Explicit SoS lower bounds from high-dimensional expanders
aa r X i v : . [ c s . CC ] S e p Explicit SoS lower bounds from high-dimensionalexpanders
Irit Dinur ∗ Yuval Filmus † Prahladh Harsha ‡ Madhur Tulsiani § Abstract
We construct an explicit family of 3XOR instances which is hard for O ( p log n ) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which in-volve a random component, our systems can be constructed explicitly in deterministicpolynomial time.Our construction is based on the high-dimensional expanders devised by Lubotzky,Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our anal-ysis is based on two notions of expansion for these complexes: cosystolic expansion,and a local isoperimetric inequality due to Gromov.Our construction offers an interesting contrast to the recent work of Alev, Jeron-imo and the last author (FOCS 2019). They showed that 3XOR instances in which thevariables correspond to vertices in a high-dimensional expander are easy to solve. Incontrast, in our instances the variables correspond to the edges of the complex. We describe a new family of instances of 3XOR, based on high-dimensional expanders,that are hard for the Sum-of-Squares (SoS) hierarchy of semidefinite programming relax-ations, which is the most powerful algorithmic framework known for optimizing overconstraint satisfaction problems. Unlike previous constructions of 3XOR hard instancesfor SoS, our construction is explicit, as it is based on the explicit construction of high-dimensional expanders due to Lubotzky, Samuels and Vishne [LSV05a, LSV05b], whichwe refer to henceforth as LSV complexes.
Theorem 1.1.
There exists a constant µ ∈ (
0, 1 ) and an infinite family of 3XOR instances on nvariables, constructible in deterministic polynomial time, satisfying the following:- No assignment satisfies more than − µ fraction of the constraints.- Relaxations obtained by O ( p log n ) levels of the SoS hierarchy fail to refute the instances. ∗ Weizmann Institute of Science, ISRAEL. email: [email protected] . Research supported byERC-CoG grant number 772839. † Technion Israel Institute of Technology, ISRAEL. email: [email protected] . This project hasreceived funding from the European Union’s Horizon 2020 research and innovation programme under grantagreement No 802020-ERC-HARMONIC. ‡ Tata Institute of Fundamental Research, INDIA. email: [email protected] . Research supported bythe Department of Atomic Energy, Government of India, under project no. RTI4001 and in part by the Swar-najayanti fellowship. § Toyota Technological Institute at Chicago. email: [email protected] . Research supported by NSF grantCCF-1816372.
1e also remark that our construction can be used to obtain explicit integrality gapinstances for various other optimization problems, using reductions in the SoS hierar-chy [Tul09]. In particular, while our instances on the LSV complexes exhibit an integralitygap of 1 − µ vs. 1 for the SoS hierarchy, reductions can be used to obtain explicit 3XORinstances with a gap of 1/2 + ε vs. 1 − ε for any ε >
0. Indeed, this yields explicit hardinstances with optimal gaps for all approximation resistant predicates based on pairwiseindependent subgroups [Cha16].
Structured instances from High-dimensional expanders.
High-dimensional expanders(HDXs) are a high-dimensional analog of expander graphs. In recent years they havefound a variety of applications in theoretical computer science, such as efficient CSP op-timization [AJT19], improved sampling algorithms [ALGV19, ALG20, AL20], quantumLDPC codes [EKZ20, KT20], novel lattice constructions [KM18], direct sum testing [GK19],and others. Explicit constructions of HDXs have also led to improved list-decoding algo-rithms [DHKNT19, AJQST20] and to sparser agreement tests [DK17, DD19]. In this work,we show how these explicit constructions can be used to construct explicit hard instancesfor SoS.High-dimensional expanders are bounded-degree (hyper)graphs (or rather, simplicialcomplexes) with certain expansion properties. A simplicial complex is a non-empty collec-tion of down-closed sets. Given a simplicial complex X , we will refer by X ( i ) the family ofall i -dimensional sets in X (i.e., sets of size i + X is the maximal dimension of any set in it. It will be convenient to refer to the sets ofdimension 0, 1, 2, 3 as vertices, edges, triangles, tetrahedra, respectively. Thus, a graph G = ( V , E ) is a 1-dimensional complex, while in this work we will be using complexes ofdimension at least 2. Given a 2-dimensional complex X = ( X ( ) , X ( ) , X ( )) , there aretwo natural ways to construct a 3XOR instance based on X — a vertex-variable construc-tion and an edge-variable construction. Let β : X ( ) → F be any F -valued function onthe set X ( ) of triangles. Vertex-variable construction:
The 3XOR instance corresponding to ( X , β ) consists of thefollowing constraints: x u + x v + x w = β { u , v , w } for each { u , v , w } ∈ X ( ) . Edge-variable construction:
The 3XOR instance corresponding to ( X , β ) consists of thefollowing constraints: x { u , v } + x { v , w } + x { w , u } = β { u , v , w } for each { u , v , w } ∈ X ( ) .The vertex-variable construction whose underlying structure is a high-dimensional ex-pander has been studied by Alev, Jeronimo and the last author [AJT19]. They gave anefficient algorithm for approximating vertex-variable constraint satisfaction problems (notnecessarily 3XOR) on an underlying high-dimensional expander. Their result is a gener-alization to higher dimensions of the corresponding result for graphs that “CSPs are easyon expanders” [BRS11, GS11]. They prove this by showing that certain types of randomwalks on vertices converge very fast on high-dimensional expanders. However, the sameanalysis fails to show a similar result for the edge-variable construction, as the correspond-ing random walk on edges of a high-dimensional expander does not mix. Our work showsthat this difference isn’t just a technical limitation of their analysis; it is inherent. The edge-variable variant is truly hard, at least for SoS. This demonstrates an interesting subtlety inthe structure of high-dimensional expanders, and how it relates to optimization.To understand our edge-variable construction better, it will be convenient to set upsome notation. Let C i denote the set of all F -valued functions on X ( i ) . For each 0 ≤ i < d ,2onsider the operator δ i : C i → C i + defined as follows: δ i f ( s ) : = ∑ u ∈ s f ( s − { u } ) .This is usually referred to as the coboundary operator. Let B i be the image of δ i − , and let Z i be the kernel of δ i . Clearly, B i , Z i ⊆ C i . Furthermore, it is not hard to see that B i ⊆ Z i .It easily follows from the definitions that the edge-variable construction corresponding to ( X , β ) is a satisfiable instance iff β ∈ B .Typically, soundness of SoS-hard instances is proved by choosing β at random from C .In contrast, we construct our explicit instances by choosing the function β more carefully,and relying on a certain type of expansion property of the complex. Recall that B ⊂ Z ,and the instance is satisfiable iff β ∈ B . Complexes for which B = Z are said to have triv-ial second cohomology. We will be working with complexes with non-trivial second coho-mology, i.e., B = Z . This lets us choose a β ∈ Z \ B to prove soundness. It is known thatthe explicit constructions of HDXs due to Lubotzky, Samuels and Vishne [LSV05a, LSV05b]have non-trivial second cohomology. In fact, these complexes have the stronger property(due to a theorem of Evra and Kaufman [EK16]) that all β ∈ Z \ B are not only not in B , but in fact far from any function in B . This latter property follows from the cosystolicexpansion of the complex, and forms the basis for the soundness of our instances.How do we prove the completeness of our instance, namely, that SoS fails to detect thatit is a negative instance? The LSV construction is a quotient of the so-called affine buildingwhich is, from a topological point of view, a simple “Euclidean-like” object with trivialcohomologies. The hardness of our instance comes from the inherent difference betweenthe LSV complex and the building, which cannot be seen through local balls whose radiusis at most the injectivity radius of the complex, in our case Θ ( log n ) . Locally, the LSVquotient is isomorphic to the building. However, unlike the building, the LSV complexis a quotient with non-trivial cohomologies. The hardness comes from the fact that localviews cannot capture the cohomology, which is a global property. Given this observation,the proof of completeness can be carried out following the argument of Ben-Sasson andWigderson [BW01] that any short resolution proof is narrow, and Grigoriev [Gri01] andSchoenebeck [Sch08]’s transformation from resolution lower bounds to SoS lower bounds.Technically, we rely on two very different types of expansion or isoperimetry. In ourproof of completeness, we rely on an isoperimetric inequality called Gromov’s filling in-equality, that says that balls are essentially the objects with smallest boundary in anyCAT(0) space (a class of spaces that includes both Euclidean spaces and the affine build-ing). In our proof of soundness, we rely on the cosystolic expansion of the LSV complex,as proven by Evra and Kaufman [EK16], which implies that any non-trivial element in thecohomology has constant weight. Both of these statements are related to expansion, yetthey are distinct from other notions of expansion used in previous SoS lower bounds. Relation to previous SoS gap constructions.
All previous constructions of hard instancesfor SoS can be viewed in the vertex/edge-variable framework (typically vertex-variable).To the best of our knowledge, all known hard instances, proving inapproximability in theSoS hierarchy, are random instances; either both the complex X and the function β are ran-dom, or just the function β is random. Explicit hard instances for SoS are known in proof More accurately, their construction depends on the group defining the quotient. They show that a certainchoice of groups yields non-trivial second cohomology. β and using a union-bound argument to show thatwith high probability, every solution violates nearly half of the constraints. In contrast, forus, a random β is not a good choice because the local structure will quickly detect localcontradictions, ruining the completeness altogether. Open directions.
Our construction of explicit hard SoS instances based on HDXs begsseveral questions, some of which we discuss below.
Improved soundness
Our construction yields 3XOR hard instances which are at most ( − µ ) -satisfiable, owing to the cosystolic expansion of the underlying HDX (moreprecisely, CoSys ( X ) ≥ µ , see Section 2.2 for the definition of CoSys ). Coupled withreductions in the SoS hierarchy [Tul09], this yields 3XOR hard instances which are atmost ( + ε ) -satisfiable for every ε ∈ (
0, 1 ) . Can we obtain such a result directlyfrom the HDX construction (bypassing reductions), say by constructing HDXs whichsatisfy CoSys ( X ) ≥ − ε ? In addition to maintaining the HDX structure, bypass-ing reductions would also allow for perfect completeness, which is lost while usingNP-hardness reductions. Fooling more levels of the SoS hierarchy
Our hard instances fool only O ( p log n ) levelsof the SoS hierarchy, as our argument is based on the injectivity radius of the com-plexes, which is O ( log n ) , and we suffer a further square-root loss due to the use ofGromov’s isoperimetry inequality. It is possible that a much stronger lower boundholds for these instances. Can one construct explicit hard instances that fool linearlymany levels of the SoS hierarchy? HDX dimension and CSP definition
We find the contrast between the vertex-variable andedge-variable constructions baffling: while the vertex-variable construction is easy,our result demonstrates the hardness of the edge-variable construction. As we go tohigher dimensions of HDX, there are more ways to define CSPs. Which of these areeasy and which are hard?
The sum-of-squares hierarchy provides a hierarchy of semidefinite programming (SDP)relaxations, for various combinatorial optimization problems. Figure 1 describes the re-laxation given by t levels of the hierarchy for an instance I of 3XOR in n variables, with For more on Sum-of-Squares, see the recent monograph by Fleming, Kothari and Pitassi [FKP19]. constraints of the form x i + x i + x i = β i i i over F . We also use I to denote the setof all tuples { i , i , i } present as constraints. A solution to the relaxation is specified bya collection of unit vectors { u S } S ⊆ [ n ] , | S |≤ t , satisfying the constraints in the program. Theobjective equals the fraction of constraints “satisfied” by the SDP solution. maximize 12 + m · ∑ { i , i , i }∈I ( − ) β i i i · D u { i , i , i } , u ∅ E subject to (cid:10) u S , u S (cid:11) = (cid:10) u S , u S (cid:11) ∀ S ∆ S = S ∆ S , | S | , . . . , | S | ≤ t k u S k = ∀ S , | S | ≤ t Figure 1: Relaxation for 3XOR given by t levels of the SoS hierarchyTo prove a lower bound on the value of the SDP relaxation, we will use the followingresult, which shows the existence of vectors u S yielding an objective value of 1, when thegiven system of XOR constraints does not have any “low-width” refutations. Formally,we consider a system called ⊕ -resolution, where the only rule allows us to combine twoequations ℓ = b and ℓ = b to derive the equation ℓ + ℓ = b + b . A refutation is aderivation of 0 =
1. The width of a refutation is the maximum number of variables in anyequation used in the refutation. We include a proof of the following lemma in Appendix A.
Lemma 2.1 ([Sch08, Lemma 13], [Tul09, Theorem 4.2]) . Let Λ be a system of equations in nvariables over F , which does not admit any refutations of width at most t. Then there exist vectors { u S } S ⊆ [ n ] , | S |≤ t satisfying the constraints in Figure 1, such that for all equations ∑ i ∈ T x i = b T in Λ with | T | ≤ t, we have h u T , u ∅ i = ( − ) b T . A simplicial complex X is a non-empty collection of sets (known as faces) which is closeddownwards. The i -dimensional faces X ( i ) are all sets of size i +
1. The dimension ofthe complex is the maximal dimension of a face. Faces of that dimension are known asfacets. Faces of dimensions 0, 1, 2, 3 are called vertices, edges, triangles, and tetrahedra,respectively.Graphs are 1-dimensional simplicial complexes. The skeleton of a simplicial complexis the graph obtained by retaining only faces of dimension at most 1.
Links
Let X be a d -dimensional simplicial complex. The link X s of a face s ∈ X ( i ) is asimplicial complex of dimension d − ( i + ) given by X s ( j ) : = { t : s ∪ t ∈ X ( j + i + ) } . Inother words, X s contains all faces in X which contain s , with s itself removed. Balls
Let X be a simplicial complex. A ball of radius r around a vertex v is the subcom-plex induced by all vertices at distance at most r from v , as measured on the skeleton of X .That is, the subcomplex contains a face of X if it contains all the vertices of the face. Simplicial map If X and Y are two simplicial complexes, then a simplicial map ψ : Y → X is a map from Y ( ) to X ( ) that maps faces to faces.5 hains Fix a d -dimensional simplicial complex X . Let C i = C i ( X , F ) be the set of allfunctions from X ( i ) to F . Elements of C i are also known as i -chains.For an i -chain f , we define | f | to be the number of non-zero elements in f . For two i -chains f , g , we define the distance between f and g to be dist ( f , g ) = | f + g | . Inner product
For f , f ′ ∈ C i , let us denote by h f , f ′ i i the following sum modulo 2: (cid:10) f , f ′ (cid:11) i : = ∑ s ∈ X ( i ) f ( s ) f ′ ( s ) .This is not an inner product in the usual sense as we are working over a field of non-zero characteristic, but it is convenient notation. We will usually drop the subscript i . Dual space
Given any subspace V ⊂ C i , the dual of V (under h· , ·i i ) is defined as: V ⊥ : = { f ∈ C i | for all g ∈ V , h f , g i i = } . Boundaries, Cycles, Homology
The boundary operator ∂ i : C i → C i − is given by ∂ i f ( s ) : = ∑ t ∈ X ( i ) : t ⊃ s f ( t ) .It gives rise to boundaries B i and cycles Z i : B i : = im ∂ i + , Z i : = ker ∂ i .In the case of graphs, Z consists of all sums of cycles (in the usual sense).The coboundary operator δ i : C i → C i + , which is the adjoint of the boundary operator,is given by δ i f ( t ) : = ∑ s ∈ X ( i ) : s ⊂ t f ( s ) = ∑ u ∈ t f ( t − { u } ) .It gives rise to coboundaries and cocycles: B i : = im δ i − , Z i : = ker δ i .We will usually drop the subscript i when invoking ∂ , δ .It is easy to see that B i ⊂ Z i (every boundary is a cycle) and B i ⊂ Z i (every coboundaryis a cocycle). For example, in a 2-dimensional complex, the boundary of every triangle isa cycle. We call such cycles trivial cycles . Modding out by trivial cycles and cocycles, weobtain the homology and cohomology spaces H i : = Z i / B i , H i : = Z i / B i .The dimensions of these spaces (which are identical) measure the number of “holes” in aparticular dimension. Nice complexes (such as the buildings considered below) have noholes.The following claim shows that that the coboundary operator is the adjoint of theboundary operator. Claim 2.2.
Let f ∈ C i , g ∈ C i − . Then h f , δ g i i = h ∂ f , g i i − . roof. h f , δ i − g i i = ∑ t ∈ X ( i ) f ( t ) · δ i − g ( t ) = ∑ t ∈ X ( i ) f ( t ) · ∑ s ∈ X ( i − ) : s ⊂ t g ( s ) ! = ∑ s ∈ X ( i − ) ∑ t ∈ X ( i ) : t ⊃ s f ( t ) ! · g ( s ) = h ∂ i f , g i i − .The following claims shows the dimensions of homology and cohomology spaces areidentical. Claim 2.3. Z i = ( B i ) ⊥ , Z i = ( B i ) ⊥ .Proof. Z i = ker ∂ i = ker δ ∗ i − = ( im δ i − ) ⊥ = ( B i ) ⊥ . Claim 2.4. dim H i = dim H i Proof. dim H i = dim Z i − dim B i = dim C i − dim B i − dim B i [ By Claim 2.3 ]= dim Z i − dim B i [ By Claim 2.3 ]= dim H i . Cosystoles
We define, following Evra and Kaufman [EK16, Definition 2.14], the i -cosystoleof a complex X to be the minimal (fractional) size of f ∈ Z i \ B i ,CoSys i ( X ) : = min f ∈ Z i \ B i | f | / | X ( i ) | . B ( d + ) The infinite k -regular tree is the unique connected k -regular graph without cycles. Affinebuildings are higher-dimensional analogs of the infinite k -regular tree. For d =
1, theone-dimensional affine building B ( ) is the k -regular tree. For higher dimensions they are regular in the sense that all vertex links are bounded and identical in structure, they areconnected and contractible, and so have vanishing cohomologies, that is, the cohomologyspaces H , . . . , H d − are trivial, where d is the dimension.We won’t describe B ( d + ) any further; the interested reader can check [Ji12, AB08]. Acrucial property of B ( d + ) which we will need in the sequel is its being a CAT(0) space, which is a geometric definition capturing non-positive curvature; see [BH99] for moreinformation. The property of being CAT(0) has the following implication, due to Gro-mov [Gro83, Gut06, Wen08]: A complex is contractible, roughly speaking, if it can be continuously deformed to a point (techni-cally, it is homotopy-equivalent to a point). Since (co)homologies are preserved by such deformations, all(co)homologies of a contractible complex vanish. A space is CAT(0) if for every triangle x , y , z , the distance between x and the midpoint of y , z is at most thecorresponding distance in a congruent triangle in Euclidean space. heorem 2.5 (Gromov’s filling inequality for CAT(0) spaces) . For every cycle f ∈ Z there isa filling g ∈ C such that f = ∂ g and | g | = O ( | f | ) . Gromov’s filling inequality is an isoperimetric inequality. It generalizes the classicisoperimetric inequality in the plane, which states that any simple closed curve of length L encloses a region whose area is at most L /4 π .The isoperimetric inequality in the plane can be stated in an equivalent way: theboundary of any bounded region of area A is a curve whose length is at least √ π A . Thisinequality fails for unbounded regions, which could have infinite area but finite boundary(for example, consider the complement of a circle). In the same way, Gromov’s inequalitydoesn’t imply that each g ∈ C satisfies | ∂ g | = Ω ( p | g | ) . Rather, we have to replace | g | with min h ∈ Z | g + h | .Gromov’s filling inequality also applies to i -chains, with an exponent of i +
1, but wewill only need the case i = Lemma 2.6.
Balls in B ( d + ) have vanishing homologies and satisfy Gromov’s filling inequality. Whereas the affine building is an infinite simplicial complex, Lubotzky, Samuels and Vishneconstructed a growing family of finite complexes that are obtained from quotients of theaffine building. These quotients have a growing number of vertices, and locally, in a ballaround each vertex, the complex is isomorphic to the affine building. Moreover, they gavea very explicit algorithm for constructing these complexes by first constructing a Cayleygraph with an explicit set of generators, and then the higher dimensional faces are simplythe cliques in the Cayley graph.
Theorem 2.7 (Lubotzky, Samuels, Vishne [LSV05a, Theorem 1.1]) . Let q be a prime power,d ≥ . For every e > the group G = PGL d ( F p e ) has an (explicit) set of [ d ] q + [ d ] q + . . . + [ dd − ] q generators, such that the Cayley complex of G with respect to these generators is a Ramanujancomplex X covered by B ( d ) ( F ) for F = F q (( y )) . The precise definition of “Ramanujan complex” is not important for this context. Forus, there are three important aspects of this theorem: efficient construction, local structure,and global structure.-
Efficient construction:
Firstly, the fact that the complex is constructible in polyno-mial time.-
Local structure:
Next, we highlight the fact that locally the complex looks like thebuilding. The theorem states that the complex X is covered by B ( d ) . A coveringmap maps a simplicial complex Y surjectively to a simplicial complex X by mappingthe vertices ψ : Y ( ) → X ( ) such that for every k ≤ d the image of every k -face { v , . . . , v k } ∈ Y ( k ) is a k -face { ψ ( v ) , . . . , ψ ( v k ) } ∈ X ( k ) .The fact that X is covered by B ( d ) means that the neighborhood of a vertex in X andin B ( d ) look exactly the same. It turns out that for the LSV complexes this continues to8e true also for balls of larger radius around any vertex. This is a higher-dimensionalanalog of the graph property of containing no short cycles (locally looking like atree). Define the injectivity radius of X to be the largest r such that the covering map B ( d ) ψ → X is injective from balls of radius ≤ r in B ( d ) and the ball of radius ≤ r in X .We do not mention the center of the ball they are all isomorphic. Theorem 2.8 (Lubotzky and Meshulam [LM07], see also [EGL15, Corollary 5.2]) . LetX be the LSV complex above. Then the injectivity radius r ( X ) of X satisfiesr ( X ) ≥ log q | X | ( d − )( d − ) − where | X | is the number of vertices in X. - Global structure:
Finally, we look at the second cohomology group of the LSV com-plexes. Kaufman, Kazhdan and Lubotzky [KKL16] showed that the groups defin-ing the LSV quotient complexes can be chosen so that the second homology is non-empty.
Proposition 2.9 (Kaufman, Kazhdan, Lubotzky [KKL16, Proposition 3.6]) . There is aninfinite and explicit sequence of LSV complexes with a non-vanishing second cohomology.
We remark that Kaufman, Kazhdan and Lubotzky [KKL16] proved that these com-plexes exist. To show that they are also efficiently constructible, we look into theirproof to recall the construction: start with any LSV complex X viewed as a Cayleygraph of a group G . Find some element of order 2 in G (such an element alwaysexists), and then quotient X by this element, thus obtaining a complex Y that is itselfis a Ramanujan complex because it is a quotient of one. Y is clearly efficiently con-structible from X , and has half as any vertices. This construction shows (see [KKL16,Proposition 3.5]) that H ( Y ) =
0. Furthermore, the proof of [KKL16, Proposition 3.6]shows that because G has property T one can deduce also that H ( Y ) = B ( d ) (and even amore general class of complexes) are so-called “cosystolic expanders” which in par-ticular implies the following. Theorem 2.10 (Evra and Kaufman [EK16, Part of Theorem 1.9]) . Let { X n } be a familyof LSV complexes. There exists some constant µ > that depends only on q and d but noton the size n of the complex, such that every f ∈ Z ( X ) \ B ( X ) must have weight at least µ · | X ( ) | . The infinite sequence of complexes we will be working with are the LSV complexes de-scribed in Section 2.4 above. The properties we care about are (1) that they are efficiently The theorem was proven by [LM07]. They stated their theorem using a slightly different definition forinjectivity radius but one can prove that the two definitions coincide in this case. This was reproven in [EGL15]who use the definition of injectivity radius that is convenient for us. h ∈ C corresponds to a set of triangles. For the following statements,we consider two triangles to be connected if they share an edge. This can be used to defineconnected components. Note that if h can be split into connected components h , . . . , h s ,then the components correspond to disjoint sets of triangles. Moreover, no triangle in h i shares an edge with a triangle in h j when i = j , which also implies that the boundaries ∂ h i and ∂ h j correspond to disjoint sets of edges.We prove the following claims by mapping small connected sets in X ( ) to correspond-ing sets in the infinite building B . The first proposition shows that there can be no smallnon-trivial cancellations (i.e., not coming from tetrahedra). Proposition 3.1.
Let h ∈ C be a connected set of triangles such that | h | < r and ∂ h = .Then h ∈ B .Proof. Since | h | < r , there is a ball N of radius r that contains the support of h . Byassumption, the covering map ψ : B → X has injectivity radius of at least r . This meansthat there is a radius- r ball ˆ N = ψ − ( N ) in B that is isomorphically mapped by ψ to N .Look at ˆ h = ψ − ( h ) ∈ C ( ˆ N ) , the chain isomorphic to h in the building. Clearly ∂ ˆ h = ψ − ( ∂ h ) =
0, and since balls in the building have zero homologies by Lemma 2.6, wededuce that ˆ h itself must be a boundary, i.e. there must be some ˆ g ∈ C ( ˆ N ) such that ∂ ˆ g = ˆ h . Moving back to X , we see that g : = ψ ( ˆ g ) ∈ C ( X ) necessarily satisfies ∂ g = h , and so h ∈ B .This proposition states that locally (i.e., within the injective radius r ), Z looks like B .We thus have a complex whose cohomology group is non-trivial, yet locally, the homology group “looks” trivial. Note that this is a twist on what we had claimed in the introduc-tion, a complex whose cohomology group is non-trivial, yet locally, the cohomology group“looks” trivial. However, these are identical statements owing to Claim 2.4.The next proposition shows that Gromov’s filling inequality in the infinite building B can be used to yield a similar consequence for small sets in the finite complex X . Proposition 3.2.
Let h ∈ C be a connected set of triangles such that | h | < r and | h | ≤ | h + h | for all h ∈ B . Then, | ∂ h | ≥ c · | h | , where c > is an absolute constant.Proof. As before, the support of h is contained in a ball N of X which is isomorphic under ψ to a ball ˆ N in B . Let ˆ h = ψ − ( h ) ∈ C ( B ) , and let ˆ f = ∂ ˆ h . We now apply the fillingtheorem of Gromov, which holds in ˆ N due to Lemma 2.6, to deduce that there is some ˆ h that fills ˆ f , namely ∂ ˆ h = ˆ f , and whose size is at most | ˆ h | = O ( | ˆ f | ) .Now ∂ ( ˆ h − ˆ h ) = ˆ f − ˆ f =
0. Since the ball ˆ N has zero homologies by Lemma 2.6,ˆ h − ˆ h itself must be a boundary: there must be some ˆ g ∈ C ( ˆ N ) such that ∂ ˆ g = ˆ h − ˆ h .Pushing ˆ g and ˆ h back to X , we get g = ψ ( ˆ g ) and h = ψ ( ˆ h ) , which satisfy ∂ g = h − h .10t this point we have a small h that is close via a boundary to h . Finally, observe that f = ∂ h satisfies f = ψ − ( ˆ f ) . So | f | = | ˆ f | ≥ c · | ˆ h | = c · | h | ≥ c · | h | ,where the last inequality used that | h | ≤ | h + ( h − h ) | , since h − h = ∂ g ∈ B . Ω ( p log n ) levels of SoS hierarchy Let X be a d -dimensional LSV complex, with | X ( ) | = n and non-trivial second cohomol-ogy group, as per Proposition 2.9. Below, we construct an instance of 3XOR in n variablesusing this complex, and prove a lower bound on the integrality gap of the relaxation ob-tained by Ω ( p log n ) levels of the SoS hierarchy. Construction.
We construct a system of equations on X by putting a variable x { a , b } foreach edge { a , b } ∈ X ( ) of the complex, and an equation x { a , b } + x { b , c } + x { c , a } = β { a , b , c } for each triangle { a , b , c } ∈ X ( ) , where β is an arbitrary element of Z \ B .Recall that X can be constructed efficiently. Given X , we can find a vector β ∈ Z \ B using elementary linear algebra. Therefore the entire system can be constructed efficiently. Soundness.
Soundness of this system follows easily from the fact that the cosystole islarge.
Claim 3.3 (Soundness) . Every assignment to the system defined above falsifies at least µ fractionof the equations.Proof. An assignment to the variables is equivalent to an f ∈ C . Every equation satisfiedby f is a triangle in which δ f ( { a , b , c } ) = β { a , b , c } , and so the number of unsatisfied equa-tions is dist ( δ f , β ) = | δ f + β | . Since δ f ∈ B and β ∈ Z \ B , also δ f + β ∈ Z \ B , andso | δ f + β | / | X ( ) | ≥ CoSys ( X ) ≥ µ . In other words, the assignment falsifies at least a µ fraction of the equations.The main work is to prove completeness, namely to show that the system looks locallysatisfiable. Completeness.
Our main result is that this system appears satisfiable to the Sum-of-Squares hierarchy with O ( p log n ) levels. Grigoriev [Gri01] and Schoenebeck [Sch08]showed that to prove such a statement it suffices to analyze the refutation width of thesystem of equations (see Lemma 2.1). If the refutation width is at least w , then w /2 levelsof the Sum-of-Squares hierarchy cannot refute the system.A system of linear equations over F can be refuted using a proof system known as ⊕ -resolution , in which the only inference rule is: given ℓ = b and ℓ = b , deduce ℓ ⊕ ℓ = b ⊕ b ; here ℓ , ℓ are XORs of variables, and b , b are constants. A refutation has thestructure of a directed acyclic graph (DAG) where each non-leaf node has two incomingedges. A refutation is a derivation which starts with the given linear equations, placed atthe leaves of a DAG, and reaches the equation 0 = width of a11inear equation ℓ = b is the number of variables appearing in ℓ . The width of a refutationis the maximum width of an equation in any of the nodes of the DAG.In the remainder of this section, we prove the following theorem, which together withLemma 2.1 implies Theorem 1.1. Theorem 3.4.
The construction above requires width at least Ω ( √ r ) to refute in ⊕ -resolution,where r = Θ ( log n ) is the injectivity radius of the complex. The proof follows classical arguments of Ben-Sasson and Wigderson [BW01] regard-ing lower bounds on resolution width, which were also used in the proof of Schoene-beck [Sch08]. Whereas Ben-Sasson and Wigderson relied on boundary expansion, we relyon Gromov’s filling inequality (and so lose a square root).Suppose we are given a refutation for this system, and consider the correspondingDAG. Each leaf ν in the DAG is labeled by a triangle T ν ∈ X ( ) . Define h ν : = T ν ∈ C , b ν : = β T ν ∈ F .For each inner node ν in the DAG, let ν , ν be its two incoming nodes. Define inductively, h ν : = h ν + h ν ∈ C , b ν : = b ν + b ν ∈ F . Proposition 3.5.
For every node ν , b ν = h β , h ν i .Proof. This is immediate by following inductively the structure of the DAG.As in [BW01], we next define a complexity measure for each node of the DAG. Whilein [BW01] the complexity measure is based on the number of “leaf equations” used toderive the one at a given node, we will need to discount sets of triangles correspondingto tetrahedra, as these cannot lead to contradictions. Recall that B = im ∂ is the set oftriangle chains that “come from” tetrahedra chains, which we consider as the “trivial”cycles. We define a complexity measure at each node, κ ( ν ) : = dist ( h ν , B ) = min h ∈ B | h ν + h | that measures the distance of h ν from these trivial cycles. The complexity measure κ satis-fies the following sub-additivity property. Proposition 3.6. If ν is an inner node in the DAG with ν , ν its two incoming nodes, then κ ( ν ) ≤ κ ( ν ) + κ ( ν ) . Proof.
Let h , h ∈ B be such that κ ( ν ) = | h ν + h | and κ ( ν ) = | h ν + h | . Recall that h ν = h ν + h ν . Then, we have κ ( ν ) + κ ( ν ) = | h ν + h | + | h ν + h | ≥ | h ν + h ν + h + h | = | h ν + h + h | ≥ κ ( ν ) .We also need the fact that the complexity of a node with a contradiction must be non-zero. Proposition 3.7. If κ ( ν ) = then b ν = . roof. If κ ( ν ) = h ν ∈ B . Hence b ν = h β , h ν i = β ∈ Z = ( B ) ⊥ (Claim 2.3).Next, we consider the width of each node in the DAG. For a node ν , let f ν : = ∂ h ν ∈ C .Thus f ν indicates the set of variables appearing in the left-hand side of the equation onnode ν . So the width of the system is the maximum, over all nodes ν in the DAG, of | f ν | .We can now prove Theorem 3.4 using the above complexity measure, and results fromSection 3.1. Proof of Theorem 3.4.
Let ν ∗ denote the root of the DAG. By virtue of being a refutation, b ν ∗ = f ν ∗ =
0. In other words, ∂ h ν ∗ = f ν ∗ =
0, which means that h ν ∗ ∈ Z . Since b ν ∗ =
1, we also have by Proposition 3.7 that κ ( ν ∗ ) > h ∈ B be such that κ ( ν ∗ ) = | h ν ∗ + h | , and let h , . . . , h s be the disjoint connectedcomponents of h ν ∗ + h . We will first show that κ ( ν ∗ ) = | h ν ∗ + h | ≥ r . Assuming κ ( ν ∗ ) < r ,we have that | h | + · · · + | h s | = | h ν ∗ + h | < r .Also, since ∂ h + · · · + ∂ h s = ∂ ( h ν ∗ + h ) = ∂ h ν ∗ = ∂ h i = i ∈ [ s ] , since connected components have disjointboundaries. Applying Proposition 3.1 to each h i , we get that h i ∈ B for each i ∈ [ s ] .However, this implies h ν ∗ + h ∈ B and hence κ ( ν ∗ ) =
0, which is a contradiction.Using sub-additivity (Proposition 3.6), κ ( ν ∗ ) ≥ r , and the fact that the leaves of theDAG satisfy κ ( ν ) =
1, we get that there must be some internal node ν for which r /2 ≤ κ ( ν ) < r . We can find such a node by starting at the root and always going to the childwith higher complexity, until reaching a node ν such that κ ( ν ) < r . We will prove that forsuch a node, we must have | f ν | = Ω ( √ r ) .As before, let h ∈ B now be such that κ ( ν ) = | h ν + h | , and let h , . . . , h s be the disjointconnected components of h ν + h . We have that | h i | ≤ | h ν + h | < r for each i ∈ [ s ] . By theminimality of | h ν + h | , we also have that for any h ′ ∈ B and any i ∈ [ s ] , | h i | + | h ν + h − h i | = | h ν + h | ≤ | h ν + h + h ′ | ≤ | h i + h ′ | + | h ν + h − h i | .Thus, | h i | is also minimal for each i , and we can apply Proposition 3.2 to each connectedcomponent h i , to obtain | f ν | = | ∂ ( h ν + h ) | = | ∂ h | + · · · + | ∂ h s | ≥ c · | h | + · · · + c · | h s | ≥ c · ( | h | + · · · + | h s | ) = c · | h ν + h | ≥ ( c / √ ) · √ r . Acknowledgements
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A Proof of Lemma 2.1
Lemma 2.1 (Restated) ([Sch08, Lemma 13], [Tul09, Theorem 4.2])
Let Λ be a system of equa-tions in n variables over F , which does not admit any refutations of width at most t. Thenthere exist vectors { u S } S ⊆ [ n ] , | S |≤ t satisfying the constraints in Figure 1, such that for all equations ∑ i ∈ T x i = b T in Λ with | T | ≤ t, we have h u T , u ∅ i = ( − ) b T .Proof. We assume that Λ is closed under width-2 t ⊕ -resolution, replacing Λ by its closureif necessary, and also that it contains the trivial equation 0 =
0. We will now construct theunit vector u S .Define a relation ∼ on subsets of [ n ] of size at most t as follows: S ∼ T iff there existsan equation ∑ i ∈ S ∆ T x i = b in Λ for some b ∈ F . It is easy to check that the relationis reflexive and symmetric. It is also transitive since for S ∼ S , S ∼ S , we can addthe corresponding equations to obtain one of the form ∑ i ∈ S ∆ S x i = b for some b ∈ F .Since | S | , | S | ≤ t , this equation has at most 2 t variables and must be in Λ by the closureproperty. Thus, we have an equivalence relation which partitions all sets of size at most t into equivalence classes, say C , . . . , C s . Choose an arbitrary representative R i for each class C i , and let R ( S ) denote the representative for the class containing S . For convenience, wechoose R ( ∅ ) = ∅ .We now construct the SDP vectors. Let e , . . . , e s be an arbitrary orthonormal set ofvectors, and assign u R i = e i for all i ∈ [ s ] . Note that for any S with | S | ≤ t , there must bea unique equation of the form ∑ i ∈ S ∆ R ( S ) x i = b S in Λ , since two different equations can beused to obtain a width-2 t refutation. We assign the vector for S as u S : = ( − ) b S · u R ( S ) .The vectors are unit-length by construction. Note that if S ∆ S = S ∆ S , we must have S ∼ S ⇔ S ∼ S . If S S , then we have that h u S , u S i = h u S , u S i =
0. Otherwise,we have R ( S ) = R ( S ) , R ( S ) = R ( S ) , and equations of the form ∑ i ∈ S j ∆ R ( S j ) x i = b S j , j ∈ {
1, 2, 3, 4 } .We must also have b S + b S = b S + b S , since otherwise we obtain two different equationswith variables in S ∆ S = S ∆ S , yielding a refutation. This suffices to satisfy the SDPconstraints, since h u S , u S i = ( − ) b S + b S · D u R ( S ) , u R ( S ) E = ( − ) b S + b S = ( − ) b S + b S = h u S , u S i .16inally, for any equation ∑ i ∈ T x i = b T in Λ with | T | ≤ t , we get h u T , u ∅ i = ( − ) b T , sincewe must have T ∼ ∅ and R ( T ) = ∅∅