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We investigate the set of cycles of a connected graph G = (V, X) such that \V\ > 5 andfor each three vertices x,y, z with d(x, y) = 2 and z e N(x) Π N(y), where d(x, y) is the distance between the vertices χ and y, and for any it 6 V the set W(u) is the subset of vertices from V adjacent to u. We show that if G does not coincide with the complete bipartite graph K m ,m> m > 2, then there exists a cyclic path of any length n, 3 < n < |V|, and what is more, for each vertex χ there exists a set of cycles C 4 , C 5 ,..., Cp, ρ = [Η of lengths 4,5,... ,-p such that ζ 6 C 4 and K(C n ) C K(C n+1 ) orV(C n ) C V(C n+2 ) and V(C n +i) C V(C n+2 ) for each n, 4 < n < \V\ 9 where V(d) is the set of vertices of the cycle (7,·, i = 4,...,p.

We use randomised rounding to obtain an upper bound for the optimum value of the programwhere b > 0, c ≥ 0 are rational vectors and A is an arbitrary rational matrix. Our bound generalises some known bounds for covering integer programs (that is, the same programs with the restriction that all elements of A are non-negative).

We consider the problem on estimating optima of the so-called packing-covering programs, which contain constrains of packing and covering types simultaneously. We introduce the notion of 5-relaxation of such programs and show that the randomized rounding permits to obtain simple sufficient conditions for tight approximations of the integer-valued optima by the optima of the δ-relaxations. We suggest an efficient randomized algorithm for finding an approximate solution of integer packing-covering linear integer programs and point out that this algorithm can be transformed to a deterministic algorithm by the well-known techniques of de-randomization.

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