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

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1 Beating the Random Ordering is Hard: Inapproximaility of Maximum Acyclic Sugraph VENKATESAN GURUSWAMI University of Washington RAJSEKAR MANOKARAN Princeton University PRASAD RAGHAVENDRA University of Washington Astract 1 Introduction We prove that approximating the MAX ACYCLIC SUB- GRAPH prolem within a factor etter than 1/ is Unique Games hard. Specifically, for every constant ε > 0 the following holds: given a directed graph G that has an acyclic sugraph consisting of a fraction 1 ε of its edges, if one can efficiently find an acyclic sugraph of G with more than 1/ + ε of its edges, then the UGC is false. Note that it is trivial to find an acyclic sugraph with 1/ the edges, y taking either the forward or ackward edges in an aritrary ordering of the vertices of G. The existence of a ρ-approximation algorithm for ρ > 1/ has een a asic open prolem for a while. Our result is the first tight inapproximaility result for an ordering prolem. The starting point of our reduction is a directed acyclic sugraph DAG in which every cut is nearly-alanced in the sense that the numer of forward and ackward edges crossing the cut are nearly equal; such DAGs were constructed in [. Using this, we are ale to study MAX ACYCLIC SUBGRAPH, which is a constraint satisfaction prolem CSP over an unounded domain, y relating it to a proxy CSP over a ounded domain. The latter is then amenale to powerful techniques ased on the invariance principle [1, 17. Our results also give a super-constant factor inapproximaility result for the MIN FEEDBACK ARC SET prolem. Using our reductions, we also otain SDP integrality gaps for oth the prolems. Work done while on leave at the Institute for Advanced Study, School of Mathematics, Princeton, NJ Research supported in part y a Packard Fellowship, and NSF grant CCR to The IAS supported y NSF grants MSPA-MCS , and ITR Work done while the author was visiting Princeton University. Supported y NSF grant CCF Given a directed acyclic graph G, one can efficiently order topological sort its vertices so that all edges go forward from a lower ranked vertex to a higher ranked vertex. But what if a few, say fraction ε, of edges of G are reversed? Can we detect these errors and find an ordering with few ack edges? Formally, given a directed graph whose vertices admit an ordering with many, i.e., 1 ε fraction, forward edges, can we find a good ordering with fraction α of forward edges for some α 1? This is equivalent to finding a sugraph of G that is acyclic and has many edges, and hence this prolem is called the MAX ACYCLIC SUB- GRAPH MAS prolem. It is trivial to find an ordering with fraction 1/ of forward edges: take the etter of an aritrary ordering and its reverse. This gives a factor 1/ approximation algorithm for MAX ACYCLIC SUBGRAPH. This is also achieved y picking a random ordering of the vertices. Despite much effort, no efficient ρ-approximation algorithm for a constant ρ > 1/ has een found for MAX ACYCLIC SUBGRAPH. The existence of such an algorithm has een a longstanding and central open prolem in the theory of approximation algorithms. In this work, we prove a strong hardness result that rules out the existence of such an approximation algorithm assuming the Unique Games conjecture. Our main result is the following. Theorem 1.1. Conditioned on the Unique Games conjecture, the following holds for every constant γ > 0. Given a directed graph G with m edges, it is NP-hard to distinguish etween the following two cases: 1. There is an ordering of the vertices of G with at least 1 γm forward edges or equivalently, G has an acyclic sugraph with at least 1 γm edges.. For every ordering of the vertices of G, there are at most 1/ + γm forward edges or equivalently, every sugraph of G with more than 1/ + γm edges contains a directed cycle. 1

2 To the est of our knowledge, the aove is the first tight hardness of approximation result for an ordering/permutation prolem. As an immediate consequence, we otain the following hardness result for the complementary prolem of MIN FEEDBACK ARC SET, where the ojective is to minimize the numer of ack edges. Corollary 1.. Conditioned on the Unique Games conjecture, for every C > 0, it is NP -hard to find a C- approximation to the MIN FEEDBACK ARC SET prolem. Comining the unique game integrality gap instance of Khot-Vishnoi [9 with the reduction, we otain SDP integrality gaps for MAX ACYCLIC SUBGRAPH prolem. Our integrality gap instances also apply to a related SDP relaxation studied y Newman [15. This SDP relaxation was shown to otain an approximation etter than half on random graphs which were previously used to otain integrality gaps for a natural linear program [ Related work MAX ACYCLIC SUBGRAPH is a classic optimization prolem, figuring in Karp s early list of NP-hard prolems [6; the prolem remains NP-hard on graphs with maximum degree 3, when the in-degree plus out-degree of any vertex is at most 3. MAX ACYCLIC SUBGRAPH is also complete for the class of permutation optimization prolems, MAX SNP[π, defined in [16, that can e approximated within a constant factor. It is shown in [14 that MAX ACYCLIC SUBGRAPH is NP-hard to approximate within a factor greater than Turning to algorithmic results, the prolem is known to e efficiently solvale on planar graphs [10, 5 and reducile flow graphs [18. Berger and Shor [1 gave a polynomial time algorithm with approximation ratio 1/ + Ω1/ d max where d max is the maximum vertex degree in the graph. When d max = 3, Newman [14 gave a factor 8/9 approximation algorithm. The complementary ojective of minimizing the numer of ack edges, or equivalently deleting the minimum numer of edges in order to make the graph a DAG, leads to the MIN FEEDBACK ARC SET FAS prolem. This prolem admits a factor Olog n log log n approximation algorithm [19 ased on ounding the integrality gap of the natural covering linear program for FAS; see also [3. Using this algorithm, one can get an approximation ratio of 1 +Ω1/log n log log n for MAX ACYCLIC SUBGRAPH. Recently, Charikar, Makarychev, and Makarychev [ gave a factor 1/ + Ω1/ log n-approximation algorithm for MAX ACYCLIC SUBGRAPH, where n is the numer of vertices. In fact, their algorithm is stronger: given a digraph with an acyclic sugraph consisting of a fraction 1/+δ of edges, it finds a sugraph with at least a fraction 1/+Ωδ/ log n of edges. This algorithm and, in particular, an instance showing tightness of its analysis from [, play a crucial role in our work. 1. Organization We egin with an outline of the key ideas of the proof in Section. In Section 3, we review the definitions of influences, noise operators and restate the unique games conjecture. The groundwork for the reduction is laid in Section 4 and Section 5, where we define influences for orderings, and multiscale gap instances respectively. We present the dictatorship test in Section 6, and convert it to a UG hardness result in Section 7. Finally, SDP integrality gaps for MAX ACYCLIC SUBGRAPH are presented in Section 8. Proof Overview In this section, we outline the central ideas of the proof. To keep the description concise, we will set up some asic notation. For sake of revity, let us denote [m = {1,..., m}. Given an ordering O of the vertices of a directed graph G = V, E, let ValO refer to the fraction of the edges E that are oriented correctly in O. At the heart of all Unique Games ased hardness results lies a dictatorship testing result for an appropriate class of functions. A function F : [m R [m is said to e a dictator if Fx = x i for some fixed i. A dictatorship test DICT is a randomized algorithm such that, given a function F : [m R [m, it makes a few queries to the values of F and distinguishes etween whether F is a dictator or far from every dictator. While Completeness of the test refers to the proaility of acceptance of a dictator function, Soundness is the maximum proaility of acceptance of a function far from a dictator. The approximation prolem one is showing UG hardness for determines the nature of the dictatorship test needed for the purpose. Now let us turn to the specific prolem at hand : MAX ACYCLIC SUBGRAPH. Designing the appropriate dictatorship test for this prolem amounts to the following: Construct a directed graph over the set of vertices V = [m R such that : For a Dictator ordering O of V, ValO 1 For any ordering O which is far from a dictator, ValO 1. Unlike the case of functions, it is unclear as to what is the right notion of Dictators for orderings. For every ordering O of [m R, define m R functions F [p,q : [m R {0, 1} as follows: { F [p,q 1 if p Ox q x = 0 otherwise The i th coordinate is said to e influential if it has a large influence > τ on any of the functions F [p,q. Here influence

3 refers to the natural notion of influence for real valued functions on [m R see Section 3. An ordering O is said to e τ-pseudorandom far from a dictator if it has no influential coordinates > τ. For this notion to e useful, it is necessary that a given ordering O does not have too many influential coordinates. Towards this, in Lemma 4.3 we show that the numer of influential coordinates is ounded after certain smoothening. Further this notion of influence is well suited to deal with orderings of multiple long codes instead of one - a crucial requirement in translating dictatorship tests to UG hardness. Armed with the notion of influential coordinates, we otain a directed graph on [m R a dictatorship test for which the following holds: Theorem.1. Soundness If O is any τ-pseudorandom ordering of [m R, then ValO 1 + o τ 1. This dictatorship test yields tight UG hardness for the MAX ACYCLIC SUBGRAPH prolem. Using the Khot- Vishnoi [9 SDP gap instance for unique games, we otain an SDP integrality gap for the same. Now we descrie the design of the dictatorship test in greater detail. At the outset, the approach is similar to recent work on Constraint Satisfaction ProlemsCSPs [17. Fix a constraint satisfaction prolem Λ. Starting with an integrality gap instance Φ for the natural semi-definite program for Λ, [17 constructs a dictatorship test DICT Φ. The Completeness of DICT Φ is equal to the SDP value SDPΦ, while the Soundness is close to the integral value INTΦ. Since the result of [17 applies to aritrary CSPs, a natural direction would e to pose the MAX ACYCLIC SUB- GRAPH as a CSP. MAX ACYCLIC SUBGRAPH is fairly similar to a CSP, with each vertex eing a variale taking values in domain [n and each directed edge a constraint etween variales. However, the domain, [n, of the CSP is not fixed, ut grows with input size. We stress here that this is not a superficial distinction ut an essential characteristic of the prolem. For instance, if MAX ACYCLIC SUBGRAPH was reducile to a -CSP over a domain of fixed size, then we could otain a approximation ratio etter than a random assignment [4. Towards using techniques from the CSP result, we define the following variant of MAX ACYCLIC SUBGRAPH: Definition.. A t-ordering of a directed graph G = V, E consists of a map O : V [t. The value of a t-ordering O is given y Val t O = Pr Ou < Ov u,v E + 1 Pr Ou = Ov u,v E In the t-order prolem, the ojective is to find an t-ordering of the input graph G with maximum value. On the one hand, the t-order prolem is a CSP over a fixed domain that is similar to MAS. However, to the est of our knowledge, for the t-order prolem, there are no known SDP gaps, which constitute the starting point for results in [17. For any fixed constant t, Charikar, Makarychev and Makarychev [ construct directed acyclic graphs i.e., with value of the est ordering equal to 1, while the value of any t-ordering of G is close to 1. For the rest of the discussion, let us fix one such graph G on m vertices. Notice that the graph G does not serve as SDP gap example for either the MAS or the t-order prolem. As the graph G has only m vertices, and an ordering of value 1, it has a good t-ordering for t = m. Viewing G as an instance of the m-order CSP corresponding to predicate <, we otain a directed graph, G, on [m R. As a m-order CSP, the dictator m-orderings yield value 1 on G. In turn, this implies that the Dictator orderings have value 1 on G. Turning to the soundness proof, consider a τ-pseudorandom ordering O. Otain a t-ordering O y the following coarsening process : Divide the ordering O in to t equal locks, and map the vertices in the i th lock to value i. The crucial oservation relating O and O is as follows: For a τ-pseudorandom ordering O, Val t O ValO. Clearly, ValO Val t O is ounded y the fraction of edges whose oth endpoints fall in the same lock, during the coarsening. We use the Gaussian noise staility ounds of [1, to ound the fraction of such edges. From the aove oservation, in order to prove that ValO 1, it is enough to ound Val t O. Notice that O is a solution to t-order prolem - a CSP over finite domain. Consequently, the soundness analysis of [17 can e used to show that Val t O is at most the value of the est t-ordering for G, which is close to 1. Summarizing the key ideas, we define the notion of influential coordinates for orderings, and then use it to construct a dictatorship test for orderings. Using gaussian noise staility ounds, we relate the value of a pseudorandom ordering to a related CSP, and then apply techniques from [17. 3 Preliminaries For a positive integer t, t denotes the the t dimensional simplex. We will use oldface letters z to denote vectors z = z 1,..., z R. Let o τ 1 denote a quantity that tends to zero as τ 0, while keeping all other parameters fixed. For notational convenience, an ordering O of a directed graph G = V, E is represented y a map O : V Z. On the other hand, a t-ordering O consists of a map O : V [t. For a t-ordering O, the map need not e injective or surjective, ut an ordering O is required to e injective. In an orderingor t-ordering O, an edge e = u, v is a forward edge if Ou < Ov. For a graph G, the 3

4 quantities ValG,Val t G refer to the maximum fraction of forward edges in an ordering,t-ordering respectively. Oservation 3.1. For all directed graphs G, and integers t t, Val t G Val t G ValG While the first part of the inequality is trivial, we will elaorate on the latter half. Given a t -ordering O, construct a full ordering O y using a random permutation of the elements within each of the t locks, while retaining the natural order etween the locks. It is easy to see that the expected value of the random ordering O is exactly equal to ValO. 3.1 Noise Operators and Influences Let Ω denote the finite proaility space corresponding to the uniform distriution over [m. Let {χ 0 = 1, χ 1, χ,..., χ m 1 } e an orthonormal asis for the space L Ω. For σ [m R, define χ σ z = k [R χ σ i z k. Every function F : Ω R R can e expressed as a multilinear polynomial as Fz = σ ˆFσχ σ z. The L norm of F in terms of the coefficients of the multilinear polynomial is F = σ ˆF σ Definition 3.. For a function F : Ω R R, define Inf k F = E z [Var z k[f = σ k 0 ˆF σ. Here Var z k[f denotes the variance of Fz over the choice of the k th coordinate z k. Definition 3.3. For a function F : Ω R R, define the function T ρ F as follows: T ρ Fz = E[F z z = ρ σ ˆFσχσ z σ [m R where each coordinate z k of z = z 1,..., z R is equal to z k with proaility ρ and with the remaining proaility, z k is a random element from the distriution Ω. Lemma 3.4. Consider two functions F, G : [m R [0, 1 with E[F = E[G = µ, and Inf k T 1 ɛ F, Inf k T 1 ɛ G τ for all k. Let x, y e random vectors in [m R whose marginal distriutions are uniform over [m R ut are aritrarily correlated. For every ɛ > 0, there exists small enough µ such that the following holds : E [T 1 ɛfxt 1 ɛ Gy µ 1+ɛ/ + o τ 1 x,y Proof. The lemma essentially follows from the Majority is Stalest theorem see Theorem 4.4 in [13. We ound each factor individually as follows: T 1 ɛ F = 1 ɛ σ ˆF σ σ [k R 1 ɛ σ ˆFσ1 ɛ σ ˆFσ σ [k R E[T 1 ɛ FxT 1 ɛ T 1 ɛ Fx Since the influences of T 1 ɛ F are low, we can apply Theorem 4.4 from [13 to ound the last expression y noise staility in gaussian space Γ 1 ɛ µ. E[T 1 ɛ FT 1 ɛ T 1 ɛ F Γ 1 ɛ µ + o τ 1 Using standard estimates see Theorem B. in [13, Γ 1 ɛ µ is ounded y µ 1+ɛ/ for µ small enough compared to ɛ. Applying a similar ound for G and applying Cauchy-Schwartz gives the result: E x [T 1 ɛ FxT 1 ɛ Gy µ 1+ɛ/ + o τ 1 T 1 ɛ F T 1 ɛg for µ small enough Due to space constraints, we defer the proof of the following simple lemma to the full version. Lemma 3.5. Given a function F : [m R [0, 1, if H = T 1 ɛ F then R k=1 Inf 1 kh e ln 1/1 ɛ 1 ɛ 3. Semidefinite Program We use the following natural SDP relaxation of the MAX ACYCLIC SUBGRAPH prolem. Given a directed graph G = V, E with V = n, the program has n variales {u 1,..., u n } for each vertex u V. In the intended solution, the variale u i = 1 and u j = 0 for all j i if and only if u is assigned position i. [ Max. E u i v j + 1 e u i v i i<j i MAS SDP such that u i v j 0, u i u j = 0 u, v V, i, j [n u i = 1 u V i [n u i v i = 0 i [n i [n 3.3 Unique Games u, v V Definition 3.6. An instance of Unique Games represented as Υ = A B, E, Π, [R, consists of a ipartite graph over node sets A,B with the edges E etween them. Also part of the instance is a set of laels [R = {1,..., R}, and a set of permutations π a : [R [R for each edge e = a, E. An assignment Λ of laels to vertices is said to satisfy an edge e = a,, if π a Λa = Λ. The ojective is to find an assignment Λ of laels that satisfies the maximum numer of edges. 4

5 For a vertex a A B, we shall use Na to denote its neighorhood. For the sake of convenience, we shall use the following version of the Unique Games Conjecture [8 which is equivalent to the original conjecture [7. Conjecture 3.7. Unique Games Conjecture [8, 7 For all constants δ > 0, there exists large enough constant R such that given a ipartite unique games instance Υ = A B, E, Π = {π e : [R [R : e E}, [R with numer of laels R, it is NP-hard to distinguish etween the following two cases: 1 δ-satisfiale instances: There exists an assignment Λ of laels such that for 1 δ fraction of vertices a A, all the edges incident at a are satisfied. Instances that are not δ-satisfiale: No assignment satisfies more than a δ-fraction of the constraints Π. 4 Orderings In this section, we develop the notions of influences for orderings and prove some asic results aout them. Definition 4.1. Given an ordering O of vertices V, its t- coarsening is a t-ordering O otained y dividing O into t-contiguous locks, and assigning lael i to vertices in the i th lock. Formally, if M = V /t then F [p,q O {v Ov < Ou} O u = + 1 M For an ordering O of points in [m R, Define functions : [m R {0, 1} for integers p, q as follows: { F [p,q O x = 1 if Ox [p, q 0 otherwise We will omit the suscript and write F [p,q instead of F [p,q O, when it is clear. Definition 4.. For an ordering O of [m R, define the set of influential coordinates S τ O as follows: S τ O = {k Inf k T 1 ɛ F [p,q τ for some p, q Z} An ordering O is said to e τ-pseudorandom if S τ O is empty. Lemma 4.3. Few Influential Coordinates For any ordering O of [m R, we have S τ O 400 ɛτ 3 Proof. Although an ordering is an injective map O : [m R Z, the range of O can e compressed to {1,..., m R } without affecting the set of functions {F [p,q O } associated with O. After compression, it is enough to consider functions {F [p,q O } only for p, q {0,..., mr + 1}. For integers p, q, δ 1, δ such that δ i < τ 8 mr, let f = T 1 ɛ F [p,q and g = T 1 ɛ F [p+δ1,q+δ. Now, Inf k f g f g F [p,q F [p+δ1,q+δ = Pr z [F [p,q z F [p+δ1,q+δ z τ/4 Hence, using a + a, we get: Inf k f ĝ σ + ˆfσ ĝσ σ k 0 Inf k g + τ/ σ k 0 Thus, if Inf k f τ, then Inf k g τ/4. It is easy to see that there is a set N = {F [p,q } of size at most 100/τ such that for every F [p,q there is a F [r,s N such that max p r, q s < τmr 8. Further, y Lemma 3.5, the functions T 1 ɛ F [p,q 4 have at most ɛτ coordinates with influence more than τ/4. Hence, S τ O 400 ɛτ 3. Claim 4.4. For any τ-pseudorandom ordering O of [m R, its t-coarsening O is also τ-pseudorandom. Proof. Since the functions {F [, O } are a suset of the functions {F [, O }, S τ O S τ O. 5 Multiscale Gap Instances In this section, we will construct acyclic directed graphs with no good t-ordering. These graphs will e crucial in designing the dictatorship test Section 6. Definition 5.1. For η > 0 and a positive integer t, a η, t- Multiscale Gap instance is a weighted directed graph G = V, E with the following properties: ValG = 1 and Val t G 1 + η There exists a solution {u i u V, 1 i V } to SDP with ojective value at least 1 η such that for all u, v V and 1 i, j V, we have u i = 1 V. The cut norm of a directed graph, G, represented y a skew-symmetric matrix W is: G C = max xi,y j {0,1} ij x iy j w ij. We will need the following theorem from [ relating the cut norm of a directed graph G to ValG. Theorem 5. Theorem 3.1, [. If a directed graph G on n vertices has a maximum acyclic sugraph with at least a 1 + δ fraction of the edges, then, G δ C Ω log n. The following lemma and its corollary construct Multiscale Gap instances starting from graphs that are the tight cases of the aove theorem. 5

6 Lemma 5.3. Given η > 0 and a positive integer t, for every sufficiently large n, there exists a directed graph G = V, E on n vertices such that ValG = 1, Val t G 1 + η. Proof. Charikar et al Section 4, [ construct a directed graph, G = V, E, on n vertices whose cut norm is ounded y O 1/ log n. The graph is represented y the skew-symmetric matrix W, where w ij = n πj ik k=1 sin n+1. It is easy to verify that for every 0 < t < n, n k=1 sin πtk n+1 0. Thus, w ij 0 whenever i < j, implying that the graph is acyclic in other words, ValG = 1. We ound Val t G as follows. Let Val t G = 1 + δ and let O : V [t e the optimal t-ordering. Construct a graph H on t vertices with a directed edge from Ou to Ov for every edge u, v E with Ou Ov. Now, using Theorem 5., the cut norm of H is ounded δ from elow y Ω log t. Moreover, since O is a partition of V, the cut norm of G is at least the cut norm of H. δ Thus, Ω log t H C G C O 1/ log nthus, δ O log t log n implying that Val t G 1 + O log t log n. Choosing n to e a sufficiently gives the required result. Corollary 5.4. For every η > 0 and positive integer t, there exists a Multiscale Gap instance with a corresponding SDP solution {u i u V, 1 i V } satisfying u i = 1/ V for all u V, 1 i V. Proof. Let G = V, E e the graph otained y taking 1/η disjoint copies of the graph guaranteed y Lemma 5.3 and let m = V. Note that the graph still satisfies the required properties: ValG = 1, Val t G 1 + η. Let O e the ordering of [m that satisfies every edge of G. Let D denote the distriution over laellings otained y shifting O y a random offset cyclically. For every u V, i [m, Pr[Du = i = 1/m. Further, every directed edge is satisfied with proaility at least 1 η. Being a distriution over integral laellings, D gives raise to a set of vectors satisfying the constraints in Definition 5.1. G along with these vectors form the required η, t-multiscale gap instance. 6 Dictatorship Test Let G = V, E e a η, t-multiscale gap instance on m vertices, where m is divisile y t. Let {u i u V, i [m} denote the corresponding SDP solution. Define distriutions P e as follows: Definition 6.1. For an edge e = u, v E in a η, t- multiscale gap instance G, define the local integral distriution P e over [m as P e i, j = u i v j. Using the multiscale gap instance G, construct a dictatorship test DICT G on orderings O of [m R as follows: DICT G Test: Pick an edge e = u, v E at random from the Multiscale gap instance G. Sample z e = {z u, z v } from the product distriution P R e, i.e. For each 1 k R, z e k = {z u k, z v k } is sampled using the distriution P e i, j = u i v j. Otain z u, z v y perturing each coordinate of z u and z v independently. Specifically, sample the k th coordinates z u k, z v k as follows: With proaility = z u k, and with the remaining pro- u is a new sample from Ω. 1 ɛ, z u k aility z k Introduce a directed edge z u z v. alternatively test if O z u < O z v Theorem 6.. Soundness Analysis For every ɛ > 0, there exists sufficiently large m, t such that : For any τ- pseudorandom ordering O of [m R, ValO Val t G + Ot ɛ + oτ 1 where o τ 1 0 as τ 0 keeping all other parameters fixed. Let F [p,q : [m R {0, 1} denote the functions associated with the t-ordering O. For the sake of revity, we shall write F i for F [i,i. The result follows from Lemma 6.4 and Lemma 6.3 shown elow. Lemma 6.3. For every ɛ > 0, there exists sufficiently large m, t such that : For any τ-pseudorandom ordering O of [m R ValO Val t O + Ot ɛ + oτ 1 where O is the t-coarsening of O. Proof. As O is a coarsening of O, clearly ValO Val t O. Note that the loss due to coarsening, is ecause for some edges e = z, z which are oriented correctly in O, fall in to same lock during coarsening, i.e O z = O z. Thus we can write ValO Val t O + 1 O Pr z u = O z v Pr O z u = O z v = i [t = i [t [ E E E e=u,v z u,z v zu, z v [ E E e=u,v z u,z v F i z u F i z v T 1 ɛ Fuz i u T 1 ɛ Fvz i v 6

7 As O is a t-coarsening of O, for each value i [t, there are exactly 1 t fraction of z for which O z = i. Hence for each i [t, E z [Fuz i = 1 t. Further, since the ordering O is τ-pseudorandom, for every k [R and i [t, Inf k T 1 ɛ Fa i τ. Hence using Lemma 3.4, for sufficiently large t, the aove proaility is ounded y t t 1 ɛ + t o τ 1 = Ot ɛ + o τ 1. Lemma 6.4. For every choice of m, t, ɛ, and any τ- pseudorandom t-ordering O of [m R, Val t O Val t G + o τ 1. Proof. The t-ordering prolem is a CSP over a finite domain, and is thus amenale to techniques of [17. Specifically, consider the payoff function P : [t [0, 1 defined y: P i, j = 1 for i < j, P i, j = 0 for i > j and P i, j = 1 otherwise. The t-ordering prolem is a Generalized CSPsee Definition 3.1, [17 with the payoff function P. For the sake of exposition, let us pretend that t = m. In this case, the vectors {u i u V, i [m} form a feasile SDP solution for the t-ordering instance G. Let DICT G denote the dictatorship test otained y running the reduction of [17 on this SDP solution for t-ordering instance G. DICT G is an instance of the t-ordering prolem, over the set of vertices [t R. A t-ordering solution O for DICT G corresponds naturally to a function F : [t R t. Now we make the following oservations : The t-ordering instance DICT G is identical to the dictatorship test descried in this section when t = m. For a τ-pseudorandom t-ordering O, for every k [R and i [t, the corresponding function F satisfies Inf k T 1 ɛ F i τ. In the terminology of [17Definitions 4.1 and 4., this is equivalent to the function F = F 1,..., F t eing γ, τ- pseudorandom with γ = 0. By Corollary. in [17, for a γ, τ-pseudorandom function F, its proaility of acceptance on the dictatorship test is at most Val t G + o γ,τ 1. Hence the aove lemma is just a restatement of Corollary. of [17 for the specific generalized CSP : t-ordering, aleit in the language of τ-pseudorandom orderings. Recall that the actual case of interest here satisfies t < m. Unfortunately, in this case, a lack ox application of the result from [17 does not suffice. However, the proof in [17 can e easily adopted without any new technical ideas. In fact, many of the technical difficulties encountered in [17 can e avoided here. For instance, the SDP solution associates with each vertex u, the uniform proaility distriution over {1... m}, unlike [17 where there are several aritrary proaility distriutions to deal with. With the value of m fixed, this removes the need for smoothing the SDP solution Lemma 3.4 in [17. Due to paucity of space, we defer the details of the proof to the full version. The following claim is a restatement of the Lemma 6.4 in terms of functions F : [m R t corresponding to pseudorandom t-orderings. Claim 6.5. For a function F : [m R Inf k T 1 ɛ F τ for all k [R, [ 1 E i=j F i z u F j z u + i<j t satisfying F i z u F j z u Val t G + o τ 1 where the expectation is over the edge e = u, v, z u, z v, z u, and z v. 7 Hardness Reduction Let G = V, E e a η, t-multiscale gap instance, and let m = V. Further let {u i u V, i [m} denote the corresponding SDP solution. Let Υ = A B, E, Π = {π e : [R [R e E}, [R e a ipartite unique games instance. Towards constructing a MAS instance G = V, E from Υ, we shall introduce a long code for each vertex in B. Specifically, the set of vertices V of the directed graph G is indexed y B [m R. Hardness Reduction: Input : Unique games instance Υ = A B, E, Π = {π e : [R [R e E}, [R and a η, t Multiscale gap instance G = V, E. Output : Directed graph G = V, E with set of vertices : V = B [m R and edges E given y the following verifier: Pick a random vertex a A. Choose two neighours, B independently at random. Let π, π denote the permutations on the edges a, and a,. Pick an edge e = u, v E at random from the Multiscale gap instance G. Sample z e = {z u, z v } from the product distriution P R e, i.e. For each 1 k R, z e k = {z u k, z v k } is sampled using the distriution P e i, j = u i v j. Otain z u, z v y perturing each coordinate of z u and z v independently. Specifically, sample the k th coordinates z u k, z v k as follows: With proaility = z u k, and with the remaining pro- u is a new sample from Ω. 1 ɛ, z u k aility z k Introduce a directed edge, π z u, π z v. 7

8 Theorem 7.1. For every γ > 0, there exists choice of parameters ɛ, η, t, δ such that: COMPLETENESS: If Υ is a 1 δ-satisfiale instance of Unique Games, then there is an ordering O for the graph G with value at least 1 γ. i.e. ValG 1 γ. SOUNDNESS: If Υ is not δ-satisfiale, then no ordering to G has value more than 1 +γ, i.e ValG 1 +γ. In the rest of the section, we will present the proof of the aove theorem. To egin with, we fix the parameters of the reduction. Parameters : Fix ɛ = γ/8 and η = γ/4. Let τ, t e the constants otained from Theorem 7.5. Finally, let us choose δ = min{γ/4, γɛ τ 8 /10 9 }. 7.1 Completeness In order to show that ValG 1 γ, we will instead show that Val m G 1 γ. From Oservation 3.1, this will imply the required result. By assumption, there exists laelings to the Unique Game instance Υ such that for 1 δ fraction of the vertices a A all the edges a, are satisfied. Let Λ : X Y [R denote one such laelling. Define an m-ordering of G as follows: Oa, z = z Λa a A, z [m R Clearly the mapping O : V [m defines an m-ordering of the vertices V = B [m R. To determine Val m O, let us compute the proaility of acceptance of a verifier that follows the aove procedure to generate an edge in E and then checks if the edge is satisfied. Arithmetizing this proaility, we can write Val m O = 1 O, Pr π z u = O, π z v + Pr O, π z u < O, π z v With proaility at least 1 δ, the verifier picks a vertex a A such that the assignment Λ satisfies all the edges a,. In this case, for all choices of, Na, πλa = Λ and π Λa = Λ. Let us denote Λa = l. By definition of the m-ordering O, we get O, πz = πz Λ = z π 1Λ = z l for all z [m R. Similarly for, O, π z = z l for all z [m R. Thus we get 1 Val m O 1 δ Pr z l u = z v l +Pr l z u < z v l With proaility at least 1 ɛ, for oth z u and z v we have z u l = z u l and z v l = z v l. Further, note that each coordinate z u l, z v l is generated according to the local distriution P e for the edge e = u, v. For the local distriution P e corresponding to an edge e = u, v E, Pr z u l Pr z u l = z v l = u i v j i=j < z v l = u i v j i<j Sustituting in the expression for Val m O we get, [ 1 Val m O 1 δ1 ɛ E u i v j + u i v j e=u,v i=j i<j Recall that the SDP vectors {u i } have an ojective value at least 1 η. Thus for small enough choice of δ, ɛ and η, we have Val m O 1 γ. 7. Soundness Let O e an ordering of G with ValO 1 + γ. Using the ordering, we will otain a laelling Λ for the unique games instance Υ. Towards this, we shall uild machinery to deal with multiple long codes. For B, define O as the restriction of the map O to vertices corresponding to the long code of. Formally, O is a map O : [m R Z given y O z = O, z. Similarly, for a vertex a A, let O a denote the restriction of the map O to the vertices Na [m R, i.e O a, z = O, z Multiple Long Codes Throughout this section, we shall fix a vertex a A and analyze the long codes corresponding to all neighours of a. For a neighour Na, we shall use π to denote the permutation along the edge a,. Let F [p,q denote the functions associated with the ordering O. Define functions : [m R R as follows: F [p,q a F a [p,q z = Pr O a, π z [p, q Na = E Na [p,q [F π z Definition 7.. Define the set of influential coordinates S τ O a as follows: S τ O a = {k Inf k T 1 ɛ F [p,q a τ for some p, q Z} An ordering O a is said to e τ-pseudorandom if S τ O a is empty. Lemma 7.3. For any influential coordinate k S τ O a, for at least τ fraction of Na, π k is influential on O. More precisely, π k S τ/ O. 8

9 Proof. As the coordinate k is influential on O a, there exists p, q such that Inf k F a [p,q τ. Recall that F [p,q a z = E Na [F [p,q π z. Using convexity of Inf this implies, E [Inf π kf [p,q τ Na All the influences Inf π kf [p,q are ounded y 1, since each of the functions F [p,q take values in the range [0, 1. Therefore for at least τ/ fraction of vertices Na, we have Inf π kf [p,q τ/. This concludes the proof. Lemma 7.4. For any vertex a A, S τ O a 800/ɛτ 4. Proof. From Lemma 7.3, for each coordinate k S τ O a there is a corresponding coordinate π k in S τ/ O for at least τ/ fraction of the neighours. Further from Lemma 4.3, the size of each set S τ/ O is at most 400/ɛτ 3. By doule counting, we get that S τ O a is at most 800/ɛτ 4. Theorem 7.5. For all ɛ, γ > 0, there exists constants t, τ > 0 such that for any vertex a A, if O a is τ-pseudorandom then ValO a Val t G + γ/4. Proof. The proof outline is similar to that of Theorem 6.. Let O a denote the t-coarsening of O a. Then we can write, ValO a Val t Oa + 1 O Pr a, π z u = Oa, π z v The t-coarsening Oa is otained y dividing the order O a in to t-locks. Let [p 1 + 1, p, [p + 1, p 3,..., [p t + 1, p t+1 denote the t locks. For the sake of revity, let us denote Fa i = F a [pi+1,pi+1 we can write: and F i = F [pi+1,pi+1 Pr Oa, π z u = Oa, π z v = E E e=u,v, i [t = i [t = i [t. In this notation, [ E F i z u,z v, z u, z v π z u F i π z v [ E E E e=u,v z u,z v zu, z v [ E E e=u,v z u,z v Fa z i u Fa z i v T 1 ɛ Faz i u T 1 ɛ Faz i v As the ordering O a is τ-pseudorandom, for every k [R and i [t, Inf k T 1 ɛ F i a τ. Hence y Lemma 3.4, the aove value is less than Ot ɛ + o τ 1. Now we shall ound the value of Val t O a. In terms of the functions F i, the expression for Val to a is as follows: [ 1 Val t Oa = E F i π z u F j π z v [ 1 = E i=j + Fπ i z u F j π z v i<j i=j F i a z u F j a z v + i<j Fa z i u Fa z j v Again, since the ordering O a is τ-pseudorandom, for every k [R and i [t, Inf k T 1 ɛ F i a τ. Hence y Claim 6.5, the aove value is ounded y Val t G + o τ 1. From the aove inequalities, we get ValO a Val t G + Ot ɛ + o τ 1, which finishes the proof. 7.. Defining a Laelling Define the laelling Λ for the unique games instance Υ as follows: For each a A, Λa is a uniformly random element from S τ O a if it is non-empty, and a random lael otherwise. Similarly for each B, assign Λ to e a random element of S τ/ O if it is nonempty, else an aritrary lael. If ValO = E a A [ValO a 1 + γ is greater than 1 + γ, then for at least γ/ fraction of vertices a A, we have ValO a 1 + γ/. Let us refer to these vertices a as good vertices. From Theorem 7.5, for every good vertex the order O a is not τ-pseudorandom. In other words, for every good vertex a, the set S τ O a is non-empty. Further y Lemma 7.3 for every lael l S τ O a, for at least τ/ fraction of the neighours Na, π l elongs to S τ/ O. For every such, the edge a, is satisfied with proaility at least 1/ S τ O a 1/ S τ/ O. By Lemma 4.3 and Lemma 7.4, this proaility is at least ɛτ 4 /800 ɛτ 3 /300. Summarizing the argument, the expected fraction of edges satisfied y the laelling Λ is at least γɛ τ 8 / By a small enough choice of δ, this yields the required result. 8 SDP Integrality Gap In this section, we construct integrality gaps for the MAS SDP relaxation using the unique games hardness reduction. Specifically we show, Theorem 8.1. For any γ > 0, there exists a directed graph G such that the value of semi-definite program MAS SDP is at least 1 γ, while ValG 1 + γ. The proof uses a ipartite variant of the Khot-Vishnoi [9 Unique Games integrality gap instance as in [17, 11. Specifically, the following is a direct consequence of [9. 9

10 Theorem 8.. [9 For every δ > 0, there exists a UG instance, Υ = A B, E, Π = {π e : [R [R e E}, [R and vectors {V k } for every B, k [R such that the following conditions hold : No assignment satisfies more than δ fraction of constraints in Π. For all, B, k, l [R, V k Vl 0 and Vk V l = 0. For all, B, k, l [R, V k l [R Vl = V k and k [R Vk = 1 The SDP [ value is at least 1 δ: E a A,, B k [R Vπk V π k 1 δ Let G e a η, t-multiscale gap instance with m vertices. Apply Theorem 8., with a sufficiently small δ to otain a UGC instance Υ and SDP vectors {V k B, k [R} {I}. Consider the instance G constructed y running the UG hardness reduction in Section 7 on the UG instance Υ. The set of vertices of G is given y B [m R. Set M = B m R and N = B. The program MAS SDP on the instance G contains M vectors {W,z i i [M} for each vertex, z B [m R and a special vector I denoting the constant 1. Define a solution to MAS SDP as follows: Set the vector I to e the corresponding vector in the instance Υ. Define, W,z i = z k =i V k i [m,, z B W,z l = 0 for all other l [M,, z B It is easy to check that the vectors {W,z i } satisfy the constraints of MAS SDP and have an SDP value close to 1. On the other hand, the soundness analysis in Section 7 implies that the integral optimum for G is at most 1 +γ. The details of the proof will appear in the full version of the paper. 9 Acknowledgments We would like to thank Alantha Newman for pointing out the similarity etween Maximum Acyclic Sugraph and CSPs, and suggesting the prolem. We would like to thank Sanjeev Arora for many fruitful discussions. References [1 B. Berger and P. W. Shor. Tight ounds for the Maximum Acyclic Sugraph prolem. J. Algorithms, 51:1 18, [ M. Charikar, K. Makarychev, and Y. Makarychev. On the advantage over random for maximum acyclic sugraph. In FOCS, pages IEEE Computer Society, 007. [3 G. Even, J. Naor, B. Schieer, and M. Sudan. Approximating minimum feedack sets and multicuts in directed graphs. Algorithmica, 0: , [4 J. Håstad. Every -csp allows nontrivial approximation. In H. N. Gaow and R. Fagin, editors, STOC, pages ACM, 005. [5 A. Karazanov. On the minimal numer of arcs of a digraph meeting all its directed cutsets. Graph Theory Newsletters, 8, [6 R. M. Karp. Reduciility among cominatorial prolems. In Complexity of Computer Computations. New York: Plenum, pages , 197. [7 S. Khot. On the power of unique -prover 1-round games. In IEEE Conference on Computational Complexity, page 5, 00. [8 S. Khot and O. Regev. Vertex cover might e hard to approximate to within -epsilon. J. Comput. Syst. Sci., 743: , 008. [9 S. Khot and N. K. Vishnoi. The unique games conjecture, integrality gap for cut prolems and emeddaility of negative type metrics into l 1. In FOCS, pages IEEE Computer Society, 005. [10 C. Lucchesi and D. H. Younger. A minimax theorem for directed graphs. Journal London Math. Society, 17: , [11 R. Manokaran, J. S. Naor, P. Raghavendra, and R. Schwartz. Sdp gaps and ugc hardness for multiway cut, 0-extension and metric laelling. In STOC 08: Proceedings of the 40th ACM Symposium on Theory of Computing, 008. [1 E. Mossel. Gaussian ounds for noise correlation of functions arxiv: math/ [ To appear in FOCS, 008. [13 E. Mossel, R. O Donnell, and K. Oleszkiewicz. Noise staility of functions with low influences: invariance and optimality. In FOCS, pages IEEE Computer Society, 005. [14 A. Newman. Approximating the maximum acyclic sugraph. Master s thesis, MIT, June 000. [15 A. Newman. Cuts and orderings: On semidefinite relaxations for the linear ordering prolem. In K. Jansen, S. Khanna, J. D. P. Rolim, and D. Ron, editors, APPROX- RANDOM, volume 31 of Lecture Notes in Computer Science, pages Springer, 004. [16 C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., 433:45 440, [17 P. Raghavendra. Optimal algorithms and inapproximaility results for every csp? In STOC 08: Proceedings of the 40th ACM Symposium on Theory of Computing, 008. [18 V. Ramachandran. Finding a minimum feedack arc set in reducile flow graphs. J. Algorithms, 9:99 313, [19 P. Seymour. Packing directed circuits fractionally. Cominatorica, 15:81 88,