Given a multigraph \(G=(V,E)\), the edge cover packing problem (ECPP) on G is to find a coloring of edges of G using the maximum number of colors such that at each vertex all colors occur. ECPP can be formulated as an integer program and is NP-hard in general. In this paper, we consider the fractional edge cover packing problem, the LP relaxation of ECPP. We focus on the more general weighted setting, the weighted fractional edge cover packing problem (WFECPP), which can be formulated as the following linear program$$\begin{aligned} \begin{array}{ll} \hbox {Maximize} \ \ \ &{} {{\varvec{1}}}^T {{\varvec{x}}} \\ \hbox {subject to} &{} A{{\varvec{x}}} \le {{\varvec{w}}} \\ &{} \quad {{\varvec{x}}} \ge {{\varvec{0}}}, \end{array} \end{aligned}$$where A is the edge–edge cover incidence matrix of G, \({\varvec{w}}=(w(e): e\in E)\), and w(e) is a positive rational weight on each edge e of G. The weighted co-density problem, closely related to WFECPP, is to find a subset \(S\subseteq V\) with \(|S|\ge 3\) and odd, such that \(\frac{2w(E^{+}(S))}{|S|+1}\) is minimized, where \(E^{+}(S)\) is the set of all edges of G with at least one end in S and \(w(E^{+}(S))\) is the total weight of all edges in \(E^{+}(S)\). We present polynomial combinatorial algorithms for solving these two problems exactly.

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共密度和分数边盖包装

给定一个多重图形\（G =（V，E）\），G上的边缘覆盖堆积问题（ECPP）将使用最大数量的颜色来查找G边缘的着色，从而使每个顶点处都出现所有颜色。ECPP可以表示为整数程序，并且通常是NP难的。在本文中，我们考虑了分数边覆盖问题，即ECPP的LP松弛。我们将重点放在更一般的加权设置上，即加权分数边缘覆盖包装问题（WFECPP），可以将其表达为以下线性程序$$ \ begin {aligned} \ begin {array} {ll} \ hbox {Maximize} \ \ \＆{} {{\ varvec {1}}} ^ T {{\ varvec {x}}} \\ \ hbox {受}＆{} A {{\ varvec {x}}} \ le {{\ varvec {w}}} \\＆{} \ quad {{\ varvec {x}}} \ ge {{\ varvec {0}}}，\ end {array} \ end {aligned} $$，其中A是G的边-边覆盖率关联矩阵，\（{\ varvec {w}} =（w（e）：e \ in E）\），并且w（e）是G的每个边e的正有理权重。与WFECPP密切相关的加权共密度问题是找到具有\（| S | \ ge 3 \）和奇数的子集\（S \ subseteq V \），使得\（\ frac {2w（E ^ {+}（S））} {| S | +1} \）最小化，其中\（E ^ {+}（S）\）是G的所有边的集合，在S中至少有一个末端，而\（w（E ^ {+}（S））\）是总数\（E ^ {+}（S）\）中所有边的权重。我们提出了用于精确解决这两个问题的多项式组合算法。