## 3 Proposed Algorithm
### Definition of Opposite Path using the Current Optimum  
In order to achieve a better solution of TSP efficiently, we modify the definition of opposite path as follows. As in Figure 3, it is supposed that, \(n\) is the number of nodes in a graph and \(m\) is the population size. In fact, this figure can decomposed into three parts to comprehend the novel definition of opposite path clearly, which are shown in Figure 4. The first part, the optimal path in the current population \(P_{co}\) = [A1, A2,..., A1], as seen in Figure 4a, is translated into line A1A4 in Figure 3. In addition, similarly, the candidate path \(P_{i}\) = [B1, B2,..., B1] as the second part of Figure 3 is also translated into line B1B3. The clearly common ground between two important parts of Figure 3 is that, they are curves in Figure 4, instead, they are lines in Figure 3. The only reason for the different expression of the same TSP paths is to simplify the most critical figure in this paper. Based on the similar reason, the third part of Figure 3, all cities in the graph as seen in Figure 4c, is ignored in Figure 3. Of cause, you can image it is ubiquity for all cities in this graph in order to understand the following procedure easily. Figure 3: Novel definition of opposite path (Backward Ellipse). Figure 2: Opposite point using the current optimum defined in domain [_a_, _b_]. \(x\) is a candidate solution, \(\bar{x}\) is the opposite of \(x\) and \(\bar{x}_{co}\) is its opposite using the current optimum. Based on the preliminary and explanation above, the opposite path (Backward Ellipse), \(P_{i}^{o}\) = [O\({}_{1}\), O\({}_{2}\),..., O\({}_{1}\)], of any path \(P_{i}\) = [B\({}_{1}\), B\({}_{2}\),..., B\({}_{1}\)] can be defined according to the following procedure. (1) A set of remaining nodes include all nodes in the graph, and a set of visited nodes is empty in initialization stage. (2) Let \(k\) = 1. The start city, A\({}_{1}\)(B\({}_{1}\)) of optimal path \(P_{co}\) is also determined as the first node of path \(P_{i}\) and its opposite path \(P_{i}^{o}\). Then A\({}_{1}\)(B\({}_{1}\)) is labeled as a visited node and deleted from the set of remaining nodes. (3) Let \(k\) = \(k\) + 1. An ellipse is determined and denoted by \(E_{k}\), in which the \((k-1)\)th node and \(k\)th node of the optimal path \(P_{co}\) are the left focus and the right focus of the ellipse, respectively, and the \(k\)th node of \(P_{i}\) is on the boundary of the ellipse \(E_{k}\). (4) The \(k\)th node O\({}_{k}\) of opposite path \(P_{i}^{o}\) is the nearest node from the set of remaining nodes to the boundary of the ellipse \(E_{k}\). Then the \(k\)th node is labeled as a visited node and deleted from the set of remaining nodes. (5) Steps 3 and 4 above are iterated until all nodes are included in the set of visited nodes. Then the opposite path, \(P_{i}^{o}\), of any path \(P_{i}\) is defined well. As stated above, the ellipse \(E_{k}\) is determined by means of the \((k-1)\)th node, the \(k\)th node of the optimal path \(P_{co}\) and the \(k\)th node of \(P_{i}\). The \(k\)th node of opposite path \(P_{i}^{o}\) is close to the boundary of the ellipse \(E_{k}\). In other words, the \(k\)th node of \(P_{i}\) and the \(k\)th node of opposite path \(P_{i}^{o}\) have the same (or similar at least) distance from the \((k-1)\)th node and the \(k\)th node of the optimal path \(P_{co}\). Further, the \((k-1)\)th node and the \(k\)th node of the optimal path \(P_{co}\) can take the place of the whole optimal path \(P_{co}\). Then the path \(P_{i}\) and its corresponding opposite path \(P_{i}^{o}\) have the same (or similar at least) distance to the whole optimal path \(P_{co}\) in the current population in a sense. Quite similarly, we can define the other two methods, opposite path (Forward Ellipse) and opposite path (Circle), as shown in Figures 5 and 6, respectively. For opposite path (Forward Ellipse), the ellipse \(E_{k}\) is determined by means of the \(k\)th node, the \((k+1)\)th node of the optimal path \(P_{co}\) and the \(k\)th node of \(P_{i}\).