1 Introduction
We describe a new family of instances of 3XOR, based on highdimensional expanders, that are hard for the SumofSquares (SoS) hierarchy of semidefinite programming relaxations, which is the most powerful algorithmic framework known for optimizing over constraint satisfaction problems. Unlike previous constructions of 3XOR hard instances for SoS, our construction is explicit, as it is based on the explicit construction of highdimensional expanders due to Lubotzky, Samuels and Vishne [LSV05a, LSV05b], which we refer to henceforth as LSV complexes.
Theorem 1.1.
There exists a constant and an infinite family of 3XOR instances on variables, constructible in deterministic polynomial time, satisfying the following:

No assignment satisfies more than fraction of the constraints.

Relaxations obtained by levels of the SoS hierarchy fail to refute the instances.
We also remark that our construction can be used to obtain explicit integrality gap instances for various other optimization problems, using reductions in the SoS hierarchy [Tul09]. In particular, while our instances on the LSV complexes exhibit an integrality gap of vs. for the SoS hierarchy, reductions can be used to obtain explicit XOR instances with a gap of vs. for any . Indeed, this yields explicit hard instances with optimal gaps for all approximation resistant predicates based on pairwise independent subgroups [Cha16].
Structured instances from Highdimensional expanders.
Highdimensional expanders (HDXs) are a highdimensional analog of expander graphs. In recent years they have found a variety of applications in theoretical computer science, such as efficient CSP optimization [AJT19], improved sampling algorithms [ALGV19, ALG20, AL20], quantum LDPC codes [EKZ20, KT20], novel lattice constructions [KM18], direct sum testing [GK19], and others. Explicit constructions of HDXs have also led to improved listdecoding algorithms [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 instances for SoS.
Highdimensional expanders are boundeddegree (hyper)graphs (or rather, simplicial complexes) with certain expansion properties. A simplicial complex is a nonempty collection of downclosed sets. Given a simplicial complex , we will refer by the family of all dimensional sets in (i.e., sets of size ). The dimension of the simplicial complex is the maximal dimension of any set in it. It will be convenient to refer to the sets of dimension 0, 1, 2, 3 as vertices, edges, triangles, tetrahedra, respectively. Thus, a graph is a 1dimensional complex, while in this work we will be using complexes of dimension at least 2. Given a 2dimensional complex , there are two natural ways to construct a 3XOR instance based on — a vertexvariable construction and an edgevariable construction. Let be any valued function on the set of triangles.
 Vertexvariable construction:

The 3XOR instance corresponding to consists of the following constraints: for each .
 Edgevariable construction:

The 3XOR instance corresponding to consists of the following constraints: for each .
The vertexvariable construction whose underlying structure is a highdimensional expander has been studied by Alev, Jeronimo and the last author [AJT19]. They gave an efficient algorithm for approximating vertexvariable constraint satisfaction problems (not necessarily 3XOR) on an underlying highdimensional expander. Their result is a generalization to higher dimensions of the corresponding result for graphs that “CSPs are easy on expanders” [BRS11, GS11]. They prove this by showing that certain types of random walks on vertices converge very fast on highdimensional expanders. However, the same analysis fails to show a similar result for the edgevariable construction, as the corresponding random walk on edges of a highdimensional expander does not mix. Our work shows that this difference isn’t just a technical limitation of their analysis; it is inherent. The edgevariable variant is truly hard, at least for SoS. This demonstrates an interesting subtlety in the structure of highdimensional expanders, and how it relates to optimization.
To understand our edgevariable construction better, it will be convenient to set up some notation. Let denote the set of all valued functions on . For each , consider the operator defined as follows:
This is usually referred to as the coboundary operator. Let be the image of , and let be the kernel of . Clearly, . Furthermore, it is not hard to see that . It easily follows from the definitions that the edgevariable construction corresponding to is a satisfiable instance iff .
Typically, soundness of SoShard instances is proved by choosing at random from . In contrast, we construct our explicit instances by choosing the function more carefully, and relying on a certain type of expansion property of the complex. Recall that , and the instance is satisfiable iff . Complexes for which are said to have trivial second cohomology. We will be working with complexes with nontrivial second cohomology, i.e., . This lets us choose a to prove soundness. It is known that the explicit constructions of HDXs due to Lubotzky, Samuels and Vishne [LSV05a, LSV05b] have nontrivial second cohomology.^{1}^{1}1More accurately, their construction depends on the group defining the quotient. They show that a certain choice of groups yields nontrivial second cohomology. In fact, these complexes have the stronger property (due to a theorem of Evra and Kaufman [EK16]) that all are not only not in , but in fact far from any function in . This latter property follows from the cosystolic expansion 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 that it is a negative instance? The LSV construction is a quotient of the socalled affine building which is, from a topological point of view, a simple “Euclideanlike” object with trivial cohomologies. The hardness of our instance comes from the inherent difference between the LSV complex and the building, which cannot be seen through local balls whose radius is at most the injectivity radius of the complex, in our case . Locally, the LSV quotient is isomorphic to the building. However, unlike the building, the LSV complex is a quotient with nontrivial cohomologies. The hardness comes from the fact that local views cannot capture the cohomology, which is a global property. Given this observation, the proof of completeness can be carried out following the argument of BenSasson and Wigderson [BW01] that any short resolution proof is narrow, and Grigoriev [Gri01] and Schoenebeck [Sch08]’s transformation from resolution lower bounds to SoS lower bounds.
Technically, we rely on two very different types of expansion or isoperimetry. In our proof of completeness, we rely on an isoperimetric inequality called Gromov’s filling inequality, that says that balls are essentially the objects with smallest boundary in any CAT(0) space (a class of spaces that includes both Euclidean spaces and the affine building). 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 nontrivial element in the cohomology has constant weight. Both of these statements are related to expansion, yet they are distinct from other notions of expansion used in previous SoS lower bounds.
Relation to previous SoS gap constructions.
All previous constructions of hard instances for SoS can be viewed in the vertex/edgevariable framework (typically vertexvariable). To the best of our knowledge, all known hard instances, proving inapproximability in the SoS hierarchy, are random instances; either both the complex and the function are random, or just the function is random. Explicit hard instances for SoS are known in proof complexity (e.g., Tseitin tautologies on expanders), however, we do not know how to transform these hard instances into ones for inapproximability. The proof of SoS hardness of these random instances relies on very strong expansion of the underlying complex [Sch08] or on certain pseudorandom properties [KMOW17], both of which are not yet known to be explicitly constructible.
In contrast, our instances are “antirandom”. They are very structured and easily distinguishable from random instances. For example, all balls around a vertex up to some radius are identical and have very specific structure. Naturally, the typical analysis that works for random instances cannot work here. For example, soundness for random instances is based on choosing a random
and using a unionbound argument to show that with high probability, every solution violates nearly half of the constraints. In contrast, for us, a random
is not a good choice because the local structure will quickly detect local contradictions, ruining the completeness altogether.Open directions.
Our construction of explicit hard SoS instances based on HDXs begs several questions, some of which we discuss below.
 Improved soundness

Our construction yields 3XOR hard instances which are at most satisfiable, owing to the cosystolic expansion of the underlying HDX (more precisely, , see Section 2.2 for the definition of ). Coupled with reductions in the SoS hierarchy [Tul09], this yields 3XOR hard instances which are at most satisfiable for every . Can we obtain such a result directly from the HDX construction (bypassing reductions), say by constructing HDXs which satisfy ? In addition to maintaining the HDX structure, bypassing reductions would also allow for perfect completeness, which is lost while using NPhardness reductions.
 Fooling more levels of the SoS hierarchy

Our hard instances fool only levels of the SoS hierarchy, as our argument is based on the injectivity radius of the complexes, which is , and we suffer a further squareroot loss due to the use of Gromov’s isoperimetry inequality. It is possible that a much stronger lower bound holds for these instances. Can one construct explicit hard instances that fool linearly many levels of the SoS hierarchy?
 HDX dimension and CSP definition

We find the contrast between the vertexvariable and edgevariable constructions baffling: while the vertexvariable construction is easy, our result demonstrates the hardness of the edgevariable construction. As we go to higher dimensions of HDX, there are more ways to define CSPs. Which of these are easy and which are hard?
2 Preliminaries
2.1 The SumofSquares hierarchy
The sumofsquares hierarchy^{2}^{2}2For more on SumofSquares, see the recent monograph by Fleming, Kothari and Pitassi [FKP19].
provides a hierarchy of semidefinite programming (SDP) relaxations, for various combinatorial optimization problems.
Figure 1 describes the relaxation given by levels of the hierarchy for an instance of 3XOR in variables, with constraints of the form over . We also use to denote the set of all tuplespresent as constraints. A solution to the relaxation is specified by a collection of unit vectors
, satisfying the constraints in the program. The objective equals the fraction of constraints “satisfied” by the SDP solution.To prove a lower bound on the value of the SDP relaxation, we will use the following result, which shows the existence of vectors yielding an objective value of 1, when the given system of XOR constraints does not have any “lowwidth” refutations. Formally, we consider a system called resolution, where the only rule allows us to combine two equations and to derive the equation . A refutation is a derivation of . The width of a refutation is the maximum number of variables in any equation used in the refutation. We include a proof of the following lemma in Appendix A.
2.2 Simplicial complexes
A simplicial complex is a nonempty collection of sets (known as faces) which is closed downwards. The dimensional faces are all sets of size . The dimension of the complex is the maximal dimension of a face. Faces of that dimension are known as facets. Faces of dimensions are called vertices, edges, triangles, and tetrahedra, respectively.
Graphs are dimensional simplicial complexes. The skeleton of a simplicial complex is the graph obtained by retaining only faces of dimension at most .
Links
Let be a dimensional simplicial complex. The link of a face is a simplicial complex of dimension given by . In other words, contains all faces in which contain , with itself removed.
Balls
Let be a simplicial complex. A ball of radius around a vertex is the subcomplex induced by all vertices at distance at most from , as measured on the skeleton of . That is, the subcomplex contains a face of if it contains all the vertices of the face.
Simplicial map
If and are two simplicial complexes, then a simplicial map is a map from to that maps faces to faces.
Chains
Fix a dimensional simplicial complex . Let be the set of all functions from to . Elements of are also known as chains.
For an chain , we define to be the number of nonzero elements in . For two chains , we define the distance between and to be .
Inner product
For , let us denote by the following sum modulo :
This is not an inner product in the usual sense as we are working over a field of nonzero characteristic, but it is convenient notation. We will usually drop the subscript .
Dual space
Given any subspace , the dual of (under ) is defined as:
Boundaries, Cycles, Homology
The boundary operator is given by
It gives rise to boundaries and cycles :
In the case of graphs, consists of all sums of cycles (in the usual sense).
The coboundary operator , which is the adjoint of the boundary operator, is given by
It gives rise to coboundaries and cocycles:
We will usually drop the subscript when invoking .
It is easy to see that (every boundary is a cycle) and (every coboundary is a cocycle). For example, in a dimensional complex, the boundary of every triangle is a cycle. We call such cycles trivial cycles. Modding out by trivial cycles and cocycles, we obtain the homology and cohomology spaces
The dimensions of these spaces (which are identical) measure the number of “holes” in a particular dimension. Nice complexes (such as the buildings considered below) have no holes.
The following claim shows that that the coboundary operator is the adjoint of the boundary operator.
Claim 2.2.
Let . Then .
Proof.
The following claims shows the dimensions of homology and cohomology spaces are identical.
Claim 2.3.
.
Proof.
. ∎
Claim 2.4.
Proof.
Cosystoles
We define, following Evra and Kaufman [EK16, Definition 2.14], the cosystole of a complex to be the minimal (fractional) size of ,
2.3 The building
The infinite regular tree is the unique connected regular graph without cycles. Affine buildings are higherdimensional analogs of the infinite regular tree. For , the onedimensional affine building is the regular tree. For higher dimensions they are regular in the sense that all vertex links are bounded and identical in structure, they are connected and contractible,^{3}^{3}3A complex is contractible, roughly speaking, if it can be continuously deformed to a point (technically, it is homotopyequivalent to a point). Since (co)homologies are preserved by such deformations, all (co)homologies of a contractible complex vanish. and so have vanishing cohomologies, that is, the cohomology spaces are trivial, where is the dimension.
We won’t describe any further; the interested reader can check [Ji12, AB08]. A crucial property of which we will need in the sequel is its being a CAT(0) space,^{4}^{4}4A space is CAT(0) if for every triangle , the distance between and the midpoint of is at most the corresponding distance in a congruent triangle in Euclidean space. which is a geometric definition capturing nonpositive curvature; see [BH99] for more information. The property of being CAT(0) has the following implication, due to Gromov [Gro83, Gut06, Wen08]:
Theorem 2.5 (Gromov’s filling inequality for CAT(0) spaces).
For every cycle there is a filling such that and .
Gromov’s filling inequality is an isoperimetric inequality. It generalizes the classic isoperimetric inequality in the plane, which states that any simple closed curve of length encloses a region whose area is at most .
The isoperimetric inequality in the plane can be stated in an equivalent way: the boundary of any bounded region of area is a curve whose length is at least . This inequality 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 inequality doesn’t imply that each satisfies . Rather, we have to replace with .
Gromov’s filling inequality also applies to chains, with an exponent of , but we will only need the case .
In the sequel, we will apply Gromov’s filling inequality not to the building itself, but rather to balls in the building. The CAT(0) property almost immediately implies that a ball in a CAT(0) space is itself CAT(0) [BH99, Exercise II.1.6]. Furthermore, it is wellknown that CAT(0) spaces are contractible, and so have vanishing homologies.
Lemma 2.6.
Balls in have vanishing homologies and satisfy Gromov’s filling inequality.
2.4 The LSV quotient
Whereas the affine building is an infinite simplicial complex, Lubotzky, Samuels and Vishne constructed a growing family of finite complexes that are obtained from quotients of the affine building. These quotients have a growing number of vertices, and locally, in a ball around each vertex, the complex is isomorphic to the affine building. Moreover, they gave a very explicit algorithm for constructing these complexes by first constructing a Cayley graph with an explicit set of generators, and then the higher dimensional faces are simply the cliques in the Cayley graph.
Theorem 2.7 (Lubotzky, Samuels, Vishne [LSV05a, Theorem 1.1]).
Let be a prime power, . For every the group has an (explicit) set of generators, such that the Cayley complex of with respect to these generators is a Ramanujan complex covered by for .
The precise definition of “Ramanujan complex” is not important for this context. For us, 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 polynomial time.

Local structure: Next, we highlight the fact that locally the complex looks like the building. The theorem states that the complex is covered by . A covering map maps a simplicial complex surjectively to a simplicial complex by mapping the vertices such that for every the image of every face is a face .
The fact that is covered by means that the neighborhood of a vertex in and in look exactly the same. It turns out that for the LSV complexes this continues to be true also for balls of larger radius around any vertex. This is a higherdimensional analog of the graph property of containing no short cycles (locally looking like a tree). Define the injectivity radius of to be the largest such that the covering map is injective from balls of radius in and the ball of radius in . We do not mention the center of the ball they are all isomorphic.
Theorem 2.8 (Lubotzky and Meshulam [Lm07], see also^{5}^{5}5The theorem was proven by [Lm07]. They stated their theorem using a slightly different definition for injectivity 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. [Egl15, Corollary 5.2]).
Let be the LSV complex above. Then the injectivity radius of satisfies
where is the number of vertices in .

Global structure: Finally, we look at the second cohomology group of the LSV complexes. Kaufman, Kazhdan and Lubotzky [KKL16] showed that the groups defining the LSV quotient complexes can be chosen so that the second homology is nonempty.
Proposition 2.9 (Kaufman, Kazhdan, Lubotzky [Kkl16, Proposition 3.6]).
There is an infinite and explicit sequence of LSV complexes with a nonvanishing second cohomology.
We remark that Kaufman, Kazhdan and Lubotzky [KKL16] proved that these complexes exist. To show that they are also efficiently constructible, we look into their proof to recall the construction: start with any LSV complex viewed as a Cayley graph of a group . Find some element of order in (such an element always exists), and then quotient by this element, thus obtaining a complex that is itself is a Ramanujan complex because it is a quotient of one. is clearly efficiently constructible from , and has half as any vertices. This construction shows (see [KKL16, Proposition 3.5]) that . Furthermore, the proof of [KKL16, Proposition 3.6] shows that because has property one can deduce also that .
Evra and Kaufman proved [EK16, Theorem 1.9] that quotients of (and even a more general class of complexes) are socalled “cosystolic expanders” which in particular implies the following.
Theorem 2.10 (Evra and Kaufman [Ek16, Part of Theorem 1.9]).
Let be a family of LSV complexes. There exists some constant that depends only on and but not on the size of the complex, such that every must have weight at least .
3 Main result
3.1 Local geometry of LSV complexes
The infinite sequence of complexes we will be working with are the LSV complexes described in Section 2.4 above. The properties we care about are (1) that they are efficiently constructible, (2) that small balls in these complexes are isomorphic to the affine building, which satisfies certain isoperimetric inequalities because it is a CAT(0) space, and (3) that each complex has a twodimensional cocycle with linear distance from the set of coboundaries. The second and third properties provide the tension between the local and the global structure of these complexes that we now harness for our hardness.
To construct an SDP solution, we will need to show that our instance based on the LSV complex “locally looks satisfiable”. To this end, we will first develop some local properties of the LSV complex.
Note that each 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 define connected components. Note that if can be split into connected components , then the components correspond to disjoint sets of triangles. Moreover, no triangle in shares an edge with a triangle in when , which also implies that the boundaries and correspond to disjoint sets of edges.
We prove the following claims by mapping small connected sets in to corresponding sets in the infinite building . The first proposition shows that there can be no small nontrivial cancellations (i.e., not coming from tetrahedra).
Proposition 3.1.
Let be a connected set of triangles such that and . Then .
Proof.
Since , there is a ball of radius that contains the support of . By assumption, the covering map has injectivity radius of at least . This means that there is a radius ball in that is isomorphically mapped by to . Look at , the chain isomorphic to in the building. Clearly , and since balls in the building have zero homologies by Lemma 2.6, we deduce that itself must be a boundary, i.e. there must be some such that . Moving back to , we see that necessarily satisfies , and so . ∎
This proposition states that locally (i.e., within the injective radius ), looks like . We thus have a complex whose cohomology group is nontrivial, yet locally, the homology group “looks” trivial. Note that this is a twist on what we had claimed in the introduction, a complex whose cohomology group is nontrivial, yet locally, the cohomology group “looks” trivial. However, these are identical statements owing to creftype 2.4.
The next proposition shows that Gromov’s filling inequality in the infinite building can be used to yield a similar consequence for small sets in the finite complex .
Proposition 3.2.
Let be a connected set of triangles such that and for all . Then, , where is an absolute constant.
Proof.
As before, the support of is contained in a ball of which is isomorphic under to a ball in . Let , and let . We now apply the filling theorem of Gromov, which holds in due to Lemma 2.6, to deduce that there is some that fills , namely , and whose size is at most .
Now . Since the ball has zero homologies by Lemma 2.6, itself must be a boundary: there must be some such that . Pushing and back to , we get and , which satisfy . At this point we have a small that is close via a boundary to . Finally, observe that satisfies . So
where the last inequality used that , since . ∎
3.2 Fooling levels of SoS hierarchy
Let be a dimensional LSV complex, with and nontrivial second cohomology group, as per Proposition 2.9. Below, we construct an instance of 3XOR in variables using this complex, and prove a lower bound on the integrality gap of the relaxation obtained by levels of the SoS hierarchy.
Construction.
We construct a system of equations on by putting a variable for each edge of the complex, and an equation
for each triangle , where is an arbitrary element of .
Recall that can be constructed efficiently. Given , we can find a vector 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 is large.
Claim 3.3 (Soundness).
Every assignment to the system defined above falsifies at least fraction of the equations.
Proof.
An assignment to the variables is equivalent to an . Every equation satisfied by is a triangle in which , and so the number of unsatisfied equations is . Since and , also , and so . In other words, the assignment falsifies at least a fraction of the equations. ∎
The main work is to prove completeness, namely to show that the system looks locally satisfiable.
Completeness.
Our main result is that this system appears satisfiable to the SumofSquares hierarchy with levels. Grigoriev [Gri01] and Schoenebeck [Sch08] showed that to prove such a statement it suffices to analyze the refutation width of the system of equations (see Lemma 2.1). If the refutation width is at least , then levels of the SumofSquares hierarchy cannot refute the system.
A system of linear equations over can be refuted using a proof system known as resolution, in which the only inference rule is: given and , deduce ; here are XORs of variables, and are constants. A refutation has the structure of a directed acyclic graph (DAG) where each nonleaf node has two incoming edges. A refutation is a derivation which starts with the given linear equations, placed at the leaves of a DAG, and reaches the equation at the root of the DAG. The width of a linear equation is the number of variables appearing in . The width of a refutation is 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 with Lemma 2.1 implies Theorem 1.1.
Theorem 3.4.
The construction above requires width at least to refute in resolution, where is the injectivity radius of the complex.
The proof follows classical arguments of BenSasson and Wigderson [BW01] regarding lower bounds on resolution width, which were also used in the proof of Schoenebeck [Sch08]. Whereas BenSasson and Wigderson relied on boundary expansion, we rely on Gromov’s filling inequality (and so lose a square root).
Suppose we are given a refutation for this system, and consider the corresponding DAG. Each leaf in the DAG is labeled by a triangle . Define
For each inner node in the DAG, let be its two incoming nodes. Define inductively,
Proposition 3.5.
For every node , .
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. While in [BW01] the complexity measure is based on the number of “leaf equations” used to derive the one at a given node, we will need to discount sets of triangles corresponding to tetrahedra, as these cannot lead to contradictions. Recall that is the set of triangle chains that “come from” tetrahedra chains, which we consider as the “trivial” cycles. We define a complexity measure at each node,
that measures the distance of from these trivial cycles. The complexity measure satisfies the following subadditivity property.
Proposition 3.6.
If is an inner node in the DAG with its two incoming nodes, then
Proof.
Let be such that and . Recall that . Then, we have
We also need the fact that the complexity of a node with a contradiction must be nonzero.
Proposition 3.7.
If then .
Proof.
If then . Hence since (creftype 2.3). ∎
Next, we consider the width of each node in the DAG. For a node , let
Thus indicates the set of variables appearing in the lefthand side of the equation on node . So the width of the system is the maximum, over all nodes in the DAG, of .
We can now prove Theorem 3.4 using the above complexity measure, and results from Section 3.1.
Proof of Theorem 3.4.
Let denote the root of the DAG. By virtue of being a refutation, while . In other words, , which means that . Since , we also have by Proposition 3.7 that .
Let be such that , and let be the disjoint connected components of . We will first show that . Assuming , we have that
Also, since
we must have that for each , since connected components have disjoint boundaries. Applying Proposition 3.1 to each , we get that for each . However, this implies and hence , which is a contradiction.
Using subadditivity (Proposition 3.6), , and the fact that the leaves of the DAG satisfy , we get that there must be some internal node for which . We can find such a node by starting at the root and always going to the child with higher complexity, until reaching a node such that . We will prove that for such a node, we must have .
As before, let now be such that , and let be the disjoint connected components of . We have that for each . By the minimality of , we also have that for any and any ,
Thus, is also minimal for each , and we can apply Proposition 3.2 to each connected component , to obtain
Acknowledgements
Part of this work was done when the authors were visiting the Simons Institute of Theory of Computing, Berkeley for the 2020 summer cluster on ”ErrorCorrecting Codes and HighDimensional Expansion”. We thank the Simons institute for their kind hospitality.
References
 [AB08] Peter Abramenko and Kenneth S. Brown. Buildings, Theory and applications, volume 248 of Graduate Texts in Mathematics. Springer, 2008.
 [AJQST20] Vedat Levi Alev, Fernando Granha Jeronimo, Dylan Quintana, Shashank Srivastava, and Madhur Tulsiani. List decoding of direct sum codes. In Proc. st Annual ACMSIAM Symp. on Discrete Algorithms (SODA), pages 1412–1425. 2020.
 [AJT19] Vedat Levi Alev, Fernando Granha Jeronimo, and Madhur Tulsiani. Approximating constraint satisfaction problems on highdimensional expanders. In Proc. th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 180–201. 2019. arXiv:1907.07833.
 [AL20] Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proc. nd ACM Symp. on Theory of Computing (STOC), pages 1198–1211. 2020.
 [ALG20] Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in highdimensional expanders and applications to the hardcore model. In Proc. st IEEE Symp. on Foundations of Comp. Science (FOCS). 2020. (To appear). arXiv:2001.00303.
 [ALGV19] Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Logconcave polynomials II: highdimensional walks and an FPRAS for counting bases of a matroid. In Proc. st ACM Symp. on Theory of Computing (STOC), pages 1–12. 2019. arXiv:1811.01816.
 [BH99] Martin R. Bridson and André Haefliger. Metric Spaces of NonPositive Curvature, volume 319 of Grundlehren der mathematischen Wissenschaften. Springer, 1999.
 [BRS11] Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In Proc. st IEEE Symp. on Foundations of Comp. Science (FOCS), pages 472–481. 2011. arXiv:1104.4680, eccc:2011/TR11065.
 [BW01] Eli BenSasson and Avi Wigderson. Short proofs are narrow  resolution made simple. J. ACM, 48(2):149–169, 2001. (Preliminary version in 31st STOC, 1999). eccc:1999/TR99022.
 [Cha16] Siu On Chan. Approximation resistance from pairwiseindependent subgroups. J. ACM, 63(3):27:1–27:32, 2016. (Preliminary version in 45th STOC, 2013). eccc:2012/TR12110.
 [DD19] Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In Proc. th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 1495–1524. 2019. arXiv:1909.00638, eccc:2019/TR19112.
 [DHKNT19] Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon TaShma. List decoding with double samplers. In Proc. th Annual ACMSIAM Symp. on Discrete Algorithms (SODA), pages 2134–2153. 2019. arXiv:1808.00425, eccc:2018/TR18198.
 [DK17] Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In Proc. th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 974–985. 2017. eccc:2017/TR17089.
 [EGL15] Shai Evra, Konstantin Golubev, and Alexander Lubotzky. Mixing properties and the chromatic number of Ramanujan Complexes. Int. Math. Res. Not., 2015(22):11520–11548, 2015. arXiv:1407.7700.
 [EK16] Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proc. th ACM Symp. on Theory of Computing (STOC), pages 36–48. 2016. arXiv:1510.00839.
 [EKZ20] Shai Evra, Tali Kaufman, and Gilles Zémor. Decodable quantum LDPC codes beyond the distance barrier using high dimensional expanders, 2020. (manuscript). arXiv:2004.07935.
 [FKP19] Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic proofs and efficient algorithm design. Found. Trends Theor. Comput. Sci., 14(12):1–221, 2019. eccc:2019/TR19106.
 [GK19] Roy Gotlib and Tali Kaufman. Testing odd direct sums using high dimensional expanders. In Dimitris Achlioptas and László A. Végh, eds., Proc. rd International Workshop on Randomization and Computation (RANDOM), volume 145 of LIPIcs, pages 50:1–50:20. Schloss Dagstuhl, 2019. eccc:2019/TR19124.
 [Gri01] Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. tcs, 259(12):613–622, 2001.
 [Gro83] Mikhael Gromov. Filling Riemannian manifolds. J. Differential Geom., 18(1):1–147, 1983.
 [GS11] Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives. In Proc. st IEEE Symp. on Foundations of Comp. Science (FOCS), pages 482–491. 2011. arXiv:1104.4746.
 [Gut06] Larry Guth. Notes on Gromov’s systolic estimate. Geom. Dedicata, 123:113–129, 2006.
 [Ji12] Lizhen Ji. Buildings and their applications in geometry and topology. In Differential geometry, volume 22 of Adv. Lect. Math. (ALM), pages 89–210. Int. Press, Somerville, MA, 2012. (Expanded version of Lizhen Ji. Buildings and their Applications in Geometry and Topology. Asian J. Math. 10 (2006), no. 1, 11–80. doi:10.4310/AJM.2006.v10.n1.a5).
 [KKL16] Tali Kaufman, David Kazhdan, and Alexander Lubotzky. Isoperimetric inequalities for Ramanujan complexes and topological expanders. Geom. Funct. Anal., 26(1):250–287, 2016. (Preliminary version in 55th FOCS, 2014). arXiv:1409.1397.
 [KM18] Tali Kaufman and David Mass. Good distance lattices from high dimensional expanders, 2018. (manuscript). arXiv:1803.02849.
 [KMOW17] Pravesh K. Kothari, Ryuhei Mori, Ryan O’Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In Proc. th ACM Symp. on Theory of Computing (STOC), pages 132–145. 2017. arXiv:1701.04521.
 [KT20] Tali Kaufman and Ran J. Tessler. Quantum LDPC codes with distance, for any , 2020. (manuscript). arXiv:2008.09495.
 [LM07] Alexander Lubotzky and Roy Meshulam. A Mooore bound for simplicial complexes. Bull. Lond. Math. Soc., 39:353–358, 2007.
 [LSV05a] Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type . European J. Combin., 26(6):965––993, 2005. arXiv:math/0406217.
 [LSV05b] ———. Ramanujan complexes of type . Israel J. Math., 149(1):267–299, 2005. arXiv:math/0406208.
 [Sch08] Grant Schoenebeck. Linear level Lasserre lower bounds for certain kCSPs. In Proc. th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 593–602. 2008. (Full version available at http://schoeneb.people.si.umich.edu/papers/LasserreNew.pdf).
 [Tul09] Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proc. st ACM Symp. on Theory of Computing (STOC), pages 303–312. 2009. eccc:2008/TR08104.
 [Wen08] Stefan Wenger. A short proof of Gromov’s filling inequality. Proc. Amer. Math. Soc., 136(8):2937–2941, 2008. arXiv:math/0703889.
Appendix A Proof of Lemma 2.1
Lemma 2.1 (Restated) ([Sch08, Lemma 13], [Tul09, Theorem 4.2]) Let be a system of equations in variables over , which does not admit any refutations of width at most . Then there exist vectors satisfying the constraints in Figure 1, such that for all equations in with , we have .
Proof.
We assume that is closed under width resolution, replacing by its closure if necessary, and also that it contains the trivial equation . We will now construct the unit vector .
Define a relation on subsets of of size at most as follows: iff there exists an equation in for some . It is easy to check that the relation is reflexive and symmetric. It is also transitive since for , , we can add the corresponding equations to obtain one of the form for some . Since , this equation has at most variables and must be in by the closure property. Thus, we have an equivalence relation which partitions all sets of size at most into equivalence classes, say . Choose an arbitrary representative for each class , and let denote the representative for the class containing . For convenience, we choose .
We now construct the SDP vectors. Let be an arbitrary orthonormal set of vectors, and assign for all . Note that for any with , there must be a unique equation of the form in , since two different equations can be used to obtain a width refutation. We assign the vector for as
The vectors are unitlength by construction. Note that if , we must have . If , then we have that . Otherwise, we have , , and equations of the form
We must also have , since otherwise we obtain two different equations with variables in , yielding a refutation. This suffices to satisfy the SDP constraints, since
Finally, for any equation in with , we get , since we must have and . ∎
Comments
There are no comments yet.