On the Inapproximability of Vertex Cover on k -Partite k -Uniform Hypergraphs (cid:63)

. Computing a minimum vertex cover in graphs and hyper-graphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely ﬁnding a small vertex cover in k - uniform k -partite hypergraphs, when the k -partition is given as input. For this problem Lov´asz [16] gave a k 2 factor LP rounding based approximation, and a matching (cid:0) k 2 − o (1) (cid:1) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the ﬁrst strong hardness results for this problem (here ε > 0 is an arbitrary constant):


Introduction
A k-uniform hypergraph G = (V, E) consists of a set of vertices V and hyperedges E where every hyperedge is a set of exactly k vertices.The hypergraph G is said to be m-colorable if there is a coloring of the vertex set V with at most m colors such that no hyperedge in E has all its vertices of the same color.We shall be interested in the stricter condition of strong colorability as defined in Aharoni et al. [1], wherein G is said to be m-strongly-colorable if there is an m-coloring such of the vertex set V such that every hyperedge E has k distinctly colored vertices.In particular a k-strongly-colorable k-uniform hypergraph is a k-partite k-uniform hypergraph, where the k-partition of the vertex set corresponds to the k color classes.
A vertex cover of a hypergraph G = (V, E) is a subset V of vertices such that every hyperedge in E contains at least one vertex from V .The problem of computing the vertex cover of minimum size in a (hyper)graph has been deeply studied in combinatorics with applications in various areas of optimization and computer science.This problem is known to be NP-hard.On the other hand, for k-uniform hypergraphs the greedy algorithm of picking a maximal set of disjoint hyperedges and including all the vertices in those hyperedges gives a factor k approximation.More sophisticated algorithmic techniques only marginally improve the approximation factor to k − o(1) [9].
Several inapproximability results have been shown for computing the minimum vertex cover.For general k, an Ω(k 1/19 ) hardness factor was first shown by Trevisan [21], subsequently strengthened to Ω(k 1−ε ) by Holmerin [10] and to a k − 3 − ε hardness factor due to Dinur, Guruswami and Khot [4].The currently best known k − 1 − ε hardness factor is due to Dinur, Guruswami, Khot and Regev [5] who build upon [4] and the seminal work of Dinur and Safra [6] who showed the best known 1.36 hardness of approximation for vertex cover in graphs (k = 2).
All of the above mentioned results are based on standard complexity assumptions.However, assuming Khot's Unique Games Conjecture (UGC) [12], an essentially optimal k − ε hardness of approximating the minimum vertex cover on k-uniform hypergraphs was shown by Khot and Regev [14].In more recent works the UGC has been used to relate the inapproximability of various classes of constraint satisfaction problems (CSPs) to the corresponding semi-definite programming (SDP) integrality gap [19], or the linear programming (LP) integrality gap [17] [15].The recent work of Kumar et al. [15] generalizes the result of [14] and shall be of particular interest in this work.
In this work we investigate the complexity of computing the minimum vertex cover in hypergraphs that are strongly colorable and where the strong coloring is given as part of the input.Variants of this problem are studied for databases related applications such as distributed data mining [7], schema mapping discovery [8] and in optimizing finite automata [11].The particular case of computing the minimum vertex cover in k-uniform k-partite (with the partition given) hypergraphs was studied by Lovász [16] who obtained a k/2 approximation for it by rounding its natural LP relaxation.Subsequently, Aharoni, Holzman and Krivelevich [1] proved a tight integrality gap of k/2 − o(1) for the LP relaxation.On the hardness side, [11] and [8] give reductions from 3SAT to it, which imply that the problem is APX-hard.However, to the best of our knowledge no better hardness of approximation was known for this problem.
In this work we show a k 4 − ε hardness of approximation factor for computing the minimum vertex cover on k-uniform k-partite hypergraphs.Actually, we prove a more general hardness of approximation factor of (m−(k−1))(k−1) m − ε for computing the minimum vertex cover in m-strongly colorable k-uniform hypergraphs.The result for k-uniform k-partite hypergraphs follows by a simple reduction.Our results are based on a reduction from minimum vertex cover in k-uniform hypergraphs for which, as mentioned above, the best known factor k − 1 − ε hardness of approximation factor was given in [5].
We also study the results of [15] in the context of the problems we consider.In [15], the authors proved that LP integrality gaps for a large class of monotone constraint satisfaction problems, such as vertex cover, can be converted into corresponding UGC based hardness of approximation results.As presented, the reduction in [15] does not guarantee that the structural properties of the integrality gap will be carried through into the final instance.Nevertheless, we observe that the integrality gap instance of [1] can be combined with the work of [15] with only a slight modification to yield an essentially optimal k/2 − o(1) factor hardness of approximation for computing the minimum vertex cover in k-uniform k-partite hypergraphs, i.e. the final instance is also guaranteed to be a k-uniform k-partite hypergraph.Similar tight inapproximability can also be obtained for a larger class of hypergraphs which we shall define later.
Main Results.We summarize the main results of this paper in the following informal statement.
Theorem.(Informal) For every ε > 0, and integers k 3 and m 2k, it is NP-hard to approximate the minimum vertex cover on m-strongly-colorable k-uniform hypergraphs to within a factor of In addition, it is NP-hard to approximate the minimum vertex cover on kuniform k-partite hypergraphs to within a factor of k 4 − ε, and within a factor of k 2 − ε assuming the Unique Games conjecture.
We now proceed to formally defining the problems we consider, followed by a discussion of the previous work and a precise statement of our results on these problems.

Problem Definitions
We now define the variants of the hypergraph vertex cover problem studied in this paper.
Definition 1.For any integer k 2, an instance G = (V, E) of the hypergraph vertex cover problem HypVC(k), is a k-uniform hypergraph (possibly weighted) where the goal is to compute a vertex cover V ⊆ V of minimum weight.
is given, such that every hyperedge in E has at most one vertex from each color class.In other words, every hyperedge contains k distinctly colored vertices.The goal is to compute the minimum weight vertex cover in G. Definition 3.For any integer k 2, an instance G = (V, E) of HypVCpartite(k) is an k-uniform k-partite hypergraph with the k-partition given as input.The goal is to compute the minimum weight vertex cover in G.Note that Hyp-VCpartite(k) is the same as StrongColored-HypVC(k, k).
The following definition generalizes the class of k-partite hypergraphs and defines the minimum vertex cover problem for that class.
3 Previous work and our results

Previous Results
Let LP 0 be the natural "covering" linear programming relaxation for hypergraph vertex cover (see, for example, Section 1 of [1]).The linear program is oblivious to the structure of the hypergraph and can be applied to any of the variants of hypergraph vertex cover defined above.The following theorem, first proved by Lovász [16] gives an upper bound on the integrality gap of the relaxation LP 0 for HypVCpartite(k).All the upper bounds on the integrality gap stated in this section are achieved using polynomial time rounding procedures for LP 0 .
Theorem 1. (Lovász [16]) For any integer k 2, for any instance G of Hyp-VCpartite(k), where OPT VC (G) is the weight of the minimum vertex cover in G and VAL LP0 (G) is the optimum value of the objective function of the relaxation LP 0 applied to G.
We observe that the relaxation LP 0 does not utilize the k-partiteness property of the input hypergraph.Therefore, the upper bound in Equation (1) holds irrespective of whether the k-partition is given as input.On the other hand, the k-partition is necessary for the efficient rounding algorithm given by the previous theorem.We note that for general k-uniform hypergraphs the gap between the size of the minimum vertex cover and value of the LP solution can be as high as k − o(1).The following theorem states that Equation ( 1) is essentially tight.
In addition, the integrality gap of where a = m 2 m+r − m 2 m+r .
Theorems 1 and 2 were generalized by [1] to split hypergraphs as defined in Definition 4. Their general result is stated below.
In addition, the integrality gap of LP 0 on instances of HypVCsplit(r, . The following theorem states the best known NP-hardness of approximation for the minimum vertex on general hypergraphs.Theorem 5. (Dinur et al. [5]) For any ε > 0 and integer k 3, it is NP-hard to approximate HypVC(k) to a factor of k − 1 − ε.
The above hardness of approximation for general k is not known to be tight.On the other hand, assuming the Unique Games Conjecture one can obtain optimal inapproximability factors of k−o(1) for HypVC(k).The following formal statement was proved by Khot and Regev [14].Theorem 6. (Khot et al. [14]) Assuming the Unique Games Conjecture of Khot [12], For any ε > 0, it is NP-hard to approximate HypVC(k) to within a factor of k − ε.
Remark 1.A recent paper by Bansal and Khot [3] shows a strong hardness result assuming the UGC for distinguishing between a k-uniform hypergraph that is almost k-partite and one which has no vertex cover containing at most a (1 − ε) fraction of vertices (for any desired ε > 0).We note that this is very different from our problem where the input is always k-partite with a given k-partition (and in particular has an easily found vertex cover with a 1/k fraction of vertices, namely the smallest of the k parts).

NP-hardness results
We prove the following theorem on the NP-hardness of approximating the minimum vertex cover on strongly colorable hypergraphs.
Theorem 7.For every ε > 0 and integer m k 3 (such that m 2k), it is NP hard to approximate StrongColored-HypVC(m, k) to within a factor of The above theorem is proved in Section 4 via a reduction from HypVC(k) to StrongColored-HypVC(m, k).A simple reduction from StrongColored-HypVC(k, k ) also shows the following hardness results for HypVCpartite(k) in Section 5.
Theorem 8.For every ε > 0 and integer k > 12, it is NP-hard to approximate HypVCpartite(k) within a factor of k 4 − ε.It is easy to see that an r-partite r-uniform hypergraph is also (p 1 , . . ., p k )-split for any positive integers k, p 1 , . . ., p k such that k i=1 p i = r.This is because the subsets in the r-partition can be suitably merged to produce another partition that satisfies the desired splitting property.Therefore, Theorem 8 immediately implies the following corollary.The above hardness of approximation results do not quite match the algorithmic results in Theorem 4. The next few paragraphs illustrate how recent results of [15] can be combined with the integrality gaps given in Theorems 1 and 4 to yield tight inapproximability for the corresponding problems.
Unique Games hardness In recent work Kumar, Manokaran, Tulsiani and Vishnoi [15] have shown that for a large class of monotone constraint problems, including hypergraph vertex cover, integrality gaps for a natural LP relaxation can be transformed into corresponding hardness of approximation results based on the Unique Games Conjecture.
The reduction in [15] is analyzed using the general bounds on noise correlation of functions proved by Mossel [18].For this purpose, the reduction perturbs a "good" solution, say x * , to the LP relaxation for the integrality gap G I = (V I , E I ), so that x * satisfies the property that all variables are integer multiples of some ε > 0. Therefore, the number of distinct values in x * is m ≈ 1/ε.The reduction is based on a "dictatorship test" over the set [m] × {0, 1} r (for some parameter r) and the hardness of approximation obtained is related to the performance of a certain (efficient) rounding algorithm on x * , which returns a solution no smaller than the optimum on G I .As described in [15] the reduction is not guaranteed to preserve structural properties of the integrality gap instance G I , such as strong colorability or k-partiteness.
We make the simple observation that the dictatorship test in the above reduction can analogously be defined over V I × {0, 1} r which then preserves strong colorability and partiteness properties of G I into the final instance.The gap obtained depends directly on the optimum in G I .This observation, combined with the result of [15] and the integrality gap for HypVCpartite(k) stated in Theorem 1 yields the following optimal UGC based hardness result.
Theorem 10.Assuming the Unique Games Conjecture, it is NP-hard to approximate HypVCpartite(k) to within a factor of k 2 − ε for any ε > 0. We do not prove the above theorem in its entirety, and instead we describe the dictatorship test over V I × {0, 1} r in Appendix C and refer the reader to [15] for the proof.In Appendix B we give two equivalent LP relaxations for HypVCpartite(k) and state the integrality gap for them given by [1].The latter of these relaxations, LP, given in Figure 2 is used in [15] to construct the dictatorship test.The integrality gap given in [1] satisfies the property that every value is an integral multiple of a certain ε > 0, which enables us to skip the perturbation step in constructing the dictatorship test.
To obtain the desired hardness result, the dictatorship test is combined with an instance of Unique Games using (fairly standard) techniques that have been used in earlier UGC based hardness results such as [13][20][19] [2].Except for the slightly different dictatorship test, the rest of the proof is the same as in [15].4 Reduction from HypVC(k) to StrongColored-HypVC(m, k) and Proof of Theorem 7 Let k and m be two positive integers such that m k 2. In this section we give a reduction from an instance of HypVC(k) to an instance of StrongColored-HypVC(m, k).
Reduction.Let the H = (U, F ) be an instance of HypVC(k), i.e.H is a kuniform hypergraph with vertex set U , and a set F of hyperedges.The reduction constructs an instance G = (V, E) of StrongColored-HypVC(m, k) where G is an k-uniform, m-strongly colorable hypergraph, i.e.V = ∪ m i=1 V i , where V i are m disjoint subsets (color classes) such that every hyperedge in E has exactly one vertex from each subset.The main idea of the reduction is to let new vertex set V be the union of m copies of U , and for every hyperedge e ∈ F , add all hyperedges which contain exactly one copy (in V ) of every vertex in e , and at most one vertex from any of the m copies of U (in V ).Clearly every hyperedge 'hits' any of the m copies of U in V at most once which naturally gives an mstrong coloring of V .It also ensures that if there is a vertex cover in G which is the union of a subset of the copies of U , then it must contain at least m − k + 1 of the copies.Our analysis shall essentially build upon this idea.
To formalize the reduction we first need to define a useful notation.The steps of the reduction are as follows.
1.For i = 1, . . ., m, let For every hyperedge e in F , for every subset I ⊆ [m] such that |I| = k , for every (I, e )-matching σ ∈ Γ I,e we add the hyperedge e = e(e , I, σ) which is defined as follows. ∀i The above reduction outputs the instance G = (V, E) of StrongColored-HypVC(m, k).Note that the vertex set V is of size m|U | and for every hyperedge e ∈ F the number of hyperedges added in E is m k • k!.Therefore the reduction is polynomial time.In the next section we present the analysis of this reduction.

Analyzing the reduction
Theorem 11.Let C be the size of the optimal vertex cover in H = (U, F ), and let C be the size of the optimal vertex cover in G = (V, E).Then, Using the above theorem we can complete the proof of Theorem 7 as follows.
Proof.(of Theorem 7) Theorem 11 combined with the k−1−ε inapproximability for HypVC(k) given by [5] and stated in Theorem 5, implies an inapproximability of, (m It is easy to see that the above expression can be simplified to yield as the inapproximability factor for StrongColored-HypVC(m, This proves Theorem 7. Proof.(of Theorem 11) We first show that there is a vertex cover of size at most mC in G, where C is the size of an optimal vertex cover U * in H.To see this consider the set V * ⊆ V , where For every hyperedge e ∈ F , e ∩U * = ∅, and therefore e∩U * ×{i} = ∅, for some i ∈ [m], for all e = e(e , I, σ).Therefore, V * ∩ e = ∅ for all e ∈ E. The size of V * is mC which proves the upper bound in Theorem 11.In the rest of the proof we shall prove the lower bound in Theorem 11.
Let S be the optimal vertex cover in G. Our analysis shall prove a lower bound on the size S in terms of the size of the optimal vertex cover in H. Let . Before proceeding we introduce the following useful quantity.For every Y ⊆ [m], we let A Y ⊆ U be the set of all vertices which have a copy in S i for some i ∈ Y .Formally, The following simple lemma follows from the construction of the edges E in G. Proof.Fix any subset I as in the statement of the lemma.Let e ∈ F be any hyperedge in H.For a contradiction assume that A I ∩ e = ∅.This implies that the sets S i (i ∈ I) do not have a copy of any vertex in e .Now choose any σ ∈ Γ I,e and consider the edge e(e , I, σ) ∈ E. This edge can be covered only by vertices in V i for i ∈ I.However, since S i does not contain a copy of any vertex in e for i ∈ I the edge e(e , I, σ) is not covered by S which is a contradiction.This completes the proof.
The next lemma combines the previous lemma with the minimality of S to show a strong structural statement for S, that any S i is "contained" in the union of any other k sets S j .It shall enable us to prove that most of the sets S i are large.Proof.Let I be any choice of a set of k indices in [m] as in the statement of the lemma.From Lemma 1 we know that A I is a vertex cover in H and is therefore non-empty.Let j ∈ [m] be an arbitrary index for which we shall verify the lemma for the above choice of I.If j ∈ I, then the lemma is trivially true.Therefore, we may assume that j ∈ I.For a contradiction we assume that, From the minimality of S, we deduce that there must be a hyperedge, say e ∈ E such that e is covered by (u, j ) and by no other vertex in S; otherwise S\{(u, j )} would be a smaller vertex cover in G. Let e = e(e , I , σ) for some e ∈ F , I ⊆ [m] | = k) and σ ∈ Γ I ,e .Now, since (u, j ) covers e, we obtain that j ∈ I and σ(j ) = u ∈ e .Combining this with the fact that j ∈ I, and that |I| = |I | = k, we obtain that I \ I = ∅.
Let j ∈ I \I .We claim that (u, j) ∈ S j .To see this, observe that if (u, j) ∈ S j then u ∈ A I which would contradict our assumption in Equation (7).
We now consider the following hyperedge ẽ = ẽ(e , Ĩ, σ) ∈ E where the quantities are defined as follows.The set Ĩ simply replaces the index j in I with the index j, i.e.Ĩ = (I \ {j }) ∪ {j}.
Analogously, σ ∈ Γ Ĩ,e is identical to σ except that it is defined on j instead of j where σ(j) = σ(j ) = u.Formally, Equations ( 8) and ( 9) imply the following, Since (u, j ) ∈ S uniquely covers e, Equation ( 10) implies that ẽ is not covered by any vertex in S i for all i ∈ [m] \ {j, j }.Moreover, since j ∈ Ĩ no vertex in S j covers ẽ.On the other hand, by our assumption in Equation ( 7) (u, j) ∈ S j , which along with Equation (11) implies that no vertex in S j covers ẽ.Therefore, ẽ is not covered by S. This is a contradiction to the fact that S is a vertex cover in G and therefore our assumption in Equation ( 7) is incorrect.This implies that S j ⊆ A I × {j }.This holds for every j , thus proving the lemma.
Note that the above lemma immediately implies the following corollary.
Corollary 2. For every It is easy to see the following simple lemma.

Lemma 3. For any vertex
Proof.Suppose the above does not hold.Then I u (or any subset of I u of size k) would violate Corollary 2, which is a contradiction.This completes the proof.
The above lemma immediately implies the desired lower bound on the size of S. We prove Theorem 8 by giving a simple reduction from an instance G = (V, E) of StrongColored-HypVC(k, k ) to an instance G = (V , E ) of HypVCpartite(k) where the parameters will be chosen later.
For any hyperedge e ∈ E, construct a corresponding hyperedge e ∈ E which contains all the vertices in e in addition to b i if e ∩ V i = ∅ for all i ∈ [k].It is easy to see that G is a k-partite hypergraph with the k-partition given by the subsets V i ∪ {b i }.As a final step, set the weight of the dummy vertices b 1 , . . ., b k to be much larger than |V | so that no dummy vertex is chosen in any optimal vertex cover in G .This is because V is always a vertex cover in G.Note that the hypergraph can be made unweighted by the (standard) technique of replicating each dummy vertex many times and multiplying the hyperedges appropriately.
Since no optimal vertex cover in G contains a dummy vertex we deduce that an optimal vertex cover in G is an optimal vertex cover in G and vice versa.From Theorem 7, for any ε > 0, we obtain a hardness factor of, for approximating HypVCpartite(k).Let α := (k −1) k .The above expression is maximized in terms of k when (1−α)α 2 attains a maximum where α ∈ [0, 1].
Clearly, the maximum is obtained when α = (k −1) k = 1  2 , thus yielding as the hardness of approximation factor: which proves Theorem 8.

A Proof of Theorem 9
Let integers r, k, p 1 , . . ., p k be as given the statement of Theorem 9, such that t = max{p 1 , . . ., p k } 3. Without loss of generality assume that p k = t.We reduce from from an instance G = (V, E) of HypVC(t) to an instance G = (V , E ) of HypVCsplit(r, k, p 1 , . . ., p k ) in the following manner.
The vertex set is V = V ∪{b 1 , . . ., b r−t }, where b 1 , . . ., b r−t are referred to as dummy vertices.For every hyperedge e ∈ E, add the hyperedge e∪{b 1 , . . ., b r−t } to E .Also, we set the weight of each dummy vertex in {b 1 , . . ., b r−t } to be |V | so that none of them is chosen in an optimal vertex cover for G , as V is always a vertex cover for G .Note that the hypergraph can be made unweighted by adding multiple copies of each dummy vertex and appropriately replicating the hyperedges.
Observe that every hyperedge in G consists of t vertices from V and r − t dummy vertices.Therefore, one can create a partition of V into sets V 1 , V 2 , . . ., V k where V k = V and the rest of the subsets can be created by appropriately partitioning {b 1 , . . ., b r−t }.Therefore G is an instance of HypVCsplit(r, k, p 1 , . . ., p k ).Clearly, V k is a vertex cover in G and since the dummy vertices have a very large weight we may assume that the optimal vertex cover in G is a subset of V .Therefore an optimal vertex cover in G is an optimal vertex cover in G and vice versa.This gives a t − 1 − ε hardness of approximation for HypVCsplit(r, k, p 1 , . . ., p k ) where t = max{p 1 , . . ., p k }.Combining this with Corollary 1 proves Theorem 9.

B LP Relaxations for Hypergraph Vertex Cover
In this section we give two natural linear programming (LP) relaxations for Hypergraph Vertex Cover and show that they are equivalent.The first one, LP 0 , is the natural linear programming relaxation for vertex cover, while the second relaxation LP was used in [15] to convert integrality gaps for it into corresponding UGC based hardness of approximation results.On the other hand, [1] give integrality gaps for LP 0 applied to instances of different variants of the hypergraph vertex cover problem.This motivates us to present a fairly simple argument that the two relaxations are indeed equivalent and the integrality gaps for LP 0 hold for LP as well.Note that the relaxations are oblivious to the structure of the hypergraph.We shall also state a lower bound on the integrality gap of the relaxations for instances of HypVCpartite(k) i.e. on k-uniform k-partite hypergraphs, which was shown in [1].
Let G = (V, E) be an input k-uniform hypergraph.Let x v be a real variable for every vertex v ∈ V .The first relaxation LP 0 the natural relaxation for vertex cover in hypergraphs and is given in Figure 1.Before proceeding, let us define the set is the set of all valid assignments to the k vertices of any hyperedge in E in any integral solution to the LP for G = (V, E).For the relaxation, we can constrain the k-tuple of variables for every hyperedge to lie inside the convex hull of Q(k).Keeping this subject to, in mind we write the relaxation LP which is given in Figure 2. We now prove the equivalence of the above relaxations.
Lemma 5.The relaxations LP 0 and LP are equivalent.
Proof.We observe the constraints ( 17)-( 20) are equivalent to (x v1 , . . ., x v k ) ∈ conv(Q(k)), where conv(A) is the convex hull of a set A of vectors.Furthermore, the set {0, 1} k ∩ {(y 1 , . . ., y k ) ∈ R k | i∈[k] y i = 1} consists of exactly the k unit coordinate vectors.Therefore, the corner points of the (bounded Observing that the constraints ( 14) and ( 15) are equivalent to we obtain that the two relaxations 0 and LP are equivalent, thus proving Lemma 5.
We now restate the integrality gap of [1], augmenting Theorem 2 with the fact that the lower bound holds also for the relaxation LP.C Construction of the Dictatorship Test for HypVCpartite(k) Before we begin, let us restate a more detailed form of Theorem 12 which abstracts out some useful properties of the integrality gap instance.
Theorem 13.For a positive integer k (k 2), there is an instance G k = (V k , E k ) of HypVCpartite(k) on O(mk 2 ) vertices (where m is any integer) such that there exists a solution {x * v } v∈V k to LP applied on G k such that, where OPT VC (G k ) is the optimal size of the vertex cover in G k , and For convenience, we drop the subscript and denote as G = (V, E) the integrality gap instance of StrongColored-HypVC(k), which we shall utilize for the rest of the reduction along with the solution {x * v } v∈V given in Theorem 13 to the relaxation LP for the minimum vertex cover in G. Before proceeding to the reduction we need the following definitions.Definition 6.For every hyperedge e ∈ E let P x * e be the distribution induced on Q(k) by choosing σ ∈ Q(k) with probability λ e σ , where the values {λ e σ } e∈E,σ∈Q(k) are obtained along with x * as the solution to the relaxation LP.Let M δ (P x * e ) be a distribution over {0, 1} k obtained by sampling z ∈ {0, 1} k from P x * e and then independently letting every coordinate z i remain unchanged with probability 1 − δ and setting it to be 1 with probability δ.

Note that the support of
) is also a subset of Q(k).The following lemma were proved in [15].Lemma 6.Since x * satisfies the property that every x * v is an integral multiple of ε = 1 mk , there exists a set of values {λ e k } e∈E which along with x * forms a solution to the relaxation LP for G, with the property that for every e ∈ E, the minimum probability of any atom in P x * e is at least ε the rest of this section we assume that the property given by the above lemma is satisfied by the values {λ e σ } e∈E,σ∈Q(k) associated with the solution x * .The following lemma is a simple consequence which we state without proof.The integrality gap instance G = (V, E) and the solution x * are fixed along with the parameter m to depend only on k.Additionally, δ > 0 is a small enough constant and r is a parameter to the procedure and is the size of the domain of the dictatorship test.It corresponds to the size of the label set in the eventual Unique Games reduction.We use the distributions P x * e and M δ (P x * e ) over {0, 1} k as given in Definition 6.The following steps describe the construction of the instance D.

The set of vertices
where v ∈ V and y ∈ {0, 1} r .2. Let a v := x * v (1 − δ) + δ, for v ∈ V .Let µ a be the a-biased probability measure on {0, 1} r , where every coordinate is chosen independently to be 1 with probability a.The weight wt D of any vertex (v, y) ∈ V D is given by, wt D (v, y) := µ av (y) |V | .
Note that (v,y)∈V D wt D ((v, y)) = 1. 3. The set of hyperedges E D is the union of all the hyperedges output with positive probability in the following randomized procedure.a. Pick a random hyperedge e = (v 1 , . . ., v k ) from E. b.Sample r independent copies (z j v1 , z j v2 , . . ., z j v k ) for j = 1, . . ., r from the distribution M δ (P x * e ).Let z vi ∈ {0, 1} r be defined as z vi := (z 1 vi , z 2 vi , . . ., z r vi ) for i = 1, . . ., k.We first observe that the hypergraph G D is indeed k-partite.To see this, recall that G is k-partite.Since V D = V × {0, 1} r , any partition of V extends naturally to a partition of V D .Let {V i } k i=1 be the k disjoint subsets of V comprising the k-partition.Then, {V i × {0, 1} r } k i=1 gives the k-partition of V D .Moreover, since any hyperedge e ∈ E D is of the form ((v 1 , z v1 ), (v 2 , z v2 ), . . ., (v k , z v k )), where (v 1 , . . ., v k ) ∈ E, it is easy to see that e hits each set V i × {0, 1} r (1 i k) exactly once.Therefore, G D is k-partite.
Next we shall formally state the completeness and the soundness of the dictatorship test given by D. Their proofs are completely analogous to the corresponding ones in [15].While we shall present the proof of the completeness, we refer the reader to [15] for a proof of the soundness of the test.We need the notions of a function being a "dictator" and "far from a dictator".Definition 7. A set S ⊆ V × {0, 1} r = V D is said to be a dictator if there is an index i ∈ [r] such that S = {(v, y) | y i = 1}.
Given a set S ⊆ V × {0, 1} r , let S v := S ∩ ({v} × {0, 1} r ) the function f S v : {0, 1} r → {0, 1} be the complement of S v , i.e. f S v = {y ∈ {0, 1} r | (v, y) ∈ S v }.For a given i ∈ [r] and v ∈ V , let Inf d i (f S v ) be the degree d influence of the i-th coordinate, with respect to the measure µ av .The quantity Inf d i (f ) for any function f measures the likelihood of the value of the function f changing if the ith coordinate in the input is changed.We refer the reader to [18] for a formal definition of Inf d i (f ).Using this we have the following notion of a function being far from a dictator.Proof.Let i ∈ [r] be such that S = {(v, y) | y i = 1}.Let e ∈ E be any hyperedge, where e = (v 1 , . . ., v k ).Since the tuple (z i v1 , z i v2 , . . ., z i v k ) sampled in step 3b is in the support of M δ (P x * e ) and therefore an element of Q(k), we deduce that there is some j ∈ [r] such that z i vj = 1 i.e. (v j , z vj ) ∈ S.This implies that S covers all hyperedges obtained after choosing e in step 3a.As this holds for any choice of e ∈ E, we obtain that S is a vertex cover in G D .Also, wt D (S) = We state the following theorem regarding the soundness of the dictatorship test and refer the reader to Theorem 4.10 in [15].Theorem 14. (Soundness.)For every δ > 0, there exists d, τ > 0 such that if S ⊆ V × {0, 1} r is a vertex cover of G D and is (τ, d)-pseudo-random, then, Research supported in part by a Packard Fellowship and US-Israel BSF-2008293.

Definition 4 .
For any integer k 2 and positive integers p 1 , . . ., p k , a hypergraph

Corollary 1 .
For every ε > 0 and integer r > 12, and positive integers k, p 1 , . . ., p k such that k i=1 p i = r it is NP-hard to approximate HypVCsplit(r, k, p 1 , . . ., p k ) to within a factor of r 4 − ε.Using the above corollary and a simple reduction from HypVC we prove the following theorem in Appendix A. Theorem 9.For every ε > 0, and positive integers r, k, p 1 , . . ., p k such that k i=1 p i = r 3 and t := max{p 1 , . . ., p k } 3, it is NP-hard to approximate HypVCsplit(r, k, p 1 , . . ., p k ) to within a factor of max r 4 , t − 1 − ε.

Definition 5 .
Given a hyperedge e = {u 1 , . . ., u k } in F , and a subset I ⊆ [m] where |I| = k, a mapping σ : I → {u 1 , . . ., u k } is said to be a "(I, e )-matching" if σ is a one-to-one map.Let Γ I,e be the set of all (I, e )-matchings.Clearly, |Γ I,e | = k! , for all I ⊆ [m], |I| = k and e ∈ F .

Lemma 1 .
Let I ⊆ [m] be any subset such that |I| = k.Then A I is a vertex cover of the hypergraph H.

Lemma 2 .
Let I ⊆ [m] be any set of indices such that |I| = k.Then, for any j ∈ [m], S j ⊆ A I × {j }.

Lemma 4 .
Let C be the size of the optimal vertex cover in H. Then,|S| (m − (k − 1))C.Proof.For convenience, let q = [m] |.Note that, by Lemma 1 A [m] is a vertex cover in H. Therefore, qC.From Lemma 3 we deduce that every vertex u ∈ A [m] has a copy (u, i) in at least m − (k − 1) of the sets S i .Therefore, S contains m − (k − 1) copies of every vertex in A [m] which yields,|S| (m − (k − 1))q (m − (k − 1))C,thus completing the proof.The above also completes the proof of the lower bound of Theorem 11.5 Reduction from StrongColored-HypVC(k, k ) to HypVCpartite(k) and Proof of Theorem 8

Lemma 7 .
For any hyperedge e ∈ E, the minimum probability of any atom in the distribution M x * e is at least εδ k (2 k )! .For a given parameter r, the construction of a dictatorship test shall use the integrality gap G = (V, E) of HypVCpartite(k) along with the solution x * for the relaxation LP given by Theorem 13 to output an instance D of HypVCpartite(k) consisting of a weighted k-uniform k-partite hypergraph G D = (V D , E D ) where every vertex in V D is indexed by an element of {0, 1} r .Informally, the instance D serves as a dictatorship test in the following manner: -(Completeness) Every dictator boolean function on {0, 1} r gives a vertex cover in G D of weight ≈ (VAL LP (x * , G))/(|V |).-(Soundness) Every vertex cover of weight substantially smaller than (OPT VC (G k ))/(|V |) is "close to a dictator".C.1 Construction of G D = (V D , E D ) as an instance of HypVCpartite(k) d. Add the following hyperedge inD , ((v 1 , z v1 ), (v 2 , z v2 ), . . ., (v k , z v k )). 4. Output the hypergraph G D = (V D , E D ).

Definition 8 .
For τ, d 0, as set S ⊆ V × {0, 1} r is said to be (τ, d)-pseudo random if for every v ∈ V and i ∈ [r], Inf d i (f S v ) τ .We first prove the completeness of the dictatorship test.Lemma 8. (Completeness) Suppose S ⊆ V × {0, 1} r = V D is a dictator.Then S is a vertex cover in G D and wt D (S) VAL LP (x * , G) |V | + δ.