Discussion Papers471R, Cowles Foundation for Research in Economics, Yale University, 1977.19WZ20] Marcin Wrochna and Stanislav Zivný. Improved hardness for H -colourings of G -colourable graphs. In Proceedings of the 2020 ACM-SIAM Symposium on DiscreteAlgorithms, SODA 2020 , pages 1426–1435. SIAM, 2020.[Zhu20] Dmitriy Zhuk. A proof of the CSP dichotomy conjecture.