Additive Sparsification of CSPs

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Živný [SIDMA’20]. Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{ 0,1}k→ { 0,1} of a fixed arity k, we show that CSP(P) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P) admits an additive sparsifier for any predicate P : Dk→ { 0,1} of a fixed arity k on an arbitrary finite domain D.


Introduction
Graph sparsification is the problem of, given a graph G = (V, E) with quadratically many (in |V |) edges, finding a sparse subgraph G ε = (V, E ε ⊆ E) such that important properties of G are preserved in G ε . Sparse in this context usually means with sub-quadratically many edges, though in this work we require (and can achieve) linearly many edges. One of the most studied properties of preservation is the size of cuts. If G = (V, E, w) is an undirected weighted graph with w : E → R >0 , given some S ⊆ V , the cut of S in G is

Motivation
The relatively recent notion of additive sparsification has not yet been explored to the same extent as the notion of multiplicative sparsification has been. We believe that this notion has a lot of potential for applications as the sparsifiers are not weighted, unlike multiplicative sparsifiers. Indeed, the main restriction of multiplicative sparsifiers in applications appears to be the number of distinct weights required in sparsifiers. For some graphs (such as the "barbell graph" -two disjoint cliques joined by a single edge), any nontrivial multiplicative sparsifier requires edges of different weights. In any case, the authors find the notion of additive sparsification interesting in its own right, independently of applications.
The goal of our work is to understand how the notion of additive sparsification developed in [7] for cuts behaves on (hyper)graphs with other predicates (beyond cuts), deriving inspiration from the generalisations of cuts to other predicates in the multiplicative setting established in [20,14]. In particular, already Boolean binary predicates include interesting predicates such as the uncut edges (using the predicate P (x, y) = 1 iff x = y), covered edges (using the predicate P (x, y) = 1 iff x = 1 or y = 1), or directed cut edges (using the predicate P (x, y) = 1 iff x = 0 and y = 1). While such graph problems are well-known and extensively studied, it is not clear whether one should expect them to be sparsifiable or not. For instance, as mentioned before, not all (even Boolean binary) predicates are sparsifiable multiplicatively [20]. Are there some predicates that are not additively sparsifiable?

Contributions Boolean predicates
Our main result, Theorem 9 in Section 3, shows that all hypergraphs with constant uniformity k, directed or undirected, admit additive sparsification with respect to all Boolean predicates P : {0, 1} k → {0, 1}; the number of hyperedges of the sparsifier with error ε > 0 is O nε −2 log 1 ε , where the O(·) hides a factor that depends on k. This result has three ingredients. First, we observe that the result in [7] also holds true for directed hypergraphs. Second, we use a reduction via the k-partite k-fold covers of hypergraphs to the already solved case of Boolean Cut. Finally, we use linear algebra to prove the correctness of the reduction. While the reduction via the k-partite k-fold cover was used in previous works on multiplicative sparsification [20,14], the subsequent non-trivial linear-algebraic analysis (Proposition 13) is novel and constitutes our main technical contribution, as well as our result that, unlike in the multiplicative setting, all Boolean predicates can be (additively) sparsified. We also show that our results immediately apply to the more general setting where different hyperedges are associated with different predicates (cf. Remark 10). This corresponds to CSPs with a fixed constraint language (of a finite size) rather than just a single predicate.

Non-Boolean predicates
We introduce a notion of sparsification that generalises the Boolean case to predicates on non-Boolean domains, i.e. a notion capturing predicates of the form P : D k → {0, 1}, where D is an arbitrary fixed finite set with |D| ≥ 2. We call this type of sparsification "all-but-one" sparsification since the additive error term includes the maximum volume of |D| − 1 (out of |D|) parts, where the volume of a subset is the sum of the degrees in the subset. (The precise definition can be found in Section 4.) By building on the techniques used to establish our main result, we show that all hypergraphs (again, directed or undirected) admit additive all-but-one sparsification with respect to all predicates. This is stated as Theorem 20 in Section 4. We also show, in Section 5, that our notion of all-but-one sparsification is, in some sense, optimal.

Comparison to previous work
As mentioned above, our sparsifiability result is obtained by a reduction via the k-partite k-fold cover to the cut case established in [7]. A reduction via the k-partite k-fold cover was also used (for k = 2) in previous work on multiplicative sparsification [20,14]. In particular, the correctness of the reduction for Boolean binary predicates in [20] is done via an ad hoc case analysis for 11 concrete predicates. In the generalisation to binary predicates on arbitrary finite domains in [14], the correctness is proved via a combinatorial property of bipartite graphs without a certain 4-vertex graph 1 as a subgraph and a reduction to cuts with more than two parts.
In our case, we use the same black-box reduction via the k-partite k-fold cover. Thus the reduction itself is pretty straightforward, although the analysis is not. In fact, we find it surprising and unexpected that the k-partite k-fold cover works in the additive setting. Our key contribution is the proof of its correctness. A few simple reductions get us to the most technically involved case, in which k is even and the k-ary predicate satisfies P (1, . . . , 1) = 0.

75:4 Additive Sparsification of CSPs
Additive sparsifiability of such predicates is established in Proposition 13. Unlike in the multiplicative setting, it is not clear how to do this in a straightforward way similar to [20,14]. Instead, we associate with a given predicate P a vector v P in an appropriate vector space, identify special vectors that can be shown additively sparsifiable directly, show that linear combinations preserve sparsifiability, and argue that v P can be generated by the special vectors. The latter is the most technical part of the proof. While there are several natural ideas how to achieve this in a seemingly simpler way (such as arguing that the special vectors form a basis), we have not managed to produce a simpler or shorter proof.
The result in [7] also works for non-constant k. We emphasise that we deal with constant k, which is standard in the CSP literature in that the predicate (or a set of predicates) is fixed and not part of the input. For constant k, the representation of predicates is irrelevant (cf. Remark 18). Thus we do not keep track of (and have not tried to optimise) the precise dependency of the reduction on the predicate arity k (or the domain size q = |D|).

Related work
The already mentioned spectral sparsification [33] is a stronger notion than cut sparsification as it requires that not only cuts but also the Laplacian spectrum of a given graph should be (approximately) preserved [32,34,21,31,7].
Our focus in this article is on edge sparsifiers (of cuts and generalisations via local predicates). There are also vertex sparsifiers, in which one reduces the number of vertices. Vertex sparsifiers have been studied for cut sparsification (between special vertices called terminals) [22,26,25,15] as well as for spectral sparsification [24].
Sparsification in general is about finding a sparse sub(hyper)graph while preserving important properties of interest. In addition to cut sparsifiers, another well studied concept is that of spanners. A spanner of a graph is a (sparse) subgraph that approximately preserves distances of shortest paths. Spanners have been studied in great detail both in the multiplicative [5,28,3,17,6,9,30] and additive [2,18,12,8,35,16] setting. Emulators are a generalisation of spanners in which the sparse graph is not required to be a subgraph of the original graph. We refer the reader to a nice recent survey of Elkin and Neimain for more details [19]. Some of the proofs are deferred to the full version of this paper [29].

Preliminaries
For an integer k, we denote by [k] the set {0, 1, . . . , k − 1}. All graphs and hypergraphs 2 in this paper are unweighted. For an assignment a : V → S from the set of vertices of a (hyper)graph to some set S containing 0, we denote by Z a = {v ∈ V : a(v) = 0} the set of vertices mapped to 0.
If 0 ≤ i ≤ r k − 1 is an integer, we denote by rep r,k (i) the representation of i in base r as a vector in R k , where the first coordinate stands for the most significant digit, and the last coordinate for the least significant digit. For the special case r = 2, we use the notation bin k (i) for the binary representation of i.
We now define the value of an assignment on a hypergraph with a fixed predicate.
▶ Definition 1. Let G = (V, E) be a directed k-uniform hypergraph and let P : be a k-ary predicate on a finite set D. Given an assignment a : The notion of additive sparsification was first introduced in [7] for cuts in graphs and hypergraphs. In order to define it, we will need the Cut : Given a hypergraph G = (V, E) and a set U ⊆ V , we denote by vol G (U ) the volume of U , defined as the sum of the degrees in G of all vertices in U .
▶ Definition 2. Let G = (V, E) be an undirected k-uniform hypergraph, and denote |V | = n. We say that G admits additive cut sparsification with error ε using where d G is the average degree of G.
Note that (1) can also be written as which explains the use of the term "additive" for the error. Bansal, Svensson and Trevisan [7] showed the following sparsification result: ▶ Theorem 3 (Additive Cut Sparsification [7, Theorem 1.3]). Let G = (V, E) be an undirected n-vertex k-uniform hypergraph, and ε > 0. Then G admits additive cut sparsification with error ε using O n k ε −2 log( k ε ) hyperedges.
▶ Remark 4. We call a predicate P symmetric if it is order invariant (as in Definition 1). Since Theorem 3 deals with only undirected hypergraphs, it is not clear how to generalise it to non-symmetric predicates directly, since the value of such predicates on undirected hypergraphs is not defined. Therefore, our course of action will be first to prove it for the case of directed hypergraphs, and then generalise it to other predicates on directed hypergraphs. In fact, by doing this we also prove the result for undirected hypergraphs with symmetric predicates, since hyperedges can be given arbitrary directions without changing the average degree of G, or the volume in G, or the value of the predicate in any assignment.
▶ Remark 5. Throughout this paper we only discuss the existence of sparsifiers and do not mention the time complexity to find them. However, the (implicit) time complexity results from [7] apply in our more general setting as well since the sparsifiers we find are in fact the same sparsifiers for all predicates, including cuts (cf. Remark 17).
An important tool we use to prove our results is the k-partite k-fold cover of a hypergraph. This construction is a well known one, and has been used for multiplicative sparsification (for k = 2) in [20] and [14].

75:6
Additive Sparsification of CSPs If G is undirected we define the cover in the same way except If k = 2 then γ(G) corresponds to the well-known bipartite double cover of G [13].

Sparsification of Boolean Predicates
As mentioned in Section 1, we begin by observing that Theorem 3 also works for directed hypergraphs. The simple proof of this fact can be found in the full version [29]. (We emphasise that we treat k as a constant, cf. Remark 18.) Then G admits additive cut sparsification with error ε using O nε −2 log 1 ε hyperedges. From now on, whenever we say a "hypergraph", we mean a "directed hypergraph" with n vertices. By Remark 4, the results also apply to undirected hypergraphs (whenever it makes sense, i.e. if the associated predicate is symmetric). We also omit the word additive when discussing sparsification. The following notion of sparsification is a natural generalisation of cut sparsification (Definition 2) to arbitrary predicates.
▶ Definition 8. Let P be a k-ary Boolean predicate and G = (V, E) a k-uniform hypergraph. We say that G admits P -sparsification with error ε using O(f (n, ε)) hyperedges if there exists where d G is the average degree of G.
The following theorem is our main result, extending Proposition 7 to all k-ary predicates with Boolean domains.
▶ Theorem 9 (Main). For every k-uniform hypergraph G (k is a constant), every k-ary Boolean predicate P : {0, 1} k → {0, 1}, and every ε > 0, G admits P -sparsification with error ε using O nε −2 log 1 ε hyperedges. Theorem 9 can be informally restated as "every k-uniform hypergraph is sparsifiable with respect to all k-ary Boolean predicates" or "for every Boolean predicate P of constant arity, CSP(P ) is sparsifiable". ▶ Remark 10. It is possible to consider an even more general case where each hyperedge in G has its own predicate. In this case, we can apply Theorem 9 to each of the hypergraphs obtained by taking only hyperedges corresponding to a specific predicate, and so get a sparsifier for each such predicate. Taking the union of all their hyperedges, we get a new hypergraph G ε , which is a sparsifier of the original hypergraph. Indeed, it has O nε −2 log 1 ε hyperedges since it is the union of a constant number of hypergraphs. (The number of predicates P : {0, 1} k → {0, 1} is constant, since k is constant.) It also satisfies (2) for any given assignment up to some constant factor, since all the sparsifiers it is composed of do. This constant factor can be eliminated by choosing ε 0 = ε m for an appropriate m that depends only on k.
The main work in the proof of Theorem 9 is for even values of k; a simple reduction (Proposition 16) then reduces the case of k odd to the even case.
In order to prove Theorem 9 for even k, we use the k-partite k-fold cover of G and apply Proposition 7 to various assignments of it. For a k-ary Boolean predicate P : {0, 1} k → {0, 1}, we consider the vector v P ∈ R 2 k , defined by v P [i] = P (bin k (i)). For instance, for the Cut predicate on a 3-uniform hypergraph, we have v Cut = (0, 1, 1, 1, 1, 1, 1, 0).
For a given hypergraph G and an assignment a, we consider the vector : (a(v 1 ), . . . , a(v k )) = bin k (i)}|. In other words, each coordinate of v G,a counts the hyperedges in G whose vertices are assigned some specific set of values by a.  Under these notations, we get Val G,P (a) = ⟨v P , v G,a ⟩, where ⟨·, ·⟩ is the standard inner product in R 2 k . We begin by proving the following useful lemma.
▶ Lemma 12. Let G = (V, E) be a k-uniform hypergraph, P 1 , . . . , P m be k-ary Boolean predicates (m is a constant). Suppose that for every ε > 0 and 1 ≤ i ≤ m, G admits P isparsification with error ε using O nε −2 log 1 ε hyperedges, and that the same subhypergraph Under these conditions, G admits P -sparsification with error ε using O nε −2 log 1 ε hyperedges. Proof. Let ε > 0 and denote ε i = ε m|λi| (if λ i = 0 take ε i = 1 instead) and ε 0 = min{ε 1 , . . . , ε m }. Let G ε0 = (V, E ε0 ) be the common witness subhypergraph for ε 0 promised by the assumption. We know that every P i satisfies for every assignment a : V → {0, 1}. We also have Therefore, for every assignment a we get where the second line is due to the triangle inequality, the third is due to (3) and the fourth is by the definition of ε 0 . Furthermore, since m and all λ i are constants, and so G ε0 is a witness for the P -sparsification of G. ◀ The core of the proof of Theorem 9 is in the next proposition, which establishes the result for Boolean predicates on even uniformity hypergraphs, with a small restriction.
Let a : V → {0, 1}. For every subset T ⊆ [k], we look at the assignment a T : V γ → {0, 1} defined by a T (v (i) ) = 0 if i ∈ T and a(v) = 0, and a T (v (i) ) = 1 otherwise. We therefore have Define the vector u T ∈ R 2 k as follows: otherwise .
In other words, the vector u T is 1 in index j if and only if there exists an index i ∈ T in which the binary representation of j has a zero, with the exception of u [k] [0] = 0. Denote by P T the predicate corresponding to u T , that is P T (bin k (j)) = 1 ⇐⇒ u T [j] = 1. Observe that Val γ(G),Cut (a T ) = Val G,P T (a),  (1, 1, 0, 0, 1, . . . , 0, 1), and the red dashed path to a value 0 in the coordinate with binary representation (1, 0, 1, 1, 0, , . . . , 1, 1). Note that if T = [k] then any hyperedge corresponding to a path only on Z  Figure 2). The same is true for any hypergraph, and in particular for G ε , that is Putting these results in (4), we get so G admits P T sparsification with error ε using O nε −2 log 1 ε hyperedges for every T ⊆ [k], and for every ε the sparsification is witnessed by the same subhypergraph G ε . (Notice that Proposition 7, when applied to γ(G) which has kn vertices, gives us a subhypergraph with O knε −2 log 1 ε hyperedges, and recall that k is a constant.) Our next goal is to show that the vector v P is a linear combination of the vectors u T for all T ∈ [k]. To show that, we show that every vector e r in the standard basis of R 2 k , with r ̸ = 2 k − 1, is a linear combination of these vectors. This is sufficient since the last coordinate of v P is 0 by the assumption. First we need to order the various sets T . We order them in the following decreasing lexicographic order T 0 , T 1 , . . . , Let e r be a vector in the standard basis of R 2 k . We introduce the following coefficients for 0 ≤ m ≤ 2 k − 1: We shall prove that ▷ Claim 14. The sum of all coefficients is 0; i.e., . . b k be the binary representation of r. Since r < 2 k − 1, there exists some 1 ≤ i ≤ r for which b i = 0. We can partition the coefficients into pairs, such that λ r,m1 , λ r,m2 is a pair if and only if m 1 , m 2 differ in the i-th coordinate only. This is clearly a partition.
For each pair, f 1 gives m 1 , m 2 different parity values, and f 2 gives them the same value (since b i = 0), so λ r,m1 , λ r,m2 have opposite signs, so their sum is zero. This is true for every pair, so the overall sum is zero, and the claim is proved. ◁ We prove (5) coordinate-wise. First we look at the coordinate r. Consider the set W of all vectors u Tm for which the coordinate r is 0. If we show that the sum of the corresponding coefficients of the vectors in W is −1, using Claim 14 we will deduce the result in this case. We distinguish 2 cases: Case (I): r = 0. By the definition of u T , in this case the set W contains two vectors, u [k] and u ∅ . The corresponding coefficients are λ r,0 = − 1 2 and λ r,2 k −1 = − 1 2 (since k is even), which sum up to −1.
Case (II): r > 0. As in the proof of Claim 14, let b 1 b 2 . . . b k be the binary representation of r, and choose a coordinate 1 ≤ i ≤ k for which b i = 1. Partition the vectors in W into pairs where u Tm 1 , u Tm 2 is a pair if and only if m 1 , m 2 differ in the i-th coordinate only. This is clearly a partition of all vectors, and by the definition of u T , each such pair is either contained in W or disjoint from W so this is indeed a partition of W . (Note that u Tm 1 [r] is determined by T m1 ∩ zeros(r) which is in fact zeros(m 1 ) ∩ zeros(r), and the same for m 2 . Since m 1 , m 2 differ in the i-th coordinate only, and r is not zero in this coordinate, this coordinate contributes nothing to the intersections, and so both these intersections are empty or non-empty together. The intersection never equals [k] since r > 0.) For every such pair in W , if it does not contain the negation of bin(r), then there is some other index j ̸ = i in which r, m 1 , m 2 are all 1. (This is because in all other coordinates m 1 , m 2 are equal, and since they are not the negation of r, there is some coordinate j ̸ = i in which they are equal to the j-th coordinate of r. These coordinates cannot be all 0, since this would imply u Tm 1 , u Tm 2 / ∈ W .) This implies that f 2 gives m 1 , m 2 the same value, and clearly f 1 gives them different parity values, so λ r,m1 + λ r,m2 = 0. However, for the pair which contains the negation of r (this pair is clearly in W ), suppose without loss of generality the negation is m 1 . Then f 2 gives m 1 , m 2 the values 1, 0 respectively, and f 1 gives m 1 an even value and m 2 an odd value (since k is even), and so λ r,m1 = λ r,m2 = − 1 2 , and the overall sum is −1. This finishes the proof of (5) in the coordinate r. Now let r ′ ̸ = r be some other coordinate, and let c 1 c 2 . . . c k be its binary representation. First, if r ′ = 2 k − 1 then for all m we have u Tm [r ′ ] = 0 by definition, so the linear combination Boolean Xor of two bits is their sum modulo 2. The And of two integers is defined the same way with the Boolean And function which is defined as And(i, j) = 1 ⇐⇒ i = j = 1. of this coordinate is 0. So suppose r ′ < 2 k − 1. As before let W be the set of all vectors u Tm for which the coordinate r ′ is 0. We show that the sum of the corresponding coefficients is zero, and again deduce the result using Claim 14. Now, there exists some index i for which b i ̸ = c i . Again we have two cases: Partition the vectors in W into pairs where u Tm 1 , u Tm 2 is a pair if and only if m 1 , m 2 differ in the i-th coordinate only. This is clearly a partition of all the vectors, and by the definition of u T , each such pair is either contained in W or disjoint from W , so this is indeed a partition of W . For every such pair in W , f 1 gives m 1 , m 2 different parity values, and f 2 gives them the same value (since b i = 0), so λ r,m1 , λ r,m2 have opposite signs, so their sum is zero. This is true for every pair in W , so the overall sum is zero.
Case ( This finishes the proof of (5), and so v P is a linear combination of the vectors u T . From the result above and Lemma 12 we deduce that G admits P -sparsification with error ε using O nε −2 log 1 ε hyperedges, as required. ◀ To complete the picture for even k, we reduce to Proposition 13 by a simple "complementarity trick"; the proof can be found in the full version [29].
▶ Proposition 15. Let k be an even number, and G a k-uniform hypergraph. Let P be a k-ary Boolean predicate. Then for every ε > 0, G admits P -sparsification with error ε using O nε −2 log 1 ε hyperedges.
The final piece in the jigsaw, proved in in the full version [29], shows how to reduce sparsification of k-uniform hypergraphs with k odd to the case of (k + 1)-uniform hypergraphs by adding a universal vertex and extending the original predicate by one dimension.
▶ Proposition 16. Let k be an odd number, and G = (V, E) a k-uniform hypergraph. Let P be a k-ary Boolean predicate. Then for every ε > 0, G admits P -sparsification with error ε using O nε −2 log 1 ε hyperedges.
Propositions 15 and 16 complete the proof of Theorem 9.
▶ Remark 17. In the proof of Proposition 13 the hypergraph G ε was chosen independently of the predicate P . Since Propositions 15 and 16 reduce to that case, we have in fact shown that for every ε > 0, Theorem 9 is witnessed by the same subhypergraph G ε for all different predicates P . This will be important in the proof of Theorem 20.
▶ Remark 18. We note that our main result, Theorem 9, extends Theorem 3 in the regime where k is a constant, which is the main focus of this paper. However, Theorem 3 also works for non-constant k [7]. If k is not a constant, it can be seen from the proof of Lemma 12 that the number of hyperedges of the sparse subhypergraph is multiplied by a factor of O(m 2 ) (since O(m) is the proportion between ε and ε 0 given that the coefficients λ i are constant).
In Proposition 13 we have m = 2 k , and so for k not constant we get an additional factor of 4 k . Furthermore, in Propositions 7 and 13 we obtain extra factors of k, by considering the k-partite k-fold cover. While the regime with non-constant k is interesting for cuts, for arbitrary predicates one needs to be careful about representation as the natural (explicit) representation of (non-symmetric) predicates requires exponential space in the arity k.

Sparsification of Non-Boolean Predicates
We now focus on non-Boolean predicates; i.e., predicates of the form P : . We say that a k-uniform hypergraph G = (V, E) admits all-but-one P -sparsification with error ε using O(f (n, ε)) hyperedges if there exists a subhypergraph G ε = (V, E ε ⊆ E) with |E ε | = O(f (n, ε)) such that for every assignment a : V → D we have where M a is the largest set among the sets {v ∈ V : a(v) = i}, N a is the set with the largest volume among the sets {v ∈ V : a(v) = i} for 0 ≤ i ≤ q − 2, and d G is the average degree in G.
Observe that the maximum in Definition 19 is over i = 0, . . . , q − 2 without i = q − 1, hence the name "all-but-one". We note that there is nothing special about q − 1 and any value from [q] could be chosen in Definition 19. Under Definition 19, Theorem 9 generalises (proof of can be found in the full version [29]).
Note that in the case of q = 2 we have P : {0, 1} k → {0, 1}, and Definition 19 and Theorem 20 coincide with Definition 8 and Theorem 9. This is because when q = 2 the definitions of M a , N a coincide with the definition of Z a in the Boolean case.

Optimality of All-But-One Sparsification
One might wonder if there is a different, perhaps stronger way to define sparsification for predicates on non-Boolean domains. The following example shows that all-but-one sparsification is optimal. For a hypergraph G = (V, E) and a fixed assignment a : V → [q] denote S i = {v ∈ V : a(v) = i} (so S 0 = Z a ). The definition of all-but-one sparsification lets us take a bound which depends on the sizes and volumes of all the sets S i except for S q−1 . In fact, if we try to take a bound which depends on fewer of these sets, the definition fails to generalise even the most basic case of the Cut predicate. To see this, it is sufficient to consider the graph case, i.e. k = 2. Let us suppose, without loss of generality, that our bound does not depend on S q−2 , S q−1 . Consider the predicate Cut : [q] 2 → {0, 1} defined by Cut(x, y) = 1 ⇐⇒ x ̸ = y. A simple (but lengthy) argument in the full version [29] shows that cliques do not have a Cut-sparsifier using such a definition. In fact, the same argument works for any predicate P with P (q − 2, q − 1) = P (q − 1, q − 2) = 1 and P (q − 2, q − 2) = P (q − 1, q − 1) = 0. Thus if a definition does not depend on more than just S q−2 , S q−1 , it specifically does not depend on these two, so the same argument still works. Therefore, no definition with a bound which depends on "less" is possible, under the current assumptions.