Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max k-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max k-cover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .

For a graph G, let ff(G) denote the size of the largest independent set in G, and let #(G) denote the Lov'asz #-function on G. We prove that for some c ? 0, there exists an infinite family of graphs such that #(G) ? ff(G)n=2 c p log n , where n denotes the number of vertices in a graph. This disproves a known conjecture regarding the # function. As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest. 1 Introduction Lov'asz [21] introduced the # function in order to study the so called "Shannon Capacity" of graphs. For every graph G, the # function enjoys the following sandwich property: ff(G) #(G) (G) where ff(G) is the size of the largest independent set in G, and (G) is the clique cover number of G ((G) = ( G), the chromatic number of the complement of G). This sandwich prop...

Tree pattern matching and subset matching in deterministic O(n log

In the hitting set problem one is given subsets of a finite set N and one has to find an X ae N of minimum cardinality that "hits" (intersects) all of them. The problem is NP-hard. It is not known whether there exists a polynomial-time approximation algorithm for the hitting set problem with a finite performance ratio. Special cases of the hitting set problem are described for which finite performance ratios are guaranteed. These problems arise in a geometric setting. We consider special cases of the following problem: Given n compact subsets of , find a set of straight lines of minimum cardinality so that each of the given subsets is hit by at least one line. The algorithms are based on several techniques of representing objects bypoints, not necessarily points on the objects, and solving (in some cases, only approximately) the problem of hitting the representative points. Finite performance ratios are obtained when the dimension, the number of types of sets to be hit and the number of directions of the hitting lines are bounded.

We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem. Without any constraint violation, the ε-approximation problem for many problems of this type is itself NP-hard. Our methods provide polynomial-time ε-approximations while attempting to minimize the packing constraint violation. Our methods

We present new vector quantization algorithms based on the theory devel- oped in [LiV]. The new approach is to formulate a vector quantization problem as a 0-1 integer linear program. We first solve its relaxed linear program by linear programming techniques. Then we transform the linear program solution into a provably good solution for the vector quantization problem. These methods lead to the first known polynomial-time full-search vector quantization codebook design algorithm and tree pruning algorithm with provable worst-case performance guarantees. We also introduce the notion of pseudoraw, dom prued tree-structured vector quatizers. Initial experimental results on image compression are very encouraging.

ing the situation, assume only that on input x the verifier V uses poly(logn) bits from a proof \Pi, uses O(log n) random bits, and is non-adaptive. We design a verifier V new which expects a proof \Pi new consisting of poly(n) entries each of poly(logn) bits. V new will read O(1) of these entries. The main idea is that \Pi new is supposed to be an encoding (using a low-degree polynomial) of a proof \Pi which the old verifier V would have accepted with probability 1. The verifier checks using the low degree test that \Pi new is close to a low-degree polynomial, and gets k values from it by the method in Section 3.2, where k is the polylog number of values demanded of \Pi by V . That is, \Pi new contains whatever information is needed for the low-degree test, together with the coefficients of the line polynomial ~ \Pi(c(t)) for each polynomial c derived from a random point z k+1 . So V new reads one of these line polynomials (one "piece" of information), looks up f at O(1) points to ...

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce sub-optimal function expansions by iteratively choosing the dictionary waveforms which best match the function's structures. Matching pursuits provide a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and using a stochastic differential equation model. These invariant measures define a noise with respect to a given dictionary. ...