Partial set cover problem and set multi-cover problem are two generalizations of the set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets ( S 2^E ), a total covering ratio q, each set (S S ) is associated with a cost (c_S ), each element (e E ) is associated with a covering requirement (r_e ), the goal is to find a minimum cost sub-collection ( S ' S ) to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least (r_e ) sets of ( S ' ). Denote by (r_ = r_e:e E ) the maximum covering requirement. We present an ((O (r_ ^2n(1+ ( 1 )+ 1-q q )),1- ) )-bicriteria approximation algorithm, that is, the output of our algorithm has cost (O(r_ ^2 n(1+ ( 1 )+ 1-q q )) ) times of the optimal value while the number of fully covered elements is at least ((1- )q|E| ).
For the partial set cover problem, Kearns @cite_12 gave a greedy algorithm achieving performance ratio @math . By modifying the greedy algorithm a little, Slavik @cite_17 improved the performance ratio to @math , where @math is the percentage that elements are required to be covered. Gandhi @cite_23 proposed a primal-dual algorithm achieving performance ratio @math . Bar-Yuhuda @cite_5 studied a generalized version in which each element has a profit and the total profit of covered elements should exceed a threshold. Using local ratio method, he also obtained performance ratio @math . Konemann @cite_18 presented a Lagrangian relaxation framework and obtained performance ratio @math for the generalized partial set cover problem.
@cite_18 @cite_23 @cite_5 @cite_12 @cite_17
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An instance of the generalized partial cover problem consists of a ground set U and a family of subsets @math . Each element e∈U is associated with a profit p(e), whereas each subset @math has a cost c(S). The objective is to find a minimum cost subcollection @math such that the combined profit of the elements covered by @math is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e∈U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection @math that minimizes the cost of @math plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results. We study a generalization of covering problems called partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-partial set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-partial set cover, if each element occurs in at most f sets, then we derive a primal-dual f-approximation algorithm (thus implying a 2-approximation for k-partial vertex cover) in polynomial time. Without making any assumption about the number of sets an element is in, for instances where each set has cardinality at most three, we obtain an approximation of 4 3. We also present better-than-2-approximation algorithms for k-partial vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-partial vertex cover on planar graphs, and for covering k points in Rd by disks. In this paper we consider the natural generalizations of two fundamental problems, the Set-Cover problem and the Min-Knapsack problem. We are given a hypergraph, each vertex of which has a nonnegative weight, and each edge of which has a nonnegative length. For a given threshold ??, our objective is to find a subset of the vertices with minimum total cost, such that at least a length of ?? of the edges is covered. This problem is called the partial set cover problem. We present an O(|V|2+|H|)-time, ?E-approximation algorithm for this problem, where ?E?2 is an upper bound on the edge cardinality of the hypergraph and |H| is the size of the hypergraph (i.e., the sum of all its edges cardinalities). The special case where ?E=2 is called the partial vertex cover problem. For this problem a 2-approximation was previously known, however, the time complexity of our solution, i.e., O(|V|2), is a dramatic improvement.We show that if the weights are homogeneous (i.e., proportional to the potential coverage of the sets) then any minimal cover is a good approximation. Now, using the local-ratio technique, it is sufficient to repeatedly subtract a homogeneous weight function from the given weight function. This thesis is a study of the computational complexity of machine learning from examples in the distribution-free model introduced by L. G. Valiant (V84). In the distribution-free model, a learning algorithm receives positive and negative examples of an unknown target set (or concept) that is chosen from some known class of sets (or concept class). These examples are generated randomly according to a fixed but unknown probability distribution representing Nature, and the goal of the learning algorithm is to infer an hypothesis concept that closely approximates the target concept with respect to the unknown distribution. This thesis is concerned with proving theorems about learning in this formal mathematical model. We are interested in the phenomenon of efficient learning in the distribution-free model, in the standard polynomial-time sense. Our results include general tools for determining the polynomial-time learnability of a concept class, an extensive study of efficient learning when errors are present in the examples, and lower bounds on the number of examples required for learning in our model. A centerpiece of the thesis is a series of results demonstrating the computational difficulty of learning a number of well-studied concept classes. These results are obtained by reducing some apparently hard number-theoretic problems from cryptography to the learning problems. The hard-to-learn concept classes include the sets represented by Boolean formulae, deterministic finite automata and a simplified form of neural networks. We also give algorithms for learning powerful concept classes under the uniform distribution, and give equivalences between natural models of efficient learnability. This thesis also includes detailed definitions and motivation for the distribution-free model, a chapter discussing past research in this model and related models, and a short list of important open problems. We prove that the classical bounds on the performance of the greedy algorithm for approximating MINIMUM COVER with costs are valid for PARTIAL COVER as well, thus lowering, by more than a factor of two, the previously known estimate. In order to do so, we introduce a new simple technique that might be useful for attacking other similar problems.