Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

Hyp ercon tractiv e Inequalit y for Pseudo-Bo olean F unctions o f Bounded F ourier Widt h Gregory Gutin Ro yal H o llo w a y , Univ ersit y of London Egham, Surrey TW20 0EX United Kingdom, gutin@cs.rhul.ac. uk Anders Y eo Univ ersit y of Johannesburg PO Box 524 Auck land P ark 2006 South Africa, andersyeo@gmail.com No v ember 20, 20 21 Abstract A f u n ction f : {− 1 , 1 } n → R is called pseudo-Bo ol ean. It is w ell- known th at each pseudo-Bo ol ean function f can b e written as f ( x ) = P I ∈F ˆ f ( I ) χ I ( x ) , where F ⊆ { I : I ⊆ [ n ] } , [ n ] = { 1 , 2 , . . . , n } , and χ I ( x ) = Q i ∈ I x i and ˆ f ( I ) are non - zero reals. The degree of f is max {| I | : I ∈ F } and the width of f is the minimum integer ρ such t h at every i ∈ [ n ] app ears in at most ρ sets in F . F or i ∈ [ n ], let x i b e a ran- dom va riable taking va lues 1 or − 1 un i formly and indep endently from all other v ariables x j , j 6 = i. Let x = ( x 1 , . . . , x n ). The p -norm of f is || f || p = ( E [ | f ( x ) | p ]) 1 /p for any p ≥ 1. It is well-kno wn that || f || q ≥ || f || p whenever q > p ≥ 1. How ever, the higher norm can b e b ounded by the lo wer norm times a co efficien t not directly dep ending on f : if f is of de- gree d and q > p > 1 t hen || f || q ≤  q − 1 p − 1  d/ 2 || f || p . This ineq uali ty is called the Hyp ercontracti ve I n equalit y . W e show that one can replace d by ρ in t he H y percontractiv e I nequalit y for each q > p ≥ 2 as follo ws: || f || q ≤ ((2 r )! ρ r − 1 ) 1 / (2 r ) || f || p , where r = ⌈ q / 2 ⌉ . F or the case q = 4 and p = 2, which is imp ortan t in many app li cations, we pro ve a stronger inequality: || f || 4 ≤ (2 ρ + 1) 1 / 4 || f || 2 . 1 1 In tro duction F ourier analysis of pseudo-Bo ole an functions 1 , i.e., functions f : {− 1 , 1 } n → R , has b een used in many areas of computer science (cf. [1, 6, 14, 18, 1 9]), so cial choice theory (cf. [9, 15, 17]), co m binatorics, lea rning theory , co ding theory , and many other s (cf. [18, 19]). W e will us e the following well-kno wn and e asy to prov e fact [18]: ea c h function f : {− 1 , 1 } n → R can b e uniquely written a s f ( x ) = X I ∈F ˆ f ( I ) χ I ( x ) , (1) where F ⊆ { I : I ⊆ [ n ] } , [ n ] = { 1 , 2 , . . . , n } , and χ I ( x ) = Q i ∈ I x i and ˆ f ( I ) are non-zero re a ls. F ormula (1) is the F ourier expansion o f f and ˆ f ( I ) a re the F ourier co efficients of f . The r igh t ha nd size of (1) is a p olynomial a nd the degree max {| I | : I ∈ F } of this p olynomial will b e c a lled the de gr e e o f f . F or i ∈ [ n ], let ρ i be the n umber of sets I ∈ F such that i ∈ I . Let us call ρ = ma x { ρ i : i ∈ [ n ] } the F ourier width (or, just width ) of f . The F ourier width was introduced in [1 2] without giving it a name. The degree and width can b e viewed as dua l parameter s in the following sense. Consider a bipar tit e gr aph G with partite sets V and T , where V is the set of v aria bles in f a nd T is the set of ter ms in f in (1), and z t is an edge in G if z is a v ar iable in t ∈ T . Note tha t the degree of f is the maximum degr ee of a vertex in T and the width of f is the ma x im um degree of a vertex in V . F or i ∈ [ n ], let x i be a random v a riable taking v alues 1 or − 1 uniformly and independently from a ll other v ar iables x j , j 6 = i. Let x = ( x 1 , . . . , x n ). Then f ( x ) is a random v ar ia ble and the p - norm of f is || f || p = ( E [ | f ( x ) | p ]) 1 /p for any p ≥ 1. It is easy to show that || f || 2 2 = P I ∈F ˆ f ( I ) 2 , which is Parsev al’s Ident ity for ps e udo-Boo le an functions. It is well-known and easy to show that || f || q ≥ || f || p whenever q ≥ p ≥ 1 . Ho wev er , the hig her nor m ca n b e b ounded by the low er norm times a co efficien t not dep ending on f : if f is of de g ree d then || f || q ≤  q − 1 p − 1  d/ 2 || f || p . (2) The last inequality is ca lled the Hyp er c ontr active Ine quality . (In fact, the Hy- per con tra c tiv e Inequa lit y is o ften stated differently , but the Hyp ercontractiv e Inequality in the or iginal for m and (2) are equiv alent.) Since || f || 2 is easy to compute, the Hyp ercontractiv e Inequality is q uite useful fo r p = 2 and is often used for p = 2 and q = 4; this sp ecial c a se o f the Hyp ercont r activ e Inequal- it y has b een applied in many pap ers o n algo rithmics, so cial choice theo ry and many other area s, se e, e.g., [1, 2 , 9, 11, 12, 1 4 , 15, 17] and was given specia l pro ofs (cf. [10] a nd the extended abstract of [17]). W e will call this case the (4,2)-Hyp er c ontr active Ine quality . 1 Often functions f : { 0 , 1 } n → R are called pseudo-Bo olean [3]. In F ourier Analysis, the Boolean domain is often assumed to b e {− 1 , 1 } n rather than the more usual { 0 , 1 } n and we will foll o w this assumption in our pap er. 2 Theorem 1 proved b e lo w repla ces the co efficien t 3 d/ 2 befo re || f || 2 in the (4,2)-Hyp ercont r activ e Inequa lit y by (2 ρ + 1 − 2 ρ m ) 1 / 4 , where ρ is the width o f f and m = |F | . F or functions with 2 ρ + 1 < 9 d Theorem 1 pr o vides an impo rtan t sp ecial ca se o f the Hyp ercontractive Inequality with a smaller co efficien t. Note that in some c ases o ne can change v ariables (using a different bas is) s uch that the deg ree of f decreases sig nifican tly . How ever, this is not alwa ys p ossible and, even if it is p ossible, it might b e hard to find an appropria te ba sis. Our application of Theo rem 1 in Section 4 provides a nontrivial illustration of s uc h a situation. Note that Theorem 1 improves Lemma 7 in [12]. While in Lemma 7 [1 2], the co efficient b efore || f || 2 is (2 ρ 2 ) 1 / 4 ( ρ ≥ 2), in Theor em 1, we decrease it to (2 ρ + 1 − 2 ρ m ) 1 / 4 . W e provide examples s ho wing that this co efficien t is tight. Due to Theorem 1 , we know that the width ca n r eplace the degr ee as a pa- rameter in the co efficien t b efore || f || 2 in the (4,2)-Hyp ercontractiv e Inequality . A natural question is whether the same is true in the general case of the Hy- per con tra c tiv e Inequality for pseudo- Boolea n functions. W e show that w e can replace d by ρ for e ac h q ≥ p ≥ 2 as follows: || f || q ≤ ((2 r )! ρ r − 1 ) 1 / (2 r ) || f || p , where r = ⌈ q / 2 ⌉ . 2 (4,2)-Hyp ercon tractiv e Inequalit y In (1), let F = { I 1 , . . . , I m } , f j ( x ) = ˆ f ( I j ) χ I j ( x ) and w j = ˆ f ( I j ), j ∈ [ m ] . If ∅ ∈ F , we will assume that I 1 = ∅ . Theorem 1. L et f ( x ) b e a pseudo-Bo ole an function of width ρ ≥ 0 . Then || f || 4 ≤ (2 ρ + 1 − 2 ρ m ) 1 / 4 || f || 2 . Pr o of. If ρ = 0 then f ( x ) = c , where c is a constant and hence || f || 4 = || f || 2 = c . Thu s , a ssume that ρ ≥ 1 . Let S b e the set of quadruples ( p 1 , p 2 , p 3 , p 4 ) ∈ [ m ] 4 such that P 4 j =1 |{ i } ∩ I p j | is even for ea c h i ∈ [ n ], S ′ = { ( p 1 , p 2 , p 3 , p 4 ) ∈ S : p 1 = p 2 } and S ′′ = S \ S ′ . Note that if a pro duct f p ( x ) f q ( x ) f s ( x ) f t ( x ) contains a v ariable x i in only one o r three of the factors , then E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] = E [ P ] · E ( x i ) = 0 , wher e P is a po lynomial in r andom v ar iables x l , l ∈ [ n ] \ { i } . Thu s , E [ f ( x ) 4 ] = X ( p,q, s,t ) ∈ S E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] . Observe that if ( p, q, s, t ) ∈ S ′ then p = q a nd s = t and, thus, P ( p,q, s,t ) ∈ S ′ E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] = P m p =1 P m s =1 w 2 p w 2 s . F or a pair ( p, q ) ∈ [ m ] 2 , let N ( p, q ) = |{ ( s, t ) ∈ [ m ] 2 : ( p, q , s, t ) ∈ S ′′ }| . Let a quadruple ( p, q , s, t ) ∈ S ′′ . Since p 6 = q , there must b e an i which b elongs to just one o f the t wo sets I p and I q . Since ( p, q , s, t ) ∈ S ′′ , i must a lso b elong to just one o f the tw o sets I s and I t (t wo choices). Assume that i ∈ I s . Then by the definition of ρ , s ca n be chosen from a subset of [ m ] o f c ardinalit y a t most ρ . Once s is chosen, there is a unique choice for t . T he r efore, N ( p, q ) ≤ 2 ρ. 3 Note that ( p, q, s, t ) ∈ S ′′ if and only if ( s, t, p , q ) ∈ S ′′ which implies that there are at most N ( p, q ) tuples in S ′′ of the for m ( s, t, p, q ). W e also hav e E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] ≤ w p w q w s w t ≤ ( w 2 p w 2 q + w 2 s w 2 t ) / 2 . Thu s , X ( p,q, s,t ) ∈ S ′′ E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] ≤ X 1 ≤ p 6 = q ≤ m 2 N ( p, q ) w 2 p w 2 q 2 ≤ 2 ρ X 1 ≤ p 6 = q ≤ m w 2 p w 2 q . Hence, E [ f ( x ) 4 ] ≤ m X p =1 m X s =1 w 2 p w 2 s + 2 ρ X 1 ≤ p 6 = q ≤ m w 2 p w 2 q = (2 ρ + 1 ) m X p =1 m X s =1 w 2 p w 2 s − 2 ρ m X p =1 w 4 p . W e hav e P m p =1 w 4 p P m p =1 P m s =1 w 2 p w 2 s ≥ P m p =1 w 4 p P m p =1 P m s =1 [ w 4 p / 2 + w 4 s / 2] = P m p =1 w 4 p m P m p =1 w 4 p = 1 m . Thu s , E [ f ( x ) 4 ] ≤ (2 ρ + 1 − 2 ρ m )  P m i =1 w 2 i  2 = (2 ρ + 1 − 2 ρ m ) E [ f ( x ) 2 ] 2 . The last equality follows from Parsev al’s Identit y . The following t wo examples show the sharpnes s of this theore m. Let f ( x ) = 1 + P n i =1 x i . By Parsev a l’s Indentit y , E [ f ( x ) 2 ] = n + 1 . It is easy to c heck that E [ f ( x ) 4 ] = ( n + 1) +  4 2  n +1 2  = 3 n 2 + 4 n + 1 . Clearly , ρ = 1 and m = n + 1 and, thus, 2 ρ + 1 − 2 ρ m = 3 − 2 n +1 . Also, E [ f ( x ) 4 ] / E [ f ( x ) 2 ] 2 = 3 n 2 +4 n +1 ( n +1) 2 = 3 − 2 n +1 . Let f ( x ) = P I ⊆ [ n ] χ I ( x ) . Clearly , E [ f ( x ) 2 ] = m = 2 n . T o compute E [ f ( x ) 4 ] observe that when p, q and s a r e arbitrar ily fixed we hav e E [ f p ( x ) f q ( x ) f s ( x ) f t ( x )] 6 = 0 for a unique (one in 2 n ) choice o f t . Hence , E [ f ( x ) 4 ] = m 4 / 2 n = 2 3 n . Thus, E [ f ( x ) 4 ] / E [ f ( x ) 2 ] 2 = 2 n . Obser v e that ρ = 2 n − 1 and 2 ρ + 1 − 2 ρ m = 2 n as well. 3 Hyp ercon tractiv e Inequalit y A mu ltiset may contain multiple a ppearances of the sa me element. F or multisets we will use the sa me nota tio n as for sets, but we will stress it when we deal with m ultise ts . W e do no t a ttempt to optimize g ( r ) in the following theo rem. Theorem 2. L et f ( x ) b e a pseudo-Bo ole an function of width ρ ≥ 1 . Then for e ach p ositive inte ger r we have || f || 2 r ≤ [ g ( r ) ρ r − 1 ] 1 2 r · || f || 2 , wher e g ( r ) = (2 r )! . Pr o of. Obse r v e that E [ f ( x ) 2 r ] = P  2 r α 1 ...α m  E [ f α 1 1 ( x ) · · · f α m m ( x )] , where the sum is ta ken over all partitions α 1 + · · · + α m = 2 r of 2 r into m no n- negativ es 4 summands. Consider a non-zer o term E [ f α 1 1 ( x ) · · · f α m m ( x )] . Note that each v ari- able x i app ears in an even num b er of the factors in f α 1 1 ( x ) · · · f α m m ( x ) . W e deno te the set o f a ll such m -tuples α = ( α 1 , . . . , α m ) by E . Then E [ f ( x ) 2 r ] = X α ∈E  2 r α  m Y i =1 w α i i . (3) It is useful for us to view f α 1 1 ( x ) · · · f α m m ( x ), α ∈ E , as a pro duct o f 2 r factor s f i ( x ), i.e., E [ f α 1 1 ( x ) · · · f α m m ( x )] = E [ f t 1 ( x ) · · · f t 2 r ( x )] . Let I b e a subset of the multiset { t 1 , . . . , t 2 r } ( I is a m ultiset). W e call I is nont riv ial if it co n tains at least t wo elements (not nece s sarily distinct). A subset J of I is calle d minimal ly even if J is nontrivial, E [ Q i ∈ J f i ( x )] 6 = 0 but E [ Q i ∈ K f i ( x )] = 0 for each nontrivial s ubset K of the multiset J . If I 1 = ∅ (that is ∅ ∈ F ) a nd 1 is an element o f I without r e p etition (i.e., only one copy of 1 is in I ), then { 1 } is a lso called a minimal ly even subset. (Th us , if I contains t wo or more elements 1 then { 1 , 1 } is minimally even, but { 1 } is not; howev er, if I c on tains just o ne elemen t 1, then { 1 } is minimally even.) Let µ 1 be an element in the mu ltis e t T 1 := { t 1 , . . . , t 2 r } such that w 2 µ 1 = max { w 2 t i : t i ∈ T 1 } , and let M 1 be a minima lly even subset of T 1 containing µ 1 . F or j ≥ 2, let µ j be an e le men t in the multiset T j := { t 1 , . . . , t 2 r } \ ( ∪ j − 1 i =1 M i ) such that w 2 µ j = max { w 2 t i : t i ∈ T j } , and let M j be a minima lly even subset of T j containing µ j . Let s b e the la rgest j for which µ j is defined ab ov e. Observe that s ≤ r as a t most one of the minimally even sets M 1 , M 2 , . . . , M s has s iz e one. If s < r , for every j ∈ { s + 1 , s + 2 , . . . , r } let µ j be an element in the m ultise t T 1 such that w 2 µ j = max { w 2 q : q ∈ T 1 \ { µ 1 , . . . , µ j − 1 }} . Let α ∈ E . F or every i ∈ [ m ], let β i = β i ( α ) b e the num b er of copies of i in the multiset { µ 1 , . . . , µ r } . Let E ′ := { β ( α ) : α ∈ E } . The 2 r ter ms in Q t ∈ T 1 w t = Q m i =1 w α i i can b e split into r pairs such that ea c h pa ir co n tains exactly one element with its index in the multiset { µ 1 , . . . , µ r } and, fur ther more, in each pair , the element with its index in the multiset has at lea st as high an absolute v alue a s the other element. Therefore the following holds. m Y i =1 w α i i ≤ m Y i =1 w 2 β i ( α ) i . (4) F or a n m -tuple β ∈ E ′ , let N ( β ) be the n umber of m -tuples α ∈ E suc h that β = β ( α ) . W e will now give an uppe r b ound o n N ( β ), by showing how to construct all p ossible α with β ( α ) = β . L e t M = { µ 1 , . . . , µ r } b e the multiset containing β i copies of i . W e fir st par tition M in to any num be r of no n- empt y subsets. This can b e done in a t most r ! w ays, since we can place µ 1 in the “first” subset, µ 2 in the s ame s ubs e t or in the “ second” s ubs et, etc. E ac h of the subsets will b e a subset o f a minimal even multiset. Thus, while any mult is et, M ′ i , is not a minimally even subse t, there is an x j of o dd tota l degree in Q t ∈ M ′ i f t ( x ). Thu s , to co nstruct a minimally even subset from M ′ i , we hav e to add to M ′ i an 5 element q such that f q ( x ) contains x j , which r e stricts q to at most ρ choices. Contin uing in this manner, obser v e that we hav e at most ρ choices for the r extra elements we need to add. As the very last element we add has to b e unique we note tha t we construct at mo st r ! ρ r − 1 partitions of T 1 int o minimally even subse ts in this wa y . F or ea c h such pa rtition, we hav e α = ( α 1 , . . . , α m ), where α i is the num be r of o ccurrences of i in T 1 . Note that every α for which β ( α ) = β can b e constructed this wa y , which implies that N ( β ) ≤ ρ r − 1 r ! . (5) Let α ∈ E and β ( α ) = ( β 1 , . . . , β m ). By the construction of β ( α ) , each non-zero β i app ears in the mu ltis e t { β 1 , . . . , β m } at least as many times as in { α 1 , . . . , α m } . This implies that  2 r α  /  r β ( α )  ≤ (2 r )! /r ! . (6) By Parsev al’s Identit y , E [ f ( x ) 2 ] r = m X i =1 w 2 i ! r = X  r b 1 . . . b m  w 2 b 1 1 · · · w 2 b m m , (7) where the last sum is taken over a ll partitions b 1 + · · · + b m = r o f r into m non-negatives integral summands. Now by (3), (4), (5), (6 ) and (7), we have E [ f ( x ) 2 r ] = P α ∈E  2 r α  Q m i =1 w α i i ≤ P α ∈E  2 r α  (  r β ( α )  /  r β ( α )  ) Q m i =1 w 2 β i ( α ) i ≤ P β ∈E ′ N ( β )((2 r )! / r !)  r β  Q m i =1 w 2 β i i ≤ (2 r )! ρ r − 1 P β ∈E ′  r β  Q m i =1 w 2 β i i ≤ (2 r )! ρ r − 1 E [ f ( x ) 2 ] r . W e can g et a b etter b ound o n N ( β ) in the pro of of this theorem a s follows. Note tha t the n umber o f partitions of a set of c ardinalit y r in to non-empty subsets is called the r th Be ll num b er, B r , and there is a n upp er b ound on B r : B r <  0 . 792 r ln( r +1)  r [4]. This upp er b ound is b etter than the crude one, B r ≤ r !, that we used in the pro of of this theo rem, but o ur b ound a llo wed us to obtain a simple expression for g ( r ). Moreov er, w e b elieve that the following, muc h stronger , inequality holds . Conjecture 1. Ther e exists a c onstant c such that for every pseudo-Bo ole an function f ( x ) of wi dth ρ ≥ 1 we have || f || 2 r ≤ c √ rρ || f || 2 for e ach p ositive inte ger r . 6 If Conjecture 1 holds then it would b e b est p ossible, in a se nse, due to the following e x ample. Let f ( x ) = P n i =1 x i . By Parsev al’s Indentit y , E [ f ( x ) 2 ] = n. W e will now give a b ound for E [ f ( x ) 2 r ]. Define ( a 1 , a 2 , . . . , a 2 r ) to b e a go o d vector if a ll a i belo ng to [ n ] = { 1 , 2 , . . . , n } a nd any num b er from [ n ] app ears in the vector zero times or exactly t wice . The num b er of go od vectors is eq ua l to  n r  (2 r !) 2 r , which implies that E [ f ( x ) 2 r ] ≥  n r  (2 r !) 2 r = n ! ( n − r )! × (2 r )! 2 r r ! . Note that (2 r )! 2 r r ! = (2 r − 1)!! > ( r/ e ) r and, when n tends to infinity , n ! ( n − r )! tends to n r = E [ f ( x ) 2 ] r . Ther e f o re, the b ound in Co njectur e 1 (for ρ = 1 ) cannot b e less than c √ r for so me cons tan t c . Theorem 2 can b e easily extended as follows. Corollary 1. L et f ( x ) b e a pseudo-Bo ole an fun ctio n of width ρ ≥ 1 . Th en for e ach q > p ≥ 2 we have || f || q ≤ ((2 r )! ρ r − 1 ) 1 / (2 r ) || f || p , wher e r = ⌈ q / 2 ⌉ . Pr o of. Let r = ⌈ q / 2 ⌉ . Using Theorem 2 and the fact that || f || s ≥ || f || t for ea c h s > t > 1, we o btain || f || q ≤ || f || 2 r ≤ ((2 r )! ρ r − 1 ) 1 / (2 r ) || f || 2 ≤ ((2 r )! ρ r − 1 ) 1 / (2 r ) || f || p . 4 Application of Theorem 1 Consider the following pro blem MaxL in-AA first studied in the literature on approximation alg orithms, cf. [13, 14]. H ˚ astad [13] succinctly s umma r ized the impo rtance of the maximizatio n version o f this problem by saying that it is “in many resp ects as basic as s atisfiabilit y .” W e are g iv en a nonnegative integer k and a system S of equations Q i ∈ I j x i = b j , wher e x i , b j ∈ { − 1 , 1 } , j = 1 , . . . , m and wher e each equation is assig ned a p ositiv e integral weight w j . The question is whether there is a n assig nment of v alues { − 1 , 1 } to the v ar iables x i such that the total weigh t o f s atisfied equations is at least W / 2 + k , where W is the to tal weigh t of all e quations. If we as s ign v alues ra ndomly , the ex pected w eig h t of satisfied equatio ns is W/ 2 and, th us , W/ 2 is a lower b ound on the total weigh t of satisfied equations. Hereafter, we a ssume that no tw o equatio ns o f S hav e the same left-hand side . Maha jan et al. [16] asked whether MaxLin-AA is fixed-par a meter tractable with r e spect to the parameter k , i.e., whether 2 there exists a function h ( k ) in k only a nd a p olynomial time alg o rithm that transforms S into a new system S ′ with m ′ equations and n ′ v a riables, and parameter k ′ such that n ′ m ′ + k ′ ≤ h ( k ) and we can satisfy eq uations of S of total weight at least W/ 2 + k if and only if we can sa tisfy equa tions of S ′ of total weigh t at least W ′ / 2 + k ′ . Her e W ′ is the total weigh t o f all equa tions in S ′ . This question was answered in affirmative in a s eries of tw o pap ers [6, 5], where an exp onen tial function h ( k ) was obtained. 2 This is a definition equiv alent to the one usually used in the area of parameterized algo- rithms and complexity . F or more i nformation on the area, see, e.g., [7, 8]. 7 The author s of [5] a sk ed whether the r esult ca n b e streng then to h ( k ) b eing a po lynomial (this is a natural question in the ar ea of parameter ized algor ithms and complexity due to a pplications in prepr ocessing). It was proved in [1 2] that h ( k ) = O ( k 4 ) when (i) every eq ua tion has an o dd nu mber of v ariables , or (ii) no equation has mor e than r v ar iables, where r is a co nstan t, o r (iii) no v a riable a ppears in mo re than ρ equa tions, where ρ is a constant. Cases (ii) and (iii) can b e extended to r and ρ b eing functions of n and m , resp ectiv ely . Below we consider (iii) in some detail. Case (ii) ca n b e treated in a similar wa y using (2). Note that the answer to MaxLin-AA is Y es if a nd only if the maximum of p olynomial Q = P m j =1 c j Q i ∈ I j x i is at least 2 k , where c j = w j b j and each x i ∈ {− 1 , 1 } . Ass ign − 1 or 1 to each v ariable x i independently and uniformly at random. Then Q is a r andom v ar iable. W e will use the following lemma of Alon et al. [1]: Let X b e a r eal random v ar iable and supp ose tha t its first, second and fourth mo men ts sat- isfy E [ X ] = 0 a nd E [ X 4 ] ≤ b E [ X 2 ] 2 , where b is a p ositiv e constant. Then P ( X ≥ 1 2 p E [ X 2 ] /b ) > 0 . Observe that E [ Q ] = 0 and E [ Q 2 ] = P m j =1 c 2 j . By Theorem 1, E [ Q 4 ] ≤ (2 ρ + 1 ) E [ Q 2 ] 2 and, th us , P ( Q ≥ 1 2 q P m j =1 c 2 j / (2 ρ + 1 )) > 0 . Since P m j =1 c 2 j ≥ m we have P ( Q ≥ 1 2 p m/ (2 ρ + 1)) > 0 . Thus, if 1 2 p m/ (2 ρ + 1 ) ≥ 2 k the answer to MaxLin-AA is Yes . Otherwise, m ≤ 8(2 ρ + 1) k 2 and so m is b ounded by a po lynomial in k if ρ ≤ m α for some constant α < 1 . It is shown in [12] that we may a ssume that n ≤ m a s o therwise we can replace S by an eq uiv alent sys tem for which n ≤ m holds. This implies that nm is b ounded by a p olynomial in k if ρ ≤ m α for some constant α < 1 . Now assume that ρ ≤ m α for some co ns tan t α < 1 . T o co ns truct the r e quired system S ′ , chec k whether 1 2 p m/ ( ρ + 1) ≥ 2 k . If the a nsw er is Y es , let S ′ be an arbitrar y consistent sys tem of 2 k equations with all weigh ts equal 1 and, otherwise, S ′ = S . The parameter k ′ = k . Note that the bo und ρ ≤ m α is only possible becaus e the coefficient in Theorem 1 is so s mall. 5 F urther Researc h It would b e in ter esting to verify Conjecture 1 and decrease the co efficien t b efore || f || p in Corolla ry 1. Ac kno wl edgmen ts This research w as partially supp orted by an In ter na- tional Joint gr an t of Roy a l So ciet y . P art o f the pa per was wr itt e n when the first author was a tt e nding Discrete Analysis prog ramme of the Isa ac Newton Institute for Mathematical Sciences, Cambridge. Financial suppo rt of the In- stitute is greatly appreciated. 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