Almost-Polynomial Ratio ETH-Hardness of Approximating Densest k-Subgraph

Pasin Manurangsi∗ University of California, Berkeley Berkeley, CA, USA [email protected]

ABSTRACT Symposium on the Theory of Computing, Montreal, Canada, June 2017 (STOC’17), In the D￿￿￿￿￿￿ k￿S￿￿￿￿￿￿￿ (DkS) problem, given an undirected 8 pages. graph G and an integer k, the goal is to ￿nd a subgraph of G on DOI: 10.1145/3055399.3055412 k vertices that contains maximum number of edges. Even though Bhaskara et al.’s state-of-the-art algorithm for the problem achieves + 1 INTRODUCTION only O(n1/4 ε ) approximation ratio, previous attempts at proving In the D￿￿￿￿￿￿ k￿S￿￿￿￿￿￿￿ (DkS) problem, we are given an undi- hardness of approximation, including those under average case rected graph G on n vertices and a positive integer k 6 n. The goal assumptions, fail to achieve a polynomial ratio; the best ratios is to ￿nd a set S of k vertices such that the induced subgraph on ruled out under any worst case assumption and any average case S has maximum number of edges. Since the size of S is ￿xed, the assumption are only any constant (Raghavendra and Steurer) and problem can be equivalently stated as ￿nding a k-subgraph (i.e. O(log2/3 n) 2 (Alon et al.) respectively. subgraph on k vertices) with maximum density where density1 of In this work, we show, assuming the exponential time hypothesis the subgraph induced on S is |E(S)|/ |S | and E(S) denotes the set (ETH), that there is no polynomial-time algorithm that approxi- 2 1/(log log n)c of all edges among the vertices in S. ! " mates DkS to within n factor of the optimum, where D￿￿￿￿￿￿ k￿S￿￿￿￿￿￿￿, a natural generalization of k￿C￿￿￿￿ [32], > c 0 is a universal constant independent of n. In addition, our was ￿rst formulated and studied by Kortsarz and Peleg [34] in the result has perfect completeness, meaning that we prove that it is early 90s. Since then, it has been the subject of intense study in the ETH-hard to even distinguish between the case in which G contains context of approximation algorithm and hardness of approxima- a k-clique and the case in which every induced k-subgraph of G −1/(log log n)c tion [3, 8, 10, 13–15, 22, 24–27, 33, 41, 46]. Despite this, its approx- has density at most 1/n in polynomial time. imability still remains wide open and is considered by some to be Moreover, if we make a stronger assumption that there is some an important open question in approximation algorithms [13–15]. > constant ε 0 such that no subexponential-time algorithm can On the approximation algorithm front, Kortsarz and Peleg [34], ￿ distinguish between a satis able 3SAT formula and one which is in the same work that introduced the problem, gave a polynomial- ￿ only (1 − ε)-satis able (also known as Gap-ETH), then the ratio time O˜ (n0.3885)-approximation algorithm for DkS. Feige, Kortsarz f (n) above can be improved to n for any function f whose limit is and Peleg [24] later provided an O(n1/3−δ )-approximation for the ￿ zero as n goes to in nity (i.e. f ∈ o(1)). problem for some constant δ ≈ 1/60. This approximation ratio was the best known for almost a decade until Bhaskara et al. [13] + CCS CONCEPTS invented a log-density based approach which yielded an O(n1/4 ε )- • Theory of computation → Problems, reductions and com- approximation for any constant ε > 0. This remains the state-of- pleteness; the-art approximation algorithm for DkS. While the above algorithms demonstrate the main progresses KEYWORDS of approximations of DkS in general case over the years, many D￿￿￿￿￿￿ k￿S￿￿￿￿￿￿￿, Hardness of Approximation special cases have also been studied. Most relevant to our work is the case where the optimal k-subgraph has high density, in which ACM Reference format: better approximations are known [10, 26, 37, 47]. The ￿rst and Pasin Manurangsi. 2017. Almost-Polynomial Ratio ETH-Hardness of Ap- proximating Densest k-Subgraph. In Proceedings of 49th Annual ACM SIGACT most representative algorithm of this kind is that of Feige and Seltser [26], which provides the following guarantee: when the ∗This material is based upon work supported by the National Science Foundation input graph contains a k-clique, the algorithm can ￿nd an (1 − ε)- under Grants No. CCF 1540685 and CCF 1655215. dense k-subgraph in nO(log n/ε) time. We will refer to this problem of ￿nding densest k-subgraph when the input graph is promised to Permission to make digital or hard copies of all or part of this work for personal or have a k-clique D￿￿￿￿￿￿ k￿S￿￿￿￿￿￿￿ with perfect completeness. classroom use is granted without fee provided that copies are not made or distributed for pro￿t or commercial advantage and that copies bear this notice and the full citation Although many algorithms have been devised for DkS, relatively on the ￿rst page. Copyrights for components of this work owned by others than ACM little is known regarding its hardness of approximation. While it must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior speci￿c permission and/or a is commonly believed that the problem is hard to approximate to fee. Request permissions from [email protected]. STOC’17, Montreal, Canada 1It is worth noting that sometimes density is de￿ned as |E(S)|/|S |. For DkS, both © 2017 ACM. 978-1-4503-4528-6/17/06...$15.00 de￿nitions of density result in the same objective since |S | = k is ￿xed. However, our DOI: 10.1145/3055399.3055412 notion is more convenient to deal with as it always lies in [0, 1].

954 STOC’17, June 2017, Montreal, Canada Pasin Manurangsi within some polynomial ratio [3, 14], not even a constant factor NP- T￿￿￿￿￿￿ 1.1. There is a constant c > 0 such that, assuming ETH, hardness of approximation is known. To circumvent this, Feige [22] no polynomial-time algorithm can, given a graph G on n vertices and came up with a hypothesis that a random 3SAT formula is hard to a positive integer k 6 n, distinguish between the following two cases: refute in polynomial time and proved that, assuming this hypothesis, • There exist k vertices of G that induce a k-clique. 1/(log log n)c DkS is hard to approximate to within some constant factor. • Every k-subgraph of G has density at most n− . Alon et al. [3] later used a similar conjecture regarding random If we assume a stronger assumption that it takes exponential k-AND to rule out polynomial-time algorithms for DkS with any time to even distinguish between a satis￿able 3SAT formula and constant approximation ratio. Moreover, they proved hardnesses of one which is only (1 ε)-satis￿able for some constant ε > 0 (aka approximation of DkS under the following Planted Clique Hypothe- − f (n) sis [31, 35]: there is no polynomial-time algorithm that can distin- Gap-ETH; see Hypothesis 2.2), the ratio can be improved to n for any f o(1): guish between a typical Erdős-Rényi random graph G(n, 1/2) and ∈ one in which a clique of size polynomial in n (e.g. n1/3) is planted. T￿￿￿￿￿￿ 1.2. For every function f o(1), assuming Gap-ETH, ∈ Assuming this hypothesis, Alon et al. proved that no polynomial- no polynomial-time algorithm can, given a graph G on n vertices and time algorithm approximates DkS to within any constant factor. a positive integer k 6 n, distinguish between the following two cases: They also showed that, when the hypothesis is strengthened to • There exist k vertices of G that induce a k-clique. f (n) rule out not only polynomial-time but also super-polynomial time • Every k-subgraph of G has density at most n− . algorithms for the Planted Clique problem, their inapproximability We remark that, for DkS with perfect completeness, the afore- guarantee for DkS can be improved. In particular, if no nO(√log n)- mentioned Feige-Seltser algorithm achieves an nε -approximation O(log2/3 n) time algorithm solves the Planted Clique problem, then 2 - in time nO(1/ε) for every ε > 0 [26]. Hence, the ratios in our theo- approximation for DkS cannot be achieved in polynomial time. rems cannot be improved to some ￿xed polynomial and the ratio There are also several inapproximability results of DkS based in Theorem 1.2 is tight in this sense. on worst-case assumptions. Khot [33] showed, assuming NP * Comparison to Previous Results. In terms of inapproximabil- nε BPTIME(2 ) for some constant ε > 0, that no polynomial-time ity ratios, the ratios ruled out in this work are almost polynomial + algorithm can approximate DkS to within (1 δ) factor where and provides a vast improvement over previous results. Prior to δ > 0 is a constant depending only on ε; the proof is based on a our result, the best known ratio ruled out under any worst case construction of a “quasi-random” PCP, which is then used in place assumption is only any constant [41] and the best ratio ruled out of a random 3SAT in a reduction similar to that from [22]. 2/3 under any average case assumption is only 2O(log n) [3]. In addi- While no inapproximability of DkS is known under the Unique tion, our results also have perfect completeness, which was only Games Conjecture, Raghavendra and Steurer [41] showed that a achieved in [15] under ETH and in [3] under the Planted Clique strengthened version of it, in which the constraint graph is required Hypothesis but not in [22, 33, 41]. to satisfy a “small-set expansion” property, implies that DkS is hard Regarding the assumptions our results are based upon, the aver- to approximate to within any constant ratio. age case assumptions used in [3, 22] are incomparable to ours. The Recently, Braverman et al. [15], showed, under the exponential ε assumption NP * BPTIME(2n ) used in [33] is also incomparable time hypothesis (ETH), which will be stated shortly, that, for some to ours since, while not stated explicitly, ETH and Gap-ETH by O˜ (log n) constant ε > 0, no n -time algorithm can approximate D￿￿￿￿ default focus only on deterministic algorithms and our reductions + ￿￿￿ k￿S￿￿￿￿￿￿￿ with perfect completeness to within (1 ε) factor. are also deterministic. The strengthened Unique Games Conjecture It is worth noting here that their result matches almost exactly with used in [41] is again incomparable to ours as it is a statement that a the previously mentioned Feige-Seltser algorithm [26]. speci￿c problem is NP-hard. Finally, although Braverman et al.’s re- Since none of these inapproximability results achieve a polyno- sult [15] also relies on ETH, its relation to our results is more subtle. mial ratio, there have been e￿orts to prove better lower bounds Θ˜ Speci￿cally, their reduction time is only 2 (√m) wherem is the num- for more restricted classes of algorithms. For example, Bhaskara ber of clauses, meaning that they only need to assume that 3SAT et al. [14] provided polynomial ratio lower bounds against strong < Θ˜ (√m) SDP relaxations of DkS. Speci￿cally, for the Sum-of-Squares hierar- DTIME(2 ) to rule out a constant ratio polynomial-time ap- Ω proximation for DkS. However, as we will see in Theorem 3.1, even chy, they showed integrality gaps of n2/53 ε and nε against n (ε) − Ω˜ (m3/4) and n1 O(ε) levels of the hierarchy respectively. (See also [18, 36] in to achieve a constant gap, our reduction time is 2 . Hence, if − Θ˜ 3/4 Θ˜ which 2/53 in the exponent was improved to 1/14.) Unfortunately, 3SAT somehow ends up in DTIME(2 (m ))\DTIME(2 (√m)), their it is unlikely that these lower bounds can be translated to inapprox- result will still hold whereas ours will not even imply constant ratio imability results and the question of whether any polynomial-time inapproximability for DkS. algorithm can achieve subpolynomial approximation ratio for DkS Implications of Our Results. One of the reasons that DkS remains an intriguing open question. has received signi￿cant attention in the approximation algorithm community is due to its connections to many other problems. Most 1.1 Our Results relevant to our work are the problems to which there are reductions In this work, we rule out, under the exponential time hypothesis from DkS that preserve approximation ratios to within some polyno- 2 (i.e. no subexponential time algorithm can solve 3SAT; see Hypoth- mial . These problems include D￿￿￿￿￿￿ A￿￿M￿￿￿￿k￿S￿￿￿￿￿￿￿ [5], c esis 2.1), polynomial-time approximation algorithms for DkS (even 2These are problems whose O(ρ)-approximation gives an O(ρ )-approximation for with perfect completeness) with slightly subpolynomial ratio: DkS for some constant c.

955 Almost-Polynomial Ratio ETH-Hardness of Approximating DkS STOC’17, June 2017, Montreal, Canada

S￿￿￿￿￿￿￿ m￿E￿￿￿ S￿￿￿￿￿￿￿ [17], S￿￿￿￿￿￿ k￿F￿￿￿￿￿ [28] and ￿￿￿ as a polynomial-time reduction from 3SAT to a gap version of 3SAT, ￿￿￿￿￿￿ K￿￿￿￿￿￿￿ [40]. For brevity, we do not de￿ne these problems as stated below. While this perspective is a rather narrow viewpoint here. We refer interested readers to cited sources for their de￿ni- of the theorem that leaves out the fascinating relations between tions and reductions from DkS to respective problems. We also note parameters of PCPs, it will be the most convenient for our purpose. that this list is by no means exhaustive and there are indeed numer- T￿￿￿￿￿￿ 2.3 (PCP T￿￿￿￿￿￿ [6, 7]). For some constant ε > 0, ous other problems with similar known connections to DkS. Our there exists a polynomial-time reduction that takes a 3SAT formula φ results also imply hardness of approximation results with similar and produces a 3SAT formula ϕ such that ratios to DkS for such problems: • (Completeness) if φ is satis￿able, then ϕ is satis￿able, and, C￿￿￿￿￿￿￿￿ 1.3. For some constant c > 0, assuming ETH, there is • ( ) 6 c (Soundness) if φ is unsatis￿able, then val ϕ 1 ε. no polynomial-time n1/(log log n) -approximation algorithm for D￿￿￿￿ − ￿ ￿￿￿ A￿￿M￿￿￿￿k￿S￿￿￿￿￿￿￿, S￿￿￿￿￿￿￿ m￿E￿￿￿ S￿￿￿￿￿￿￿, S￿￿￿￿￿￿ Following the rst proofs of the PCP Theorem, considerable ￿ ￿ k￿F￿￿￿￿￿, ￿￿￿￿￿￿￿￿ K￿￿￿￿￿￿￿. Moreover, for any function f e orts have been made to improve the trade-o s between the pa- 2 rameters in the theorem. One such direction is to try to reduce o(1), there is no polynomial-time nf (n)-approximation algorithm for the size of the PCP, which, in the above formulation, translates to any of these problems, unless Gap-ETH is false. reducing the size of ϕ relative to φ. On this front, it is known that 2 PRELIMINARIES AND NOTATIONS the size of ϕ can be made nearly-linear in the size of φ [12, 20, 39]. For our purpose, we will use Dinur’s PCP Theorem [20], which has ( ) ( ) x ( ) We use exp x and log x to denote e and log2 x respectively. a blow-up of only polylogarithmic in the size of ϕ: polylogn is used as a shorthand for O(logc n) for some absolute constant c. For any set S, P(S) := {T | T S} denotes the power T￿￿￿￿￿￿ 2.4 (D￿￿￿￿’￿ PCP T￿￿￿￿￿￿ [20]). For some constant ✓ set of S. For any non-negative integer t 6 |S|, we use S := {T ε,d > 0, there exists a polynomial-time reduction that takes a 3SAT t 2 P(S)||T | = t} to denote the collection of all subsets of! S" of size t. formula φ with m clauses and produces another 3SAT formula ϕ with Throughout this work, we only concern with simple unweighted m0 = O(m polylogm) clauses such that undirected graphs. Recall that the density of a graph G = (V , E) • (Completeness) if φ is satis￿able, then ϕ is satis￿able, and, on N > 2 vertices is |E|/ N . We say that a graph is α-dense if its • (Soundness) if φ is unsatis￿able, then val(ϕ) 6 1 ε, and, 2 6− density is α. Moreover, for! every" t N, we view each element of • (Bounded Degree) each variable of ϕ appears in d clauses. 2 V t as a t-size ordered multiset of V . (L, R) V t V t is said to be 2 ⇥ Note that Dinur’s PCP, combined with ETH, implies a lower a labelled copy of a t-biclique (or Kt,t ) in G if, for every u L and bound of 2Ω(m/polylogm) on the running time of algorithms that , ( , ) 2 " R, u " and u " E. The number of labelled copies of Kt,t solve the gap version of 3SAT, which is only a factor of O(polylogm) 2 2 ( , ) in G is the number of all such L R ’s. in the exponent o￿ from Gap-ETH. Putting it di￿erently, Gap-ETH 2.1 Exponential Time Hypotheses is closely related to the question of whether a linear size PCP, one where the size blow-up is only constant instead of polylogarithmic, One of our results is based on the exponential time hypothesis exists; its existence would mean that Gap-ETH is implied by ETH. (ETH), a conjecture proposed by Impagliazzo and Paturi [29] which Under the exponential time hypothesis, nearly-linear size PCPs asserts that 3SAT cannot be solved in subexponential time: allow us to start with an instance ϕ of the gap version of 3SAT and H￿￿￿￿￿￿￿￿￿ 2.1 (ETH [29]). No 2o(m)-time algorithm can decide reduce, in subexponential time, to another problem. As long as the o(m/polylogm) whether any 3SAT formula with m clauses3 is satis￿able. time spent in the reduction is 2 , we arrive at a lower bound for the problem. Arguably, Aaronson et al. [1] popularized Another hypothesis used in this work is Gap-ETH, a strength- this method, under the name birthday repetition, by using such a ened version of the ETH, which essentially states that even approx- Ω˜ reduction of size 2 (pm) to prove ETH-hardness for free games and imating 3SAT to some constant ratio takes exponential time: dense CSPs. Without going into any detail now, let us mention that H￿￿￿￿￿￿￿￿￿ 2.2 (G￿￿￿ETH [21, 38]). There exists a constant ε > 0 the name birthday repetition comes from the use of the birthday such that no 2o(m)-time algorithm can, given a 3SAT formula ϕ with paradox in their proof and, since its publication, their work has m clauses4, distinguish between the case where ϕ is satis￿able and the inspired many inapproximability results [9, 15, 16, 19, 38, 43, 45]. case where val(ϕ) 6 1 ε. Here val(ϕ) denote the maximum fraction Our result too is inspired by this line of work and, as we will see − of clauses of ϕ satis￿ed by any assignment. soon, part of our proof also contains a birthday-type paradox.

2.2 Nearly-Linear Size PCPs and 3 THE REDUCTION AND PROOFS OF THE Subexponential Time Reductions MAIN THEOREMS The celebrated PCP Theorem [6, 7], which lies at the heart of virtu- The reduction from the gap version of 3SAT to DkS is simple. Given ally all known NP-hardness of approximation results, can be viewed a 3SAT formula ϕ on n variables x1,...,xn and an integer 1 6 ` 6 n, 5 = , 3In its original form, the running time lower bound is exponential in the number of we construct a graph Gϕ,` (Vϕ,` Eϕ,`) as follows: variables not the number of clauses; however, thanks to the sparsi￿cation lemma of Impagliazzo et al. [30], both versions are equivalent. 5For interested readers, we note that our graph is not the same as the FGLSS graph [23] 4As noted by Dinur [21], a subsampling argument can be used to make the number of the PCP in which the veri￿er reads ` random variables and accepts if no clause is of clauses linear in the number of variables, meaning that the conjecture remains the violated; while this graph has the same vertex set as ours, the edges are di￿erent since same even when m denotes the number of variables. we check that no clause between the two vertices is violated, which is not checked in

956 STOC’17, June 2017, Montreal, Canada Pasin Manurangsi

Its vertex set V ,` contains all partial assignments to ` vari- L, R 2 t,t L = A, R = B denote the set of all • ϕ {( ) K |A( ) A( ) } ables, i.e., each vertex is xi ,bi ,..., xi ,bi where L, R 2 t,t with L = A and R = B. {( 1 1 ) ( ` ` )} ( ) K A( ) A( ) xi ,...,xi are ` distinct variables and bi ,...,bi 0, 1 The number of labelled copies of Kt,t in G ,` can be written as 1 ` 1 ` 2 { } ϕ are the bits assigned to them. , ,..., , We connect two vertices xi1 bi1 xi` bi` and t,t = t,t A, B . (1) • 0 0 0 0 {( ) ( )} |K | |K ( )| xi ,bi ,..., xi ,bi by an edge i￿ the two partial as- , ’✓ {( 1 1 ) ( ` ` )} A B Aϕ signments are consistent (i.e. no variable is assigned 0 in one vertex and 1 in another), and, every clause in ϕ all of To bound t,t , we will prove the following bound on t,t A, B . |K | |K ( )| whose variables are from x ,...,x , x 0 ,...,x 0 is satis- i1 i` i1 i` ￿ed by the partial assignment induced by the two vertices. L￿￿￿￿ 3.3. Let ϕ,n, `,d and ε be as in Theorem 3.1. There exists = n λ > 0 depending only on d and ε such that, for any t 2 N and any Clearly, if val ϕ 1, the ` vertices corresponding to a satisfy- ( ) 2 2t ing assignment induce a clique. Our main technical contribution is −λ` n n ! " A, B ✓ Aϕ , t,t A, B 6 2 / ` . proving that, when val ϕ 6 1 − ε, every n -subgraph is sparse: |K ( )| ⇣ ⌘ ( ) ` ! " T￿￿￿￿￿￿ 3.1. For any d, ε > 0, there! exists" δ > 0 such that, Before we prove the above lemma, let us see how Lemma 3.2 for any 3SAT formula ϕ on n variables such that val ϕ 6 1 − ε and Lemma 3.3 imply Theorem 3.1. ( ) and each variable appears in at most d clauses and for any integer 3 4 n −δ `4 n3 ` 2 n δ,n 2 , any -subgraph of G ,` has density 6 2 . P￿￿￿￿ ￿￿ T￿￿￿￿￿￿ 3.1. Assume w.l.o.g. that λ 6 1. Pick δ = [ / / / ] ` ϕ / ! " λ2 8 and t = 4 λ n2 `2 . From Lemma 3.3 and (1), we have We remark that there is nothing special about 3SAT; we can start / ( / )( / ) with any boolean CSP and end up with a similar result, albeit the 2t 2t − 2 n − 2 n soundness deteriorates as the arity of the CSP grows. However, it is 6 24n 2 λ` n 6 2 λ` n t t,t / ` / ` crucial that the variables are boolean; in fact, Braverman et al. [15] |K | · ✓ ✓ ◆◆ ( ) · ✓ ◆ considered a graph similar to ours for 2CSPs but they were unable where the second inequality comes from our choice of t; note that to achieve subconstant soundness since their variables were not 4n −λ`2 n 6 t is chosen so that the 2 factor is consumed by 2 / from boolean . In particular, there is a non-boolean 2CSP with low value n n Lemma 3.3. Finally, consider any ` -subgraph of Gϕ,`. By the which results in the graph having a biclique of size larger than ` −!λ"`2 n t n 2t (see Appendix A), i.e., one cannot get an inapproximability ratio! " above bound, it contains at most 2 / ` labelled copies ( ) · 3 4 more than two starting from a non-boolean CSP. of Kt,t . Thus, from Lemma 3.2 and from ` > n! /" δ, its density is −λ`2 nt −2δ `4 n3 −δ `4 n3 / Once we have Theorem 3.1, the inapproximability results of at most 2 2 /( ) = 2 2 / 6 2 / as desired. ⇤ DkS (Theorem 1.1 and 1.2) can be easily proved by applying the · · theorem with appropriate choices of `. We defer these proofs to We now move on to the proof of Lemma 3.3. Subsection 3.2. For now, let us turn our attention to the proof of Theorem 3.1. To prove the theorem, we resort to the following lemma due to Alon [2], which states that every dense graph contains P￿￿￿￿ ￿￿ L￿￿￿￿ 3.3. First, notice that if x,b appears in A and ( ) many labelled copies of bicliques: x, b appears in B for some variable x and bit b, then t,t A, B = ( ¬ ) K ( ) ;; this is because, for any L with L = A and R with R = L￿￿￿￿ 3.2 ([2,L￿￿￿￿ 2.1]). Any α-dense graph G on N > 2 2 2 A( ) , A(, ) t 2 2t B, there exist u L and % R that contain x b and x b vertices has at least α 2 N labelled copies of Kt,t for all t 2 N. ( ) ( ¬ ) ( / ) respectively, meaning that there is no edge between u and % and, Equipped with Lemma 3.2, our proof strategy is to bound the thus, L, R < t,t A, B . Hence, from now on, we can assume that, if ( ) K ( ) number of labelled copies of K in G where t is to be chosen x,b appears in one of A, B, then the other does not contain x, b . t,t ϕ,` ( ) ( ¬ ) later. To argue this, we will need some additional notations: Observe that this implies that, for each variable x, its assignments 7 First, let A := x1, 0 , x1, 1 ,..., xn, 0 , xn, 1 be the can appear in A and B at most two times in total. This in turn • ϕ {( ) ( ) ( ) ( )} set of all single-variable partial assignments. Observe that implies that A + B 6 2n. A | | | | n 2t V ✓ ϕ , i.e., each u 2 V is a subset of A of size `. Let us now argue that t,t A, B 6 ` ; while this is not ϕ,` ` ϕ,` ϕ |K ( )| t the bound we are looking for yet, it will serve as a basis for our Let :! Vϕ",` ! P Aϕ be a “￿attening” function that, ! " • A ( ) t ( ) argument later. For every L, R 2 t,t A, B , observe that, since on input T 2 Vϕ,` , outputs the set of all single-variable ( ) K ( ) ( ) = = 2 A t 2 B t partial assignments that appear in at least one vertex in T . L A and R B, we have L ` and R ` . This A( ) A( ) A t B t In other words, when each vertex u is viewed as a subset implies that t,t A, B ✓ ⇥ . Hence,! " ! " K ( ) ` ` of Aϕ , we can write T simply as u 2T u. ! " ! " = A( ) t t t t Let t,t : L, R 2 Vϕ,` ⇥ Vϕ,`– ∀u 2 L, ∀% 2 A B • K {( ) ( ) ( ) | t,t A, B 6 | | | | . (2) R,u , %^ u,% 2 E ,` denote the set of all labelled copies ` ` ( ) ϕ } |K ( )| ✓ ◆ ✓ ◆ of Kt,t in G ,` and, for each A, B ✓ A , let t,t A, B := ϕ ϕ K ( ) the FGLSS graph. It is possible to modify our proof to make it work for this FGLSS graph. However, the soundness guarantee for the FGLSS graph is worse. 7This is where we use the fact that the variables are boolean. For non-boolean CSPs, 6Any satis￿able boolean 2CSP is solvable in polynomial time so one cannot start with each variable x can appear more than two times in one of A or B alone, which can a boolean 2CSP either. indeed be problematic (see Appendix A).

957 Almost-Polynomial Ratio ETH-Hardness of Approximating DkS STOC’17, June 2017, Montreal, Canada

A B U Moreover, | ` | | ` | can be further bounded as random element of r does not contain both elements of any pair in 2 ! "! " −! P" r P is at most exp 4q| U| 2 . `− A B 1 1 ⇣ | | ⌘ | | | | = A − i B − i We are now ready to formalize the above intuition and ￿nish ✓ ` ◆✓ ` ◆ `! 2 (| | )(| | ) the proof of Lemma 3.3. For the sake of convenience, denote the ( ) ÷i=0 `−1 2 sets of good, bad and terrible variables by X", Xb and Xt respec- 1 A + B = = From AM-GM Inequality 6 | | | | − i tively. Moreover, let β : ε 100d and pick λ min − log 1 − ` 2 2 /( ) { ( ( ) ! ÷= ✓ ◆ β 2 , β 64, ε 384d .Tore￿ne the bound on the size of t,t A, B , ( ) i 0 / ) / /( )} K ( ) n 2 consider the following three cases: From from A + B 6 2n 6 (3) (1) Xt > βn. Since each x ∈ Xt contributes at most one to ( | | | | ) ✓`◆ | | A + B , A + B 6 1−β 2 2n . Hence, we can improve (3) | | | | | | | | ( /2)( ) A B 6 1−β 2 n n 2t to | ` | | ` | ( `/ ) . Thus, t,t A, B is bounded Combining (2) and (3) indeed yields t,t A, B 6 . |K ( )| |K ( )| ` above! by"! " ! " ! " A Inequality (2) is very crude; we include all elements of ` and 2t 2t 2t B 1 − β 2 n ` n −λ`2 n n ` as candidates for vertices in L and R respectively. However,! " ( / ) 6 1 − β 2 6 2 / A B ✓ ` ◆ ✓( / ) ✓`◆◆ ✓ ✓`◆◆ as! " we will see soon, only tiny fraction of elements of ` , ` can where the last inequality comes from λ 6 − log 1 − β 2 actually appear in L, R when L, R ∈ t,t A, B . To argue! " this,! " let ( / ) ( ) K ( ) and ` > `2 n. us categorize the variables into three groups: / x is terrible i￿ its assignments appear at most once in total > ∈ • (2) Xb βn. Since each x Xb appears either in A or B, in A and B (i.e. x, 0 , x, 1 ∩A + x, 0 , x, 1 ∩B 6 1). | | |{( ) ( )} | |{( ) ( )} | one of A and B must contain assignments to at least β 2 n x is good i￿, for some b ∈ 0, 1 , x,b ∈ A ∩ B. Note that ( / ) • { } ( ) variables in Xb . Assume w.l.o.g. that A satis￿es this prop- this implies that x, b < A ∪ B. L ∈ ( ¬ ) erty. Let X be the set of all x Xb whose assignments x is bad i￿ either x, 0 , x, 1 ⊆ A or x, 0 , x, 1 ⊆ B. b • {( ) ( )} {( ) ( )} appear in A. Note that X L > β 2 n. The next and last step of the proof is where birthday-type para- | b | ( / ) ∈ A doxes come in. Before we continue, let us brie￿y demonstrate the Observe that an element u ` is not a valid vertex ideas behind this step by considering the following extreme cases: if it contains both x, 0 and x, 1! for" some x ∈ X L.We ( ) ( ) b If all variables are terrible, then A + B 6 n and (3) can invoke Proposition 3.4 with U = A, P = x, 0 , x, 1 • | | | | L {{( ) ( )} | be immediately tightened. x ∈ X ,q = 1 and r = `, which implies that a random b } If all variables are bad, assume w.l.o.g. that, for at least element of A does not contain any prohibited pairs in P • ` , , , ⊆ L 2 half of variables x’s, x 0 x 1 A. Consider a ran- ! " X ` β 2 n`2 A {( ) ( )} with probability at most exp − | b | 6 exp − ( / ) , dom element u of ` . Since u is a set of random ` distinct ✓ 4 A 2 ◆ 4 2n 2 2 | | ⇣ ( ) ⌘ elements of A, there! " will, in expectation, be Ω ` n vari- −2λ`2 n 6 ( / ) which is at most 2 / because λ β 64. In other ables x’s with x, 0 , x, 1 ∈ u. However, the presence of −2λ`2 n / A ( ) ( ) words, at most 2 / fraction of elements of are such x’s means that u is not a valid vertex. Moreover, it ` valid vertices. This gives us the following upper bound! " on is not hard to turn this into the following probabilistic , A, B : −Ω `2 n t t statement: with probability at most 2 ( / ), u contains |K ( )| t t 3 2t at most one of x, 0 , x, 1 for every variable x. In other −2λ`2 n A B ( ) −λ`2 n n 2 / | | | | 6 2 / −Ω( `2 n) ( ) A ✓ · ✓ ` ◆◆ · ✓ ` ◆ ✓ ✓`◆◆ words, only 2 ( / ) fraction of elements of ` are valid , vertices, which yields the desired bound on ! t,"t A B . (3) Xt < βn and X < βn. In this case, X" > 1 − 2β n. |K ( )| | | | b | | | ( ) If all variables are good, then A = B is simply an assignment Let S denote the set of clauses whose variables all lie in • 6 − to all the variables. Since val ϕ 1 ε, at least εm clauses X". Since each variable appears in at most d clauses, S > ( ) | | are unsatis￿ed by this assignment. As we will argue below, m− 2βn d > 1−ε 2 m where the second inequality comes A ( ) ( / ) every element of ` that contains two variables from some from our choice of β and from m > n 3. ∈ / unsatis￿ed clause! cannot" be in L for any L, R t,t A, B . Consider the partial assignment f : X" → 0, 1 in- ( ) K ( ) { } This means that there are Θε m > Ωε n prohibited pairs duced by A and B, i.e., f x = b i￿ x,b ∈ A, B. Since ( ) ( ) ( ) ( ) of variables that cannot appear together. Again, similar val ϕ 6 1 − ε, the number of clauses in S satis￿ed by f ( ) to the previous case, it is not hard to argue that only is at most 1 − ε m. Hence, at least εm 2 clauses in S are 2 −Ωε,d ` n A ( ) / 2 ( / ) fraction of elements of ` can be candidates unsatis￿ed by f . Denote the set of such clauses by SUNSAT. for vertices of L. ! " Fix a clause C ∈ SUNSAT and let x,% be two di￿erent To turn this intuition into a bound on t,t A, B , we need the variables in C. We claim that x,% cannot appear together |K ( )| following inequality. Its proof is straightforward and is deferred to in any vertex of L for any L, R ∈ t,t A, B . Suppose for ( ) K ( ) Subsection 3.1. the sake of contradiction that x, f x and %, f % both ( ( )) ( ( )) appear in u ∈ L for some L, R ∈ t,t A, B . Let z ∈ X" ￿￿￿￿￿￿￿￿￿￿ ⊆ U 8 ( ) K ( ) P 3.4. Let U be any set and P 2 be any set of be another variable in C. Since z, f z ∈ B, some vertex pairs of elements of U such that each element of U appears in at most ( ( )) ! " 8If C contains two variables, let z = x. Note that we can assume w.l.o.g. that C q pairs. For any positive integer 2 6 r 6 U , the probability that a | | contains at least two variables.

958 STOC’17, June 2017, Montreal, Canada Pasin Manurangsi

! R contains z, f z . Thus, there is no edge between u P￿￿￿￿ ￿￿ T￿￿￿￿￿￿ 1.1. For any 3SAT formula φ withm clauses, 2 ( ( )) and ! in G ,`, which contradicts with L, R t,t . use Theorem 2.4 to produce ϕ with m = O m polylogm clauses ϕ ( ) 2 K 0 ( ) We can now appeal to Proposition 3.4 with U = A, such that each variable appears in at most d clauses. Let ζ be a = = ζ 2 q 2d, r ` and P be the prohibited pairs described above. constant such that m0 = O m log m and let ` = m log m. Let us P `2 ( = )n / 6 consider the graph Gϕ,` with k where n is the number of This implies that with probability at most exp 8|d |A 2 ` ⇣− | | ⌘ variables of ϕ. Let N be the number of vertices of G . Observe ε`2 A # $ ϕ,` exp 192dn , a random element of ` contains no pro- = ` n 2` O ` = O ` logm0 = o m − that N 2 ` 6 n 6 m0 ( ) 2 ( ) 2 ( ). ⇣ ⌘ # $ 2 ( ) hibited pair from P. In other words, at most exp ε` If φ is satis# $￿able, ϕ is also satis￿able and it is obvious that − 192dn ￿ A ⇣ ⌘ Gϕ,` contains an induced k-clique. Otherwise, If φ is unsatis able, fraction of elements of ` can be candidates for each ele- val ϕ 6 1 ε. From Theorem 3.1, any k-subgraph of Gϕ,` has den- ment of L for L, R #t,t$ A, B . This gives the following ( ) − + + ( ) 2 K ( ) Ω `4 n3 6 Ω m log3ζ 8 m = Ω 1 log log N 3ζ 8 sharpened upper bound on t,t A, B : sity at most 2− ( / ) 2− ( / ) N − ( /( ) ), |K ( )| 1 log log N 3ζ +9 ￿ t t 2t which is at most N − /( ) when m is su ciently large. `2 3 2 ε A B ( ) ε` 384dn n Hence, if there is a polynomial-time algorithm that can distinguish exp | | | | 6 2− /( ) . ✓ ✓− 192dn ◆ · ✓ ` ◆◆ · ✓ ` ◆ ✓ · ✓`◆◆ between the two cases in Theorem 1.1 when c = 3ζ + 9, then there also exists an algorithm that solves 3SAT in time 2o m , contradict- Since we picked λ 6 ε 384d , t,t A, B is once again ( ) /( )2t|K ( )| ing with ETH. ⇤ λ`2 n n bounded above by 2− / ` . ⇣ ⌘ # $ 2 2t The proof of Theorem 1.2 is even simpler since, under Gap-ETH, , 6 λ` n n In all three cases, we have t,t A B 2− / ` , complet- we have the gap version of 3SAT to begin with. Hence, we can |K ( )| ⇣ ⌘ ing the proof of Lemma 3.3. # $ ⇤ directly apply Theorem 3.1 without going through Dinur’s PCP: 3.1 Proof of Proposition 3.4 P￿￿￿￿ ￿￿ T￿￿￿￿￿￿ 1.2. Let ϕ be any 3SAT formula withm clauses ￿￿￿￿ ￿￿ ￿￿￿￿￿￿￿￿￿￿ ￿ such that each variable appears in O 1 clauses9. Let ` = m 5 f m P P 3.4. We rst construct P 0 P such that ( ) ( ) ✓ and consider the graph G with k = n where n is the numberp each element of U appears in at most one pair in P 0 as follows. Start ϕ,` ` ` n out by marking every pair in P as active and, as long as there are of variables of ϕ. The number of vertices# $ N of Gϕ,` is 2 ` 6 active pairs left, include one in P and mark every pair that shares ` 5 # $ 0 2` en 6 2O ` log m ` = 2O mpf m log 1 f m = 2o m where an element of U with this pair as inactive. Since each element of U ` ( ( / )) ( ( ) ( / ( ))) ( ) the⇣ last⌘ inequality follows from f o 1 . appears in at most q pairs in P, we mark at most 2q pairs as inactive 2 ( ) > The completeness is again obvious. For the soundness, if val ϕ 6 per each inclusion. This implies that P 0 P 2q . ( ) = | | | |/( ) 1 ε, from Theorem 3.1, any k-subgraph of G ,` has density at most Suppose that P 0 a1,b1 ,..., a P ,b P . Let u be a ran- ϕ 0 0 −Ω 4 3 Ω 4 5 Ω 4 5 U {{ } { | | | | }} ` n mf m / f m / 10 dom element of . For each i = 1,..., P , we have 2− ( / ) 6 2− ( ( ) ) 6 N − ( ( ) ), which is at most r | 0| f N ￿ # $ N − ( ) when m is su ciently large. Hence, if there is a polynomial- U 2 2 2 | r |−2 r r time algorithm that can distinguish between the two cases in The- Pr ai ,bi * u = 1 − 6 1 6 exp . [{ } ] − # U $ − 2 U 2 ✓− 2 U 2 ◆ orem 1.2, then there also exists an algorithm that solves the gap | r | o m | | | | version of 3SAT in time 2 ( ), contradicting with Gap-ETH. ⇤ If u does not contain both# elements$ of any pairs in P, it does not contain both elements of any pairs in P 0. The probability of the 4 CONCLUSION AND OPEN QUESTIONS latter can be written as In this work, we provide a subexponential time reduction from the

P 0 P 0 i 1 gap version of 3SAT to DkS and prove that it establishes an almost- | | | | − Pr ai ,bi * u = Pr ai ,bi * u/ aj ,bj * u . polynomial ratio hardness of approximation of the latter under ETH 2 { } 3 2{ } / { } 3 6€i=1 7 ÷i=1 6 / €j=1 7 and Gap-ETH. Even with our results, however, approximability of 6 7 6 / 7 6 7 6 / 7 DkS still remains wide open. Namely, it is still not known whether In addition,6 since a1,b7 1,...,a P6 ,b P are distinct,/ it is not7 hard 4 5 | 40 | | 0 | 5 it is NP-hard to approximate DkS to within some constant factor, i 1 to see that Pr ai ,bi * u = aj ,bj * u 6 Pr ai ,bi * u . and, no polynomial ratio hardness of approximation is yet known. { } j−1{ } [{ } ] h / i Although our results appear to almost resolve the second ques- / ” P 0 , Hence, we can upper bound/ Pr i|=1| ai bi * u by h { } i tion, it still seems out of reach with our current knowledge of ” hardness of approximation. In particular, to achieve a polynomial P 0 2 P 0 2 | | r | | P r ratio hardness for DkS, it is plausible that one has to prove a long- Pr ai ,bi * u 6 exp 6 exp | | , [{ } ] ✓ ✓− 2 U 2 ◆◆ ✓− 4q U 2 ◆ standing conjecture called the sliding scale conjecture (SSC) [11]. In ÷i=1 | | | | short, SSC essentially states that L￿￿￿￿ C￿￿￿￿, a problem used as ⇤ completing the proof of Proposition 3.4. starting points of almost all NP-hardness of approximation results, is NP-hard to approximate to within some polynomial ratio. Note 3.2 Proofs of Inapproximability Results of DkS here that polynomial ratio hardness for L￿￿￿￿ C￿￿￿￿ is not even In this subsection, we prove Theorem 1.1 and 1.2. The proof of Theorem 1.1 is simply by combining Dinur’s PCP Theorem and 9We can assume w.l.o.g. that each variable appears in at most O 1 clauses [38, p.21]. 10 = ( ) = Assume w.l.o.g. that f is decreasing; otherwise take f 0 m sup f m0 instead. Theorem 3.1 with ` m polylogm, as stated below. ( ) > ( ) / m0 m

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