## 2 Materials and Methods
### The Expansion Operators  
We will discuss two Expansion operators, \"Gradient Expansion operator\" and \"Mutation Expansion operator\", for our tree search method. We consider the following typical case. Assume that \(\alpha\) and \(\beta\) are Partitions of \(S=\{1,\cdots,n\}\) satisfying  
\[\alpha\prec\beta. \tag{9}\]  
Then, based on **Fact D**, we have  
\[\Omega_{\alpha}\subset\Omega_{\beta}\]  
Figure 6: Phases of our tree search. Let  
\[S=I_{1}\cup\cdots\cup I_{m_{\beta}} \tag{10}\]  
be the decomposition of \(S=\{1,\cdots,n\}\) into a union of _Interval Components_ with respect to the Partition \(\beta\). Assume that \(\mathbf{U}\in\Omega_{\alpha}\). Since \(\Omega_{\alpha}\subset\Omega_{\beta}\), we note that, for _each_ Interval Component \(I_{k}\), only checkers of the _same_ type are placed on \(I_{k}\) by the arrangement \(\mathbf{U}\). We assume that, on the Interval Component \(I_{k}\), the _checkers_ arranged by \(\mathbf{U}\) are all of the type  
\[U_{k}. \tag{11}\]  
Assume that \(\mathbf{V}\in\Omega_{\beta}\) is a \"_qualified_ arrangement with respect to the _Partition_\(\beta\)\". Assume that, on the Interval Component \(I_{k}\), the checkers arranged by \(\mathbf{V}\) are all of the type  
\[V_{k}. \tag{12}\]  
We say that the arrangement \(\mathbf{V}\in\Omega_{\beta}\) of checkers is a \"_child-node_\" of \(\mathbf{U}\) in \(\Omega_{\beta}\), by the \"_Gradient Expansion operator_\", if the following GE Condition is satisfied. **GE (Gradient Expansion) Condition. **_There is a sequence \(GE_{UV}=\{\phi(1),\cdots,\phi(q)\}\) of positive integers taken from \(\{1,\cdots,m_{\beta}\}\), satisfying_  
\[\phi(s)+2\leq\phi(s+1),\text{for all }s=1,\cdots,(q-1), \tag{13}\]  
_such that_  
\[U_{\phi(s)}\
eq U_{\phi(s)+1},\text{for all }s=1,\cdots,q \tag{14}\]  
_For each \(=1,\cdots,q\), we have_  
\[ \begin{array}{c}(V_{\phi(s)},V_{\phi(s)+1})=(U_{\phi(s)},U_{\phi(s)})\text{ or }(V_{\phi(s)},V_{\phi(s)+1})=(U_{\phi(s)},U_{\phi(s)+1})\text{ or }\\ (V_{\phi(s)},V_{\phi(s)+1})=(U_{\phi(s)+1},U_{\phi(s)+1}),\end{array}  \tag{15}\]  
_for the arrangements \(\mathbf{U}\) and \(\mathbf{V}\)_ of checkers on the pair \((I_{\phi(s)},I_{\phi(s)+1})\) of adjacent Interval components\(I_{\phi(s)}\) and \(I_{\phi(s)+1}\). _Besides, outside these pairs_\((I_{\phi(s)},I_{\phi(s)+1})\) of adjacent Interval components, \(s=1,\cdots,q\), we have_  
\[V_{K}=U_{k},\text{ if\/is\/not\/in\/in\/in\/in\/in\/in\/in\/in\/in\/  
This \"_Gradient Expansion operator_\" is motivated by the usual _Gradient Descent_ method adopted in _Machine Learning_. See, Ref. [45], for examples. Search through the simple _Gradient Descent_ method usually converges _slowly_. (To expedite the training process of AI, Stochastic Gradient Descent was introduced. See Ref. [45].)  
Now we discuss the _\"Mutation Expansion operator_\" under the assumptions (9) to (12). We say that the arrangement \(\mathbf{V}\in\Omega_{\beta}\) of checkers is a _\"child-node\"_ of \(\mathbf{U}\) in \(\Omega_{\beta}\), by the _\"Mutation Expansion operator\"_, if the following **ME Condition** is satisfied.  
**ME (Mutation Expansion) Condition.**_There is a sequence \(GE_{UV}=\{\phi(1),\cdots,\phi(q)\}\) of positive integers taken from \(\left\{1,\cdots,m_{\beta}\right\}\), satisfying_  
\[\phi(s)+2\leq\phi(s+1)\text{, forall }s=1,\cdots,(q-1), \tag{17}\]  
_such that_  
\[U_{\phi(s)}\
eq U_{\phi(s)+1}\text{, forall }s=1,\cdots,q \tag{18}\]  
_For each \(s=1,\cdots,q\), we have_  
\[(V_{\phi(s)},V_{\phi(s)+1})=(U_{\phi(s)},U_{\phi(s)+1})or\ (V_{\phi(s)},V_{\phi(s)+1})=(U_{\phi(s)+1},U_{\phi(s)}), \tag{19}\]  
_for the arrangements \(\mathbf{U}\) and \(\mathbf{V}\) of checkers on the pair \((I_{\phi(s)},I_{\phi(s)+1})\) of adjacent Interval components \(I_{\phi(s)}\) and \(I_{\phi(s)+1}\). Besides, **outside** these pairs \((I_{\phi(s)},I_{\phi(s)+1})\) of adjacent Interval components, \(s=1,\cdots,q\), we have_  
\[V_{K}=U_{k}\text{, if }k\text{ is not in the union }\{\phi(1),\cdots,\phi(q)\}\cup\{\phi(1)+1,\cdots,\phi(q)+1\}. \tag{20}\]  
**Example 6**.