Let $({\bf U},{\bf S},d)$ be an instance of Set Cover Problem, where ${\bf U}=\{u_1,...,u_n\}$ is a $n$ element ground set, ${\bf S}=\{S_1,...,S_m\}$ is a set of $m$ subsets of ${\bf U}$ satisfying $\bigcup_{i=1}^m S_i={\bf U}$ and $d$ is a positive integer. In STOC 1993 M. Bellare, S. Goldwasser, C. Lund and A. Russell proved the NP-hardness to distinguish the following two cases of ${\bf GapSetCover_{\eta}}$ for any constant $\eta > 1$. The Yes case is the instance for which there is an exact cover of size $d$ and the No case is the instance for which any cover of ${\bf U}$ from ${\bf S}$ has size at least $\eta d$. This was improved by R. Raz and S. Safra in STOC 1997 about the NP-hardness for ${\bf GapSetCover}_{clogm}$ for some constant $c$. In this paper we prove that restricted parameter range subproblem is easy. For any given function of $n$ satisfying $\eta(n) \geq 1$, we give a polynomial time algorithm not depending on $\eta(n)$ to distinguish between {\bf YES:} The instance $({\bf U},{\bf S}, d)$ where $d>\frac{2 |{\bf S}|}{3\eta(n)-1}$, for which there exists an exact cover of size at most $d$; {\bf NO:} The instance $({\bf U},{\bf S}, d)$ where $d>\frac{2 |{\bf S}|}{3\eta(n)-1}$, for which any cover from ${\bf S}$ has size larger than $\eta(n) d$. The polynomial time reduction of this restricted parameter range set cover problem is constructed by using the lattice.
Set Cover Problem is a classical combinatorial optimization problem. The instance of the problem is (U, S), where where U = {u 1 , ..., u n } is a n element ground set, S = {S 1 , ..., S m } is a set of m subsets. The goal is to find the minimal size S ′ ⊆ S such that S i ∈S ′ S i = U. The classical result says that there is the polynomial time algorithm approximating the optimal solution to a factor t, where t = max i |{j : u i ∈ S j }| is the largest number of the subsets in S for which one element in U may belong(see [10,12,15]). The greedy algorithm with polynomial time can also be used for approximating the optimal solution to a H k = 1 + 1 2 + • • • + 1 k ≤ 1 + lnk factor(see [5]), where k = max i=1,...,m {|S i |} is the size of the largest set.
On the other hand, from the NP-completeness of 3-dimensional matching problem it is well-known that the following exact covering problem by 3-sets is NP-complete: given an instance (U, S), where U is a ground set of 3n elements and S is a collection of subsets with 3 elements, the goal is to determine if there is a sub-collection S ′ ⊆ S of pairwise disjoint subsets(an exact cover) such that S∈S ′ S = U. In STOC 1993, Bellare, Goldwasser, Lund and Russell proved that the promise problem of approximating set cover to any constant factor is NP hard in [4](see also [16]). Explicitly they proved that for any positive constant η > 1 it is NP-hard to distinguish between the following YES and NO instances. The YES instance is the instance for which there is an exact cover of size d , that is, there exist pairwise disjoint subsets S i 1 , …, S i d in S satisfying d j=1 S i j = U, and the No instance is the instance for which any cover of U from S has size at least ηd. The result of R. Raz and S. Safra in STOC 1997 ([19]) implies the NP-hardness of the promise problem GapSetCover clogm for some constant c. Feige [9] proved that there cannot be a (1ε)lnm approximate algorithm for the original set cover problem, for any ε > 0, unless NP ⊆ QP. Trevisan [20] indicated that Feiges proof also implies that there is a constant c such that the Set Cover problem with sets of size k (where k is constant) has no (lnkclnlnk)-approximate algorithm for the original set cover problem unless NP = P.
A variant of Set Cover Problem is the following vertex cover problem for k-uniform hypergraphs. An edge in a hypergraph is a subset of the vertices. A k-uniform hypergraph is G = (V, E), where V is the set of n vertices and the E is set of edges and each edge in E is a k element subset of V. The Vertex Cover Problem for k-uniform hypergraphs(k is a con-stant) is defined as follows. For any given k-uniform hypergraph, to find the minimum size subset of vertices V ′ ⊆ V such that V ′ e = ∅ for each edge e. When k = 2 it is the classical vertex cover problem. There is a polynomial time greedy algorithm approximating the Vertex Cover Problem for kuniform hypergraphs to a factor k. Approximating the vertex cover problem witnin a factor k -1ε for any ε > 0 and k ≥ 3 was proved NP-hard in [7]. Khot and Regev proved that approximating the vertex cover problem for k-uniform hypergraphs to the factor kε for any ε > 0 is NP-hard under the assumption that the Unique Game Conjecture is true(see [13]).
The main results of this paper are the following theorems. We prove the following two lemmas.
Lemma 1.1) For YES instance of the vertex cover problem for kuniform hypergraphs in Theorem 1, there exists a vector in L(B) with n non-zero coordinates.
- For NO instance of the vertex cover problem for k-uniform hypergraphs in Theorem 1, any vector in L(B) has at most 2(nη(m)d) non-zero coordinates.
Proof. Let 1 be the vector in Z n (or Z m ) with all n(or m) coordinates 1. For any instance of vertex cover problem for k-uniform hypergraphs, it is obvious B • 1 = k1. For Yes case, there exists a integral vector x ∈ Z n with at most d non-zero coordinates which equals to 1 such that B • x = 1, since there exists an exact vertex cover of size at most d. Thus B • (kx -1) = 0. It is obvious that (kx -1) has n non-zero coordinates. The conclusion in 1) is proved.
For NO case, let x = x +x -∈ Z n be any vector in the lattice L(B), where x + and x -be two vectors in Z n with all their coordinates non-negative integers. Set x ′ ∈ Z n an integral vector which equals to x + at the non-zero positions of x + and takes any positive integer at the zero positions of x + . It is obvious every coordinate of the vector B • x ′ is a positive vector, since this vector is a linear combination of all columns of the matrix B with positive coefficients. Then we have every coordinates of the integral vector B•(x ′x + ) is positive integer. Note that x ′x + has all coordinates non-negative integers. Thus the vertex corresponding to the non-zero positions of the vector
For NO instance of the vertex cover problem for k-uniform hypergraphs in Theorem 1, there exists a subspace R
Proof. Let x = (x 1 , …, x n ) be a vector in the
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