Inapproximability of Maximum Acyclic Subgraph

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

VENKATESAN GURUSWAMI ∗ RAJSEKAR MANOKARAN † PRASAD RAGHAVENDRA ‡ University of Washington Princeton University University of Washington [email protected] [email protected] [email protected]

Abstract 1 Introduction

Given a directed acyclic graph G, one can efficiently or- We prove that approximating the MAX ACYCLIC SUB- der (“topological sort”) its vertices so that all edges go for- GRAPH problem within a factor better than 1/2 is Unique ward from a lower ranked vertex to a higher ranked vertex. Games hard. Specifically, for every constant ε > 0 the fol- But what if a few, say fraction ε, of edges of G are reversed? lowing holds: given a directed graph G that has an acyclic Can we detect these “errors” and find an ordering with few subgraph consisting of a fraction (1 − ε) of its edges, if one back edges? Formally, given a directed graph whose ver- can efficiently find an acyclic subgraph of G with more than tices admit an ordering with many, i.e., 1 − ε fraction, for- (1/2 + ε) of its edges, then the UGC is false. Note that ward edges, can we find a good ordering with fraction α it is trivial to find an acyclic subgraph with 1/2 the edges, of forward edges (for some α → 1)? This is equivalent to by taking either the forward or backward edges in an ar- finding a subgraph of G that is acyclic and has many edges, bitrary ordering of the vertices of G. The existence of a and hence this problem is called the MAX ACYCLIC SUB- ρ-approximation algorithm for ρ > 1/2 has been a basic GRAPH (MAS) problem. open problem for a while. It is trivial to find an ordering with fraction 1/2 of for- Our result is the first tight inapproximability result for ward edges: take the better of an arbitrary ordering and its an ordering problem. The starting point of our reduction reverse. This gives a factor 1/2 approximation algorithm is a directed acyclic subgraph (DAG) in which every cut for MAX ACYCLIC SUBGRAPH. (This is also achieved by is nearly-balanced in the sense that the number of forward picking a random ordering of the vertices.) Despite much and backward edges crossing the cut are nearly equal; such effort, no efficient ρ-approximation algorithm for a constant DAGs were constructed in [2]. Using this, we are able to ρ > 1/2 has been found for MAX ACYCLIC SUBGRAPH. study MAX ACYCLIC SUBGRAPH, which is a constraint The existence of such an algorithm has been a longstanding satisfaction problem (CSP) over an unbounded domain, by and central open problem in the theory of approximation al- relating it to a proxy CSP over a bounded domain. The gorithms. In this work, we prove a strong hardness result latter is then amenable to powerful techniques based on the that rules out the existence of such an approximation algo- invariance principle [12, 17]. rithm assuming the Unique Games conjecture. Our main result is the following. Our results also give a super-constant factor inapproximability result for the MIN FEEDBACK ARC SET problem. Theorem 1.1. Conditioned on the Unique Games conjec- Using our reductions, we also obtain SDP integrality gaps ture, the following holds for every constant γ > 0. Given a for both the problems. directed graph G with m edges, it is NP-hard to distinguish between the following two cases: 1. There is an ordering of the vertices of G with at least (1 − γ)m forward edges (or equivalently, G has an acyclic subgraph with at least (1 − γ)m edges). ∗Work done while on leave at the Institute for Advanced Study, School of Mathematics, Princeton, NJ 08540. Research supported in part by a 2. For every ordering of the vertices of G, there are at Packard Fellowship, and NSF grant CCR-0324906 to The IAS most (1/2 + γ)m forward edges (or equivalently, ev- † supported by NSF grants MSPA-MCS 0528414, and ITR 0205594 ery subgraph of G with more than (1/2 + γ)m edges ‡Work done while the author was visiting Princeton University. Sup- ported by NSF grant CCF-0343672 contains a directed cycle).

1 To the best of our knowledge, the above is the ular, an instance showing tightness of its analysis from [2], first tight hardness of approximation result for an order- play a crucial role in our work. ing/permutation problem. As an immediate consequence, 1.2 Organization we obtain the following hardness result for the complemen- We begin with an outline of the key ideas of the proof in tary problem of MIN FEEDBACK ARC SET, where the ob- Section 2. In Section 3, we review the definitions of influ- jective is to minimize the number of back edges. ences, noise operators and restate the unique games conjec- Corollary 1.2. Conditioned on the Unique Games con- ture. The groundwork for the reduction is laid in Section 4 jecture, for every C > 0, it is NP -hard to find a C- and Section 5, where we define influences for orderings, and approximation to the MIN FEEDBACK ARC SET problem. multiscale gap instances respectively. We present the dictatorship test in Section 6, and convert it to a UG hardness Combining the unique game integrality gap instance of result in Section 7. Finally, SDP integrality gaps for MAX Khot-Vishnoi [9] with the reduction, we obtain SDP inte- ACYCLIC SUBGRAPH are presented in Section 8. grality gaps for MAX ACYCLIC SUBGRAPH problem. Our integrality gap instances also apply to a related SDP relax- 2 Proof Overview ation studied by Newman [15]. This SDP relaxation was In this section, we outline the central ideas of the proof. shown to obtain an approximation better than half on ran- To keep the description concise, we will set up some ba- dom graphs which were previously used to obtain integral- sic notation. For sake of brevity, let us denote [m] = ity gaps for a natural linear program [14]. {1, . . . , m}. Given an ordering O of the vertices of a di- 1.1 Related work rected graph G = (V,E), let Val(O) refer to the fraction of the edges E that are oriented correctly in O. MAX ACYCLIC SUBGRAPH is a classic optimization problem, figuring in Karp’s early list of NP-hard prob- At the heart of all Unique Games based hardness results lies a dictatorship testing result for an appropriate class of lems [6]; the problem remains NP-hard on graphs with max- R imum degree 3, when the in-degree plus out-degree of any functions. A function F :[m] → [m] is said to be a dictator if F(x) = x for some fixed i. A dictatorship test vertex is at most 3.MAX ACYCLIC SUBGRAPH is also i (DICT) is a randomized algorithm such that, given a func- complete for the class of permutation optimization prob- R lems, MAX SNP[π], defined in [16], that can be approxi- tion F :[m] → [m], it makes a few queries to the values of F and distinguishes between whether F is a dictator mated within a constant factor. It is shown in [14] that MAX or far from every dictator. While Completeness of the test ACYCLIC SUBGRAPH is NP-hard to approximate within a factor greater than 65 . refers to the probability of acceptance of a dictator function, 66 Soundness is the maximum probability of acceptance of a Turning to algorithmic results, the problem is known function far from a dictator. The approximation problem to be efficiently solvable on planar graphs [10, 5] and re- one is showing UG hardness for determines the nature of ducible flow graphs [18]. Berger and Shor [1] gave a the dictatorship test needed for the purpose. polynomial time algorithm with approximation ratio 1/2 + √ Now let us turn to the specific problem at hand : MAX Ω(1/ dmax) where dmax is the maximum vertex degree in ACYCLIC SUBGRAPH. Designing the appropriate dictator- the graph. When d = 3, Newman [14] gave a factor 8/9 max ship test for this problem amounts to the following: Con- approximation algorithm. struct a directed graph over the set of vertices V = [m]R The complementary objective of minimizing the num- such that : ber of back edges, or equivalently deleting the minimum number of edges in order to make the graph a DAG, leads • For a Dictator ordering O of V , Val(O) ≈ 1 to the MIN FEEDBACK ARC SET (FAS) problem. This problem admits a factor O(log n log log n) approximation • For any ordering O which is far from a dictator, 1 algorithm [19] based on bounding the integrality gap of the Val(O) ≈ 2 . natural covering linear program for FAS; see also [3]. Us- Unlike the case of functions, it is unclear as to what is the ing this algorithm, one can get an approximation ratio of right notion of Dictators for orderings. For every ordering 1 +Ω(1/(log n log log n)) for MAX ACYCLIC SUBGRAPH. 2 O of [m]R, define m2R functions F [p,q] :[m]R → {0, 1} Recently, Charikar, Makarychev, and Makarychev [2] as follows: gave a factor (1/2 + Ω(1/ log n))-approximation algorithm ( for MAX ACYCLIC SUBGRAPH, where n is the number 1 if p O(x) q F [p,q](x) = 6 6 of vertices. In fact, their algorithm is stronger: given a 0 otherwise digraph with an acyclic subgraph consisting of a fraction (1/2+δ) of edges, it finds a subgraph with at least a fraction The ith coordinate is said to be influential if it has a large in- (1/2+Ω(δ/ log n)) of edges. This algorithm and, in partic- fluence (> τ) on any of the functions F [p,q]. Here influence

2 refers to the natural notion of influence for real valued func- On the one hand, the t-Order problem is a CSP over tions on [m]R (see Section 3). An ordering O is said to be a fixed domain that is similar to MAS. However, to the τ-pseudorandom (far from a dictator) if it has no influential best of our knowledge, for the t-Order problem, there are coordinates (> τ). For this notion to be useful, it is nec- no known SDP gaps, which constitute the starting point essary that a given ordering O does not have too many in- for results in [17]. For any fixed constant t, Charikar, fluential coordinates. Towards this, in Lemma 4.3 we show Makarychev and Makarychev [2] construct directed acyclic that the number of influential coordinates is bounded (af- graphs (i.e., with value of the best ordering equal to 1), 1 ter certain smoothening). Further this notion of influence while the value of any t-ordering of G is close to 2 . For is well suited to deal with orderings of multiple long codes the rest of the discussion, let us fix one such graph G on m instead of one - a crucial requirement in translating dicta- vertices. Notice that the graph G does not serve as SDP gap torship tests to UG hardness. example for either the MAS or the t-Order problem. Armed with the notion of influential coordinates, we ob- As the graph G has only m vertices, and an ordering of tain a directed graph on [m]R (a dictatorship test) for which value ≈ 1, it has a good t-ordering for t = m. Viewing the following holds: G as an instance of the m-Order CSP (corresponding to Theorem 2.1. (Soundness) If O is any τ-pseudorandom or- predicate <), we obtain a directed graph, G, on [m]R. As R 1 dering of [m] , then Val(O) 6 2 + oτ (1). a m-order CSP, the dictator m-orderings yield value ≈ 1 on G. In turn, this implies that the Dictator orderings have This dictatorship test yields tight UG hardness for the value ≈ 1 on G. Turning to the soundness proof, consider MAX ACYCLIC SUBGRAPH problem. Using the Khot- a τ-pseudorandom ordering O. Obtain a t-ordering O∗ by Vishnoi [9] SDP gap instance for unique games, we obtain the following coarsening process : Divide the ordering O an SDP integrality gap for the same. in to t equal blocks, and map the vertices in the ith block Now we describe the design of the dictatorship test in to value i. The crucial observation relating O and O∗ is as greater detail. At the outset, the approach is similar to re- follows: cent work on Constraint Satisfaction Problems(CSPs) [17]. “For a τ-pseudorandom ordering O, Val (O∗) ≈ Fix a constraint satisfaction problem Λ. Starting with an t Val(O).” integrality gap instance Φ for the natural semi-definite pro- ∗ Clearly, Val(O) − Valt(O ) is bounded by the fraction gram for Λ, [17] constructs a dictatorship test DICTΦ. The of edges whose both endpoints fall in the same block, dur- Completeness of DICTΦ is equal to the SDP value SDP(Φ), while the Soundness is close to the integral value INT(Φ). ing the coarsening. We use the Gaussian noise stability Since the result of [17] applies to arbitrary CSPs, a nat- bounds of [12], to bound the fraction of such edges. From the above observation, in order to prove that Val(O) ≈ 1 , ural direction would be to pose the MAX ACYCLIC SUB- 2 it is enough to bound Val (O∗). Notice that O∗ is a solu- GRAPH as a CSP. MAX ACYCLIC SUBGRAPH is fairly sim- t ilar to a CSP, with each vertex being a variable taking values tion to t-order problem - a CSP over finite domain. Conse- quently, the soundness analysis of [17] can be used to show in domain [n] and each directed edge a constraint between ∗ 2 variables. However, the domain, [n], of the CSP is not that Valt(O ) is at most the value of the best t-ordering for G, which is close to 1 . fixed, but grows with input size. We stress here that this is 2 not a superficial distinction but an essential characteristic of Summarizing the key ideas, we define the notion of influ- the problem. For instance, if MAX ACYCLIC SUBGRAPH ential coordinates for orderings, and then use it to construct was reducible to a 2-CSP over a domain of fixed size, then a dictatorship test for orderings. Using gaussian noise sta- we could obtain a approximation ratio better than a random bility bounds, we relate the value of a pseudorandom order- assignment [4]. ing to a related CSP, and then apply techniques from [17]. Towards using techniques from the CSP result, we define 3 Preliminaries the following variant of MAX ACYCLIC SUBGRAPH: For a positive integer t, ∆ denotes the the t dimen- Definition 2.2. A t-ordering of a directed graph G = t sional simplex. We will use boldface letters z to denote (V,E) consists of a map O : V → [t]. The value of a vectors z = (z(1), . . . , z(R)). Let o (1) denote a quantity t-ordering O is given by τ that tends to zero as τ → 0, while keeping all other parameters fixed. For notational convenience, an ordering O Valt(O) = Pr O(u) < O(v) (u,v)∈E of a directed graph G = (V,E) is represented by a map 1 O : V → Z. On the other hand, a t-ordering O consists of a + Pr O(u) = O(v) map O : V → [t]. For a t-ordering O, the map need not be 2 (u,v)∈E injective or surjective, but an ordering O is required to be in- In the t-Order problem, the objective is to find an t-ordering jective. In an ordering(or t-ordering) O, an edge e = (u, v) of the input graph G with maximum value. is a forward edge if O(u) < O(v). For a graph G, the

3 quantities Val(G),Valt(G) refer to the maximum fraction Since the influences of T1−F are low, we can apply The- of forward edges in an ordering,t-ordering respectively. orem 4.4 from [13] to bound the last expression by noise Observation 3.1. For all directed graphs G, and integers stability in gaussian space Γ(1−)(µ). 0 t 6 t , Valt(G) 6 Valt0 (G) 6 Val(G) E[(T1−F)T1−(T1−F)] 6 Γ(1−)(µ) + oτ (1) While the first part of the inequality is trivial, we will elaborate on the latter half. Given a t0-ordering O, construct Using standard estimates (see Theorem B.2 in [13]), 1+/2 a full ordering O∗ by using a random permutation of the Γ(1−)(µ) is bounded by µ for µ small enough com- elements within each of the t0 blocks, while retaining the pared to . Applying a similar bound for G and applying natural order between the blocks. It is easy to see that the Cauchy-Schwartz gives the result: ∗ expected value of the random ordering O is exactly equal q 2 2 to Val(O). E[T1−2F(x)T1−2G(y)] ||T1−2F|| ||T1−2G|| x 6 2 2 1+/2 3.1 Noise Operators and Influences 6 µ + oτ (1) (for µ small enough) Let Ω denote the finite probability space corresponding to the uniform distribution over [m]. Let {χ = 0 Due to space constraints, we defer the proof of the fol- 1, χ1, χ2, . . . , χm−1} be an orthonormal basis for the space R Q (k) lowing simple lemma to the full version. L2(Ω). For σ ∈ [m] , define χσ(z) = k∈[R] χσi (z ). R Every function F :ΩR → R can be expressed as a multilin- Lemma 3.5. Given a function F :[m] → [0, 1], if H = ear polynomial as F(z) = P Fˆ(σ)χ (z). The L norm PR 1 1 σ σ 2 T1−F then k=1 Infk(H) 6 e ln 1/(1−) 6 of F in terms of the coefficients of the multilinear polynomial is ||F||2 = P Fˆ2(σ) 2 σ 3.2 Semidefinite Program Definition 3.2. For a function F :ΩR → R, define P 2 AX Inf (F) = E [Var (k) [F]] = Fˆ (σ). We use the following natural SDP relaxation of the M k z z σk6=0 ACYCLIC SUBGRAPH problem. Given a directed graph Here Varz(k) [F] denotes the variance of F(z) over the choice of the kth coordinate z(k). G = (V,E) with |V | = n, the program has n variables {u , . . . , u } for each vertex u ∈ V . In the intended Definition 3.3. For a function F :ΩR → , define the 1 n R solution, the variable u = 1 and u = 0 for all j 6= i if and function T F as follows: i j ρ only if u is assigned position i. X |σ| TρF(z) = E[F(z˜) | z] = ρ Fˆ(σ)χσ(z) σ∈[m]R h X 1 X i Max. E ui · vj + ui · vi (MAS SDP) where each coordinate z˜(k) of z˜ = (˜z(1),..., z˜(R)) is equal e 2 i 0, there exists small i∈[n] i∈[n] enough µ such that the following holds : 1+/2 E [T1−2F(x)T1−2G(y)] 6 µ + oτ (1) x,y 3.3 Unique Games Proof. The lemma essentially follows from the Majority is Stablest theorem (see Theorem 4.4 in [13]). We bound each Definition 3.6. An instance of Unique Games represented factor individually as follows: as Υ = (A ∪ B,E, Π, [R]), consists of a bipartite graph over node sets A,B with the edges E between them. Also 2 X 2|σ| ˆ2 ||T1−2F||2 = (1 − 2) F (σ) part of the instance is a set of labels [R] = {1,...,R}, σ∈[k]R and a set of permutations πab :[R] → [R] for each edge X |σ| ˆ 2|σ| ˆ e = (a, b) ∈ E. An assignment Λ of labels to vertices is 6 (1 − ) F(σ)(1 − ) F(σ) said to satisfy an edge e = (a, b), if πab(Λ(a)) = Λ(b). The σ∈[k]R objective is to find an assignment Λ of labels that satisfies 6 E[(T1−F)(x)T1−(T1−F)(x)] the maximum number of edges.

4 τ R For a vertex a ∈ A ∪ B, we shall use N(a) to denote For integers p, q, δ1, δ2 such that |δi| < 8 m , let f = [p,q] [p+δ1,q+δ2] its neighborhood. For the sake of convenience, we shall use T1−F and g = T1−F . Now, the following version of the Unique Games Conjecture [8] which is equivalent to the original conjecture [7]. 2 [p,q] [p+δ1,q+δ2] 2 Infk(f − g) 6 ||f − g||2 6 ||F − F ||2 Conjecture 3.7. (Unique Games Conjecture [8, 7]) For all = Pr[F [p,q](z) 6= F [p+δ1,q+δ2](z)] τ/4 z 6 constants δ > 0, there exists large enough constant R such 2 2 2 that given a bipartite unique games instance Υ = (A ∪ Hence, using a 6 2(b + (a − b) ), we get: B,E, Π = {π :[R] → [R]: e ∈ E}, [R]) with number of e   labels R, it is NP-hard to distinguish between the following 2 X 2 X ˆ two cases: Infk(f) 6 2  gˆ (σ) + f(σ) − gˆ(σ)  σk6=0 σk6=0 • (1 − δ)-satisfiable instances: There exists an assignment Λ of labels such that for 1 − δ fraction of vertices 6 2Infk(g) + τ/2 a ∈ A, all the edges incident at a are satisfied. Thus, if Infk(f) > τ, then Infk(g) > τ/4. It is easy to • Instances that are not δ-satisfiable: No assignment see that there is a set N = {F [p,q]} of size at most 100/τ 2 satisfies more than a δ-fraction of the constraints Π. such that for every F [p,q] there is a F [r,s] ∈ N such that τmR max |p − r|, |q − s| < 8 . Further, by Lemma 3.5, the [p,q] 4 4 Orderings functions T1−F have at most τ coordinates with influence more than τ/4. Hence, |S (O)| 400 . In this section, we develop the notions of influences for τ 6 τ 3 orderings and prove some basic results about them. Claim 4.4. For any τ-pseudorandom ordering O of [m]R, ∗ Definition 4.1. Given an ordering O of vertices V , its t- its t-coarsening O is also τ-pseudorandom. ∗ [·,·] coarsening is a t-ordering O obtained by dividing O into Proof. Since the functions {FO∗ } are a subset of the func- t-contiguous blocks, and assigning label i to vertices in the [·,·] ∗ tions {FO }, Sτ (O ) ⊆ Sτ (O). ith block. Formally, if M = |V |/t then

j|{v|O(v) < O(u)}|k O∗(u) = + 1 5 Multiscale Gap Instances M In this section, we will construct acyclic directed graphs R For an ordering O of points in [m] , Deﬁne functions with no good t-ordering. These graphs will be crucial in [p,q] R FO :[m] → {0, 1} for integers p, q as follows: designing the dictatorship test (Section 6). ( Deﬁnition 5.1. For η > 0 and a positive integer t, a (η, t)- [p,q] 1 if O(x) ∈ [p, q] F (x) = Multiscale Gap instance is a weighted directed graph G = O 0 otherwise (V,E) with the following properties:

[p,q] [p,q] 1 We will omit the subscript and write F instead of FO , • Val(G) = 1 and Valt(G) 6 2 + η when it is clear. • There exists a solution {ui | u ∈ V, 1 6 i 6 |V |} to Definition 4.2. For an ordering O of [m]R, define the set of SDP with objective value at least 1 − η such that for 2 1 all u, v ∈ V and 1 i, j |V |, we have |ui| = . influential coordinates Sτ (O) as follows: 6 6 |V |

[p,q] The cut norm of a directed graph, G, repre- Sτ (O) = {k | Infk(T1−F ) ≥ τ for some p, q ∈ Z} sented by a skew-symmetric matrix W is: ||G||C = P An ordering O is said to be τ-pseudorandom if Sτ (O) is maxxi,yj ∈{0,1} ij xiyjwij. We will need the following empty. theorem from [2] relating the cut norm of a directed graph G to Val(G). Lemma 4.3. (Few Inﬂuential Coordinates) For any order- R 400 Theorem 5.2 (Theorem 3.1, [2]). If a directed graph G on ing O of [m] , we have |Sτ (O)| 6 τ 3 n vertices has a maximum acyclic subgraph with at least a Proof. Although an ordering is an injective map O : 1 δ R + δ fraction of the edges, then, ||G||C > Ω . [m] → Z, the range of O can be compressed to 2 log n R [p,q] {1, . . . , m } without affecting the set of functions {FO } The following lemma and its corollary construct Multi- associated with O. After compression, it is enough to con- scale Gap instances starting from graphs that are the “tight [p,q] R sider functions {FO } only for p, q ∈ {0, . . . , m + 1}. cases” of the above theorem.

5 Lemma 5.3. Given η > 0 and a positive integer t, for ev- Using the multiscale gap instance G, construct a dicta- R ery sufficiently large n, there exists a directed graph G = torship test DICTG on orderings O of [m] as follows: (V,E) on n vertices such that Val(G) = 1 , Valt(G) 6 1 DICTG Test: 2 + η . Proof. Charikar et al (Section 4, [2]) construct a di- • Pick an edge e = (u, v) ∈ E at random from the rected graph, G = (V,E), on n vertices whose cut Multiscale gap instance G. norm is bounded by O (1/ log n). The graph is repre- • Sample ze = {zu, zv} from the product distribution sented by the skew-symmetric matrix W , where w = R (k) (k) (k) ij P , i.e. For each 1 k R, ze = {zu , zv } is Pn π(j−i)k e 6 6 k=1 sin n+1 . It is easy to verify that for every 0 < sampled using the distribution Pe(i, j) = ui · vj. Pn πtk t < n, k=1 sin n+1 > 0. Thus, wij > 0 whenever • Obtain z˜u, z˜v by perturbing each coordinate of zu th i < j, implying that the graph is acyclic (in other words, and zv independently. Specifically, sample the k (k) (k) Val(G) = 1). coordinates z˜u , z˜v as follows: With probability 1 (k) (k) We bound Valt(G) as follows. Let Valt(G) = 2 + δ (1 − 2), z˜u = zu , and with the remaining prob- and let O : V → [t] be the optimal t-ordering. Construct (k) ability z˜u is a new sample from Ω. a graph H on t vertices with a directed edge from O(u) to O(v) for every edge (u, v) ∈ E with O(u) 6= O(v). • Introduce a directed edge z˜u → z˜v. (alternatively Now, using Theorem 5.2, the cut norm of H is bounded test if O(z˜u) < O(z˜v)) δ from below by Ω log t . Moreover, since O is a parti- tion of V , the cut norm of G is at least the cut norm of H. Theorem 6.2. (Soundness Analysis) For every > 0, δ there exists sufficiently large m, t such that : For any τ- Thus, Ω ||H||C ||G||C O (1/ log n)Thus, log t 6 6 6 R pseudorandom ordering O of [m] , δ O log t implying that Val (G) 1 + O log t . 6 log n t 6 2 log n − Val(O) Val (G) + O(t 2 ) + o (1) Choosing n to be a sufficiently gives the required result. 6 t τ where o (1) → 0 as τ → 0 keeping all other parameters For every η > 0 and positive integer t, there τ Corollary 5.4. fixed. exists a Multiscale Gap instance with a corresponding SDP 2 [p,q] R solution {ui|u ∈ V, 1 i |V |} satisfying |ui| = 1/|V | Let F :[m] → {0, 1} denote the functions asso- 6 6 ∗ for all u ∈ V, 1 6 i 6 |V |. ciated with the t-ordering O . For the sake of brevity, we shall write F i for F [i,i]. The result follows from Lemma 6.4 Proof. Let G = (V,E) be the graph obtained by taking and Lemma 6.3 shown below. d1/ηe disjoint copies of the graph guaranteed by Lemma 5.3 and let m = |V |. Note that the graph still satisfies the re- Lemma 6.3. For every > 0, there exists sufficiently large 1 m, t such that : For any τ-pseudorandom ordering O of quired properties: Val(G) = 1, Valt(G) + η. Let O be 6 2 R the ordering of [m] that satisfies every edge of G. Let D de- [m] note the distribution over labellings obtained by shifting O ∗ − Val(O) 6 Valt(O ) + O(t 2 ) + oτ (1) by a random offset cyclically. For every u ∈ V, i ∈ [m], Pr[D(u) = i] = 1/m. Further, every directed edge is where O∗ is the t-coarsening of O. satisfied with probability at least 1 − η. Being a distribu- Proof. As O∗ is a coarsening of O, clearly Val(O) ≥ tion over integral labellings, D gives raise to a set of vec- ∗ Valt(O ). Note that the loss due to coarsening, is be- tors satisfying the constraints in Definition 5.1. G along cause for some edges e = (z, z0) which are oriented cor- with these vectors form the required (η, t)-multiscale gap rectly in O, fall in to same block during coarsening, i.e instance. O∗(z) = O∗(z0). Thus we can write 1 Val(O) Val (O∗) + Pr O∗(z˜ ) = O∗(z˜ ) 6 Dictatorship Test 6 t 2 u v Let G = (V,E) be a (η, t)-multiscale gap instance on m ∗ ∗ vertices, where m is divisible by t. Let {ui|u ∈ V, i ∈ [m]} Pr O (z˜u) = O (z˜v) denote the corresponding SDP solution. Define distribu- X h i i i tions P as follows: = E E E F (z˜u) ·F (z˜v) e e=(u,v) zu,zv z˜u,z˜v i∈[t] Definition 6.1. For an edge e = (u, v) ∈ E in a (η, t)- X h i i i multiscale gap instance G, define the local integral distribu- = E E T1−2Fu(zu) · T1−2Fv(zv) e=(u,v) zu,zv 2 i∈[t] tion Pe over [m] as Pe(i, j) = ui · vj.

6 As O is a t-coarsening of O, for each value i ∈ [t], there SDP solution (Lemma 3.4 in [17]). Due to paucity of space, 1 ∗ are exactly t fraction of z for which O (z) = i. Hence we defer the details of the proof to the full version. i 1 for each i ∈ [t], Ez[Fu(z) = t ]. Further, since the ordering O∗ is τ-pseudorandom, for every k ∈ [R] and The following claim is a restatement of the Lemma 6.4 i R i ∈ [t], Infk(T1−Fa) 6 τ. Hence using Lemma 3.4, in terms of functions F :[m] → ∆t corresponding to for sufficiently large t, the above probability is bounded by pseudorandom t-orderings. −1− − t · t 2 + t · oτ (1) = O(t 2 ) + oτ (1) . R Claim 6.5. For a function F :[m] → ∆t satisfying Lemma 6.4. For every choice of m, t, , and any τ- Infk(T1−F) 6 τ for all k ∈ [R], ∗ R ∗ pseudorandom t-ordering O of [m] , Valt(O ) 6 Valt(G) + oτ (1). h1 X X i E F i(z˜ )F j(z˜ ) + F i(z˜ )F j(z˜ ) 2 u u u u Proof. The t-ordering problem is a CSP over a finite do- i=j i j where the expectation is over the edge e = (u, v), zu, zv, 1 and P (i, j) = 2 otherwise. The t-ordering problem is a z˜u, and z˜v. Generalized CSP(see Definition 3.1, [17]) with the payoff function P . 7 Hardness Reduction For the sake of exposition, let us pretend that t = m. In this case, the vectors {ui|u ∈ V, i ∈ [m]} form a feasi- Let G = (V,E) be a (η, t)-Multiscale gap instance, and ble SDP solution for the t-ordering instance G. Let DICTG denote the dictatorship test obtained by running the reduc- let m = |V |. Further let {ui|u ∈ V, i ∈ [m]} denote the tion of [17] on this SDP solution for t-ordering instance G. corresponding SDP solution. Let Υ = (A ∪ B,E, Π = {π :[R] → [R]|e ∈ E}, [R]) be a bipartite unique games DICTG is an instance of the t-ordering problem, over the e R ∗ instance. Towards constructing a MAS instance G = (V, E) set of vertices [t] .A t-ordering solution O for DICTG R from Υ, we shall introduce a long code for each vertex in corresponds naturally to a function F :[t] → ∆t. Now we make the following observations : B. Specifically, the set of vertices V of the directed graph G is indexed by B × [m]R. • The t-ordering instance DICTG is identical to the dictatorship test described in this section when t = m. Hardness Reduction: • For a τ-pseudorandom t-ordering O∗, for every k ∈ Input : Unique games instance Υ = (A ∪ B,E, Π = [R] and i ∈ [t], the corresponding function F {πe :[R] → [R]|e ∈ E}, [R]) and a (η, t) Multiscale gap i satisfies Infk(T1−F ) 6 τ. In the terminology instance G = (V,E). of [17](Definitions 4.1 and 4.2), this is equivalent Output : Directed graph G = (V, E) with set of vertices : to the function F = (F 1,..., F t) being “(γ, τ)- V = B×[m]R and edges E given by the following verifier: pseudorandom” with γ = 0. • Pick a random vertex a ∈ A. Choose two neighbours • By Corollary 2.2 in [17], for a (γ, τ)-pseudorandom b, b0 ∈ B independently at random. Let π, π0 denote function F, its probability of acceptance on the dicta- the permutations on the edges (a, b) and (a, b0). torship test is at most Val (G) + o (1). t γ,τ • Pick an edge e = (u, v) ∈ E at random from the Hence the above lemma is just a restatement of Corollary Multiscale gap instance G.

2.2 of [17] for the specific generalized CSP : t-Ordering, • Sample ze = {zu, zv} from the product distribution albeit in the language of τ-pseudorandom orderings. R (k) (k) (k) Pe , i.e. For each 1 6 k 6 R, ze = {zu , zv } is Recall that the actual case of interest here satisfies t < sampled using the distribution Pe(i, j) = ui · vj. m. Unfortunately, in this case, a black box application of • Obtain z˜u, z˜v by perturbing each coordinate of zu the result from [17] does not suffice. However, the proof th and zv independently. Specifically, sample the k in [17] can be easily adopted without any new technical (k) (k) ideas. In fact, many of the technical difficulties encountered coordinates z˜u , z˜v as follows: With probability (k) (k) in [17] can be avoided here. For instance, the SDP solu- (1 − 2), z˜u = zu , and with the remaining prob- (k) tion associates with each vertex u, the uniform probability ability z˜u is a new sample from Ω. distribution over {1 . . . m}, unlike [17] where there are sev- 0 0 • Introduce a directed edge (b, π(z˜u)) → (b , π (z˜v)). eral arbitrary probability distributions to deal with. With the value of m fixed, this removes the need for smoothing the

7 Theorem 7.1. For every γ > 0, there exists choice of pa- Pe corresponding to an edge e = (u, v) ∈ E, rameters , η, t, δ such that: (l) (l) X • COMPLETENESS: If Υ is a (1−δ)-satisﬁable instance Pr zu = zv = ui · vj of Unique Games, then there is an ordering O for the i=j graph G with value at least (1−γ). i.e. Val(G) ≥ 1−γ. (l) (l) X Pr zu < zv = ui · vj i

[p,q] With probability at least (1 − δ), the verifier picks a ver- Fa (z) = Pr Oa(b, πb(z)) ∈ [p, q] tex a ∈ A such that the assignment Λ satisfies all the b∈N(a) 0 [p,q] edges (a, b). In this case, for all choices of b, b ∈ N(a), = E [Fb (πb(z))] π(Λ(a)) = Λ(b) and π0(Λ(a)) = Λ(b0). Let us denote b∈N(a) Λ(a) = l. By definition of the m-ordering O, we get −1 Definition 7.2. Define the set of influential coordinates O(b, π(z)) = (π(z))(Λ(b)) = z(π (Λ(b))) = z(l) for all Sτ (Oa) as follows: z ∈ [m]R. Similarly for b0, O(b0, π0(z)) = z(l) for all R [p,q] z ∈ [m] . Thus we get Sτ (Oa) = {k|Infk(T1−Fa ) ≥ τ for some p, q ∈ Z} 1 Val (O) ≥ (1−δ)· Pr z˜(l) =z ˜(l)+Pr z˜(l) < z˜(l) An ordering O is said to be τ-pseudorandom if S (O ) is m 2 u v u v a τ a empty. 2 With probability at least (1 − 2) , for both z˜u and z˜v we (l) (l) (l) (l) Lemma 7.3. For any influential coordinate k ∈ S (O ), have z˜u = zu and z˜v = zv . Further, note that each co- τ a (l) (l) for at least τ fraction of b ∈ N(a), π (k) is influential on ordinate zu , zv is generated according to the local distri- 2 b Ob. More precisely, πb(k) ∈ S (Ob). bution Pe for the edge e = (u, v). For the local distribution τ/2

8 ∗ Proof. As the coordinate k is influential on Oa, there exists Now we shall bound the value of Valt(Oa). In terms of [p,q] [p,q] i ∗ p, q such that Infk(Fa ) ≥ τ. Recall that Fa (z) = the functions Fb, the expression for Valt(Oa) is as follows: [p,q] Eb∈N(a)[Fb (πb(z))]. Using convexity of Inf this im- ∗ h1 X i j plies, Val (O ) = E F (π (z˜ )) ·F 0 (π 0 (z˜ )) t a 2 b b u b b v [p,q] i=j E [Infπb(k)(Fb )] ≥ τ b∈N(a) X i j i + Fb(πb(z˜u)) ·Fb0 (πb0 (z˜v)) i 0, there exists constants t, τ > τ/2 b trary label. 0 such that for any vertex a ∈ A, if Oa is τ-pseudorandom 1 If Val(O) = Ea∈A[Val(Oa)] ≥ 2 + γ is greater than then Val(Oa) 6 Valt(G) + γ/4. 1 2 + γ, then for at least γ/2 fraction of vertices a ∈ A, we have Val(O ) ≥ 1 + γ/2. Let us refer to these vertices a Proof. The proof outline is similar to that of Theorem 6.2. a 2 ∗ as good vertices. From Theorem 7.5, for every good vertex Let Oa denote the t-coarsening of Oa. Then we can write, the order Oa is not τ-pseudorandom. In other words, for every good vertex a, the set Sτ (Oa) is non-empty. Further Val(O ) Val (O∗) a 6 t a by Lemma 7.3 for every label l ∈ Sτ (Oa), for at least 1 ∗ ∗ 0 τ/2 fraction of the neighbours b ∈ N(a), πb(l) belongs + Pr Oa(b, πb(z˜u)) = Oa(b , πb0 (z˜v)) 2 to Sτ/2(Ob). For every such b, the edge (a, b) is satisfied with probability at least 1/|Sτ (Oa)| × 1/|Sτ/2(Ob)|. By ∗ The t-coarsening Oa is obtained by dividing the order Oa in Lemma 4.3 and Lemma 7.4, this probability is at least to t-blocks. Let [p1 + 1, p2], [p2 + 1, p3],..., [pt + 1, pt+1] τ 4/800 × τ 3/3200. Summarizing the argument, the denote the t blocks. For the sake of brevity, let us denote expected fraction of edges satisfied by the labelling Λ is at i [pi+1,pi+1] i [pi+1,pi+1] 2 8 Fa = Fa and Fb = Fb . In this notation, least γ τ /10240000. By a small enough choice of δ, this we can write: yields the required result.

∗ ∗ 0 Pr Oa(b, πb(z˜u)) = Oa(b , πb0 (z˜v)) 8 SDP Integrality Gap X h i i i = E E E F (πb(z˜u)) ·F 0 (πb0 (z˜v)) 0 b b e=(u,v) b,b zu,zv ,z˜u,z˜v i∈[t] In this section, we construct integrality gaps for the MAS X h i i i SDP relaxation using the unique games hardness reduction. = E E E Fa(z˜u) ·Fa(z˜v) e=(u,v) zu,zv z˜u,z˜v i∈[t] Speciﬁcally we show,

X h i i i Theorem 8.1. For any γ > 0, there exists a directed graph = E E T1−2Fa(zu) · T1−2Fa(zv) e=(u,v) zu,zv G such that the value of semi-deﬁnite program (MAS SDP) i∈[t] 1 is at least 1 − γ, while Val(G) 6 2 + γ.

As the ordering Oa is τ-pseudorandom, for every k ∈ [R] The proof uses a bipartite variant of the Khot-Vishnoi [9] i and i ∈ [t], Infk(T1−Fa) 6 τ. Hence by Lemma 3.4, the Unique Games integrality gap instance as in [17, 11]. − above value is less than O(t 2 ) + oτ (1). Speciﬁcally, the following is a direct consequence of [9].

9 Theorem 8.2. [9] For every δ > 0, there exists a UG in- [2] M. Charikar, K. Makarychev, and Y. Makarychev. On the stance, Υ = (A ∪ B,E, Π = {πe :[R] → [R] | e ∈ advantage over random for maximum acyclic subgraph. In k FOCS, pages 625–633. IEEE Computer Society, 2007. E}, [R]) and vectors {Vb } for every b ∈ B, k ∈ [R] such that the following conditions hold : [3] G. Even, J. Naor, B. Schieber, and M. Sudan. Approximat- ing minimum feedback sets and multicuts in directed graphs. • No assignment satisfies more than δ fraction of con- Algorithmica, 20(2):151–174, 1998. straints in Π. [4] J. Hastad.˚ Every 2-csp allows nontrivial approximation. In H. N. Gabow and R. Fagin, editors, STOC, pages 740–746. 0 k l k • For all b, b ∈ B, k, l ∈ [R] , Vb · Vb0 > 0 and Vb · ACM, 2005. l [5] A. Karazanov. On the minimal number of arcs of a digraph Vb = 0. meeting all its directed cutsets. Graph Theory Newsletters, 0 k P l • For all b, b ∈ B, k, l ∈ [R] , Vb · l∈[R] Vb0 = 8, 1979. |Vk|2 and P |Vk|2 = 1 [6] R. M. Karp. Reducibility among combinatorial problems. In b k∈[R] b Complexity of Computer Computations. New York: Plenum, • The SDP value is at least 1 − δ: pages 85–103, 1972. h P π(k) π0(k)i [7] S. Khot. On the power of unique 2-prover 1-round games. E 0 V · V 1 − δ a∈A,b,b ∈B k∈[R] b b0 > In IEEE Conference on Computational Complexity, page 25, 2002. Let G be a (η, t)-multiscale gap instance with m vertices. [8] S. Khot and O. Regev. Vertex cover might be hard to approx- Apply Theorem 8.2, with a sufficiently small δ to obtain a imate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335– k UGC instance Υ and SDP vectors {Vb |b ∈ B, k ∈ [R]} ∪ 349, 2008. {I}. Consider the instance G constructed by running the UG [9] S. Khot and N. K. Vishnoi. The unique games conjecture, hardness reduction in Section 7 on the UG instance Υ. The integrality gap for cut problems and embeddability of neg- R R set of vertices of G is given by B×[m] . Set M = |B|×m ative type metrics into l1. In FOCS, pages 53–62. IEEE and N = |B|. Computer Society, 2005. The program MAS SDP on the instance G contains M [10] C. Lucchesi and D. H. Younger. A minimax theorem for directed graphs. Journal London Math. Society, 17:369–374, vectors {W(b,z)|i ∈ [M]} for each vertex (b, z) ∈ B×[m]R i 1978. and a special vector I denoting the constant 1. Define a [11] R. Manokaran, J. S. Naor, P. Raghavendra, and R. Schwartz. solution to MAS SDP as follows: Set the vector I to be the Sdp gaps and ugc hardness for multiway cut, 0-extension corresponding vector in the instance Υ. Define, and metric labelling. In STOC ’08: Proceedings of the 40th ACM Symposium on Theory of Computing, 2008. (b,z) X k Wi = Vb ∀i ∈ [m], (b, z) ∈ B [12] E. Mossel. Gaussian bounds for noise cor- zk=i relation of functions (arxiv: math/0703683) (b,z) [http://front.math.ucdavis.edu/0703.5683]. To appear Wl = 0 for all other l ∈ [M], (b, z) ∈ B in FOCS, 2008. [13] E. Mossel, R. O’Donnell, and K. Oleszkiewicz. Noise sta- (b,z) It is easy to check that the vectors {Wi } satisfy the con- bility of functions with low influences: invariance and op- straints of MAS SDP and have an SDP value close to 1. On timality. In FOCS, pages 21–30. IEEE Computer Society, the other hand, the soundness analysis in Section 7 implies 2005. 1 [14] A. Newman. Approximating the maximum acyclic sub- that the integral optimum for G is at most 2 +γ. The details of the proof will appear in the full version of the paper. graph. Master’s thesis, MIT, June 2000. [15] A. Newman. Cuts and orderings: On semidefinite re- laxations for the linear ordering problem. In K. Jansen, 9 Acknowledgments S. Khanna, J. D. P. Rolim, and D. Ron, editors, APPROX- RANDOM, volume 3122 of Lecture Notes in Computer Sci- We would like to thank Alantha Newman for pointing ence, pages 195–206. Springer, 2004. out the similarity between Maximum Acyclic Subgraph and [16] C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., CSPs, and suggesting the problem. We would like to thank 43(3):425–440, 1991. Sanjeev Arora for many fruitful discussions. [17] P. Raghavendra. Optimal algorithms and inapproximability results for every csp? In STOC ’08: Proceedings of the 40th References ACM Symposium on Theory of Computing, 2008. [18] V. Ramachandran. Finding a minimum feedback arc set in reducible flow graphs. J. Algorithms, 9:299–313, 1988. [1] B. Berger and P. W. Shor. Tight bounds for the Maxi- [19] P. Seymour. Packing directed circuits fractionally. Combi- J. Algorithms mum Acyclic Subgraph problem. , 25(1):1–18, natorica, 15(2):281–288, 1995. 1997.