全1问题的新版本
New versions of the all-ones problem
摘要：
我们研究了三种新的全一问题和最小全一问题。原始的全一问题可被形象的称为顶点点亮顶点问题，而这三类新问题则分别被称为顶点点亮边问题，边点亮顶点问题，边点亮边问题.这些新问题都是由顶点点亮边问题和边点亮边问题组成.顶点-顶点问题已经被广泛地研究。例如，得到了解的存在性和求解的有效算法，证明了一般图的最小顶点-顶点问题是NP-完全的，树的最小顶点-顶点问题可以在线性时间内求解等等.本文证明了顶点-边问题，一个图有解当且仅当它是二部的，因此它只有两个可能解和最优解.文中还给出了一个线性化程序。对于边-点问题，我们证明了一个图有解当且仅当它包含偶数个顶点。通过证明最小边-顶点问题可以多项式转化为最小权完美匹配问题，得到了最小边-顶点问题一般可以在多项式时间内求解.对于图的线图，边-边问题被简化为点-点问题。
英文摘要：
We study three new versions of the All-Ones Problem and the Minimum All-Ones Problem. The original All-Ones Problem is simply called the Vertex-Vertex Problem, and the three new versions are called the Vertex-Edge Problem, the Edge-Vertex Problem and the Edge-Edge Problem, respectively. The Vertex-Vertex Problem has been studied extensively. For example, existence of solutions and efficient algorithms for finding solutions were obtained, and the Minimum Vertex-Vertex Problem for general graphs was shown to be NP-complete and for trees it can be solved in linear time, etc. In this paper, for the Vertex-Edge Problem, we show that a graph has a solution if and only if it is bipartite, and therefore it has only two possible solutions and optimal solutions. A linear program version is also given. For the Edge-Vertex Problem, we show that a graph has a solution if and only if it contains even number of vertices. By showing that the Minimum Edge-Vertex Problem can be polynomially transformed into the Minimum Weight Perfect Matching Problem, we obtain that the Minimum Edge-Vertex Problem can be solved in polynomial time in general. The Edge-Edge Problem is reduced to the Vertex-Vertex Problem for the line graph of a graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
