By Ghiani G., Laporte G.

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**Extra info for A Branch-and-cut Algorithm for the Undirected Rural Postman Problem**

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There is an O(n2 m2 )-time approximation algorithm for Min-2FP achieving a ratio of 682 575 , where n (respectively, m) is the number of vertices (respectively, edges) in the input graph. Previously, the best ratio achieved by a polynomial-time approximation algorithm for Min-2-FP was 65 [6], although Min-2-FP admits a polynomial-time approximation algorithm achieving a ratio of 42Δ−30 35Δ−21 , where Δ is the maximum degree of a vertex in the input graph [6]. 5 Final Remarks When the input graph is restricted to simple graphs, Max Edge t-Coloring is easier to approximate for large values of t as follows: Given G and t, ﬁrst compute a maximum b-matching M of G where b(v) = t for all vertices v of G, then partition E(M ) into t+1 matchings M1 , .

So, the sample space of (s1 , s2 ) is of size O(n2 ). For each sample (s1 , s2 ) in the space, we perform Steps 8 through 11 to obtain an output H(s1 , s2 ). This takes a total time of (n3 ) because Steps 8 through 11 can be done in O(n) time. We then ﬁnd the sample (s1 , s2 ) in O(n2 ) time such that |H(s1 , s2 )| is maximized, and further output H(s1 , s2 ). 4 An Application Let G be a graph. An edge cover of G is a set F of edges of G such that each vertex of G is incident to at least one edge of F .

Tanahashi becomes an available bridge with probability at least 25 . Thus, we can expect at A,ex | available bridges in M , which can then be added to H as before least 15 |Eopt to obtain H . A,ex | is signiﬁcantly large, then By the discussion in the last paragraph, if |Eopt the above modiﬁed ideas lead to a randomized algorithm whose output can be expected to contain signiﬁcantly more edges than the output of the simple A,ex | is not signiﬁcantly algorithm. So, the problematic case happens when |Eopt large.

### A Branch-and-cut Algorithm for the Undirected Rural Postman Problem by Ghiani G., Laporte G.

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