The k-f(k) dense subgraph problem((k; f(k))-DSP) asks whether there is a k-vertex subgraph of a given graph G which has at least f(k) edges. When f(k) = k(k − 1)=2, (k; f(k))-DSP is equivalent to the well-known k-clique problem. The main purpose of this paper is to discuss the problem of ÿnding slightly dense subgraphs. Note that f(k) is about k 2 for the k-clique problem. It is shown that (k; f(k))-DSP remains NP-complete for f(k) = (k 1+) where may be any constant such that 0 ¡ ¡ 1. It is also NP-complete for "relatively" slightly-dense subgraphs, i.e., (k; f(k))-DSP is NP-complete for f(k) = ek 2 =v 2 (1+O(v −1)), where v is the number of G's vertices and e is the number of G's edges. This condition is quite tight because the answer to (k; f(k))-DSP is always yes for f(k) = ek 2 =v 2 (1 − (v − k)=(vk − k)) that is the average number of edges in a subgraph of k vertices. Also, we show that the hardness of (k; f(k))-DSP remains for regular graphs: (k; f(k))-DSP is NP-complete for (v 1)-regular graphs if f(k) = (k 1+ 2) for any 0 ¡ 1; 2 ¡ 1.
We study the problem of orienting the edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be N P-hard generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, or more generally, a graph with identically weighted edges, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, min{ wmax wmin , (2−ε)}-approximation algorithm for the general case, where w max and w min are the maximum and the minimum weights of edges, respectively, and ε is some small positive real number that depends on the input.
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