The goal of an outdegree-constrained edge-modification problem is to find a spanning subgraph or supergraph $H$ of an input undirected graph $G$ such that either: (Type I) the number of edges in $H$ is minimized or maximized and $H$ can be oriented to satisfy some specified constraints on the vertices’ resulting outdegrees; or: (Type II) among all subgraphs or supergraphs of $G$ that can be constructed by deleting or inserting a fixed number of edges, $H$ admits an orientation optimizing some objective involving the vertices’ outdegrees. This paper introduces eight new outdegree-constrained edge-modification problems related to load balancing called (Type I) MIN-DEL-MAX, MIN-INS-MIN, MAX-INS-MAX, and MAX-DEL-MIN and (Type II)$p$-DEL-MAX, $p$-INS-MIN, $p$-INS-MAX, and $p$-DEL-MIN. In each of the eight problems, the input is a graph and the goal is to delete or insert edges so that the resulting graph has an orientation in which the maximum outdegree (taken over all vertices) is small or the minimum outdegree is large. We first present a framework that provides algorithms for solving all eight problems in polynomial time on unweighted graphs. Next we investigate the inapproximability of the edge-weighted versions of the problems, and design polynomial-time algorithms for six of the problems on edge-weighted trees.

程度受限的边缘修改问题的目标是找到一个跨度的子图或上图$H$ 输入无向图 $G$这样：（I）类型的边数$H$ 最小化或最大化，并且 $H$可以定向为满足对顶点生成的出学位的某些指定约束；或：（II型）的所有子图或supergraphs中$G$ 可以通过删除或插入固定数量的边来构造， $H$允许优化一些涉及顶点出度的目标的方向。本文介绍了八个新的与负载平衡相关的经度限制的边缘修改问题，称为（I型） MIN-DEL-MAX，MIN-INS-MIN，MAX-INS-MAX和MAX-DEL-MIN和（II型）$p$-DEL-MAX， $p$-INS-MIN， $p$-INS-MAX，以及 $p$-DEL-MIN。在这八个问题的每一个中，输入是一个图，目标是删除或插入边，以使生成的图具有这样的方向，即最大超出度（覆盖所有顶点）较小或最小超出度较大。我们首先提出一个框架，该框架提供用于解决多项式时间内非加权图上的所有八个问题的算法。接下来，我们研究问题的边缘加权版本的不可约性，并为边缘加权树上的六个问题设计多项式时间算法。