Approximation algorithms on k- cycle covering and k- clique covering
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Given a weighted graph G(V,E) with weight w: E→ Z^|E|_+. A k-cycle covering is an edge subset A of E such that G-A has no k-cycle. The minimum weight of k-cycle covering is the weighted covering number on k-cycle, denoted by τ_k(G_w). In this paper, we design a k-1/2 approximation algorithm for the weighted covering number on k-cycle when k is odd. Given a weighted graph G(V,E) with weight w: E→ Z^|E|_+. A k-clique covering is an edge subset A of E such that G-A has no k-clique. The minimum weight of k-clique covering is the weighted covering number on k-clique, denoted by τ_k(G_w). In this paper, we design a (k^2-k-1)/2 approximation algorithm for the weighted covering number on k-clique. Last, we discuss the relationship between k-clique covering and k-clique packing in complete graph K_n.
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