feat: ✨ New MPS refinement model

Author: vepain

pangebin/mps.tex

@@ -34,4 +34,32 @@ \subsection{The Maximum Plasmidness Score problem \MPS{}}\label{sec:method:mps}
     \objlabel{mps:obj:max_plasmidness_score} % chktex 25 % chktex 35
   \end{equation}
   where \(\displaystyle PlasmidnessScore = \sum_{i \in \Fragments{}} \plm{i}\frag{i}\).
-\end{definition}
+\end{definition}
+
+\subsubsection{Find a solution of \MPS{} maximizing the coverage score and then the GC score}\label{sec:method:mps:refinement}
+
+The solution set \(S_\MPS{}\) for \MPS{} contains solution with the same value of \(PlasmidnessScore\) and different values for \(CoverageFlow\) and \(GCScore\) because of~\Cref{mgc:cst:coverage_score_near_optimal,mps:cst:gc_probability_score_near_optimal}.
+The final refinement stage consists in fixing the value of \(PlasmidnessScore\) and finding the subset of \(S_\MPS{}\) maximizing first \(CoverageFlow\) and then \(GCScore\).
+It results in a last MILP formulation \(\MPS{}'\):
+
+\begin{definition}{\MPS{}' MILP variables}{milp:mpsprime:variables}
+  The varibales of \MPS{} are kept.
+\end{definition}
+
+\begin{definition}{\MPS{}' MILP constraints}{milp:mpsprime:constraints}
+  To the constraints in \Cref{definition:milp:mps:constraints} we add the following:
+
+  The plasmidness score must be equal to the plasmidness score \(P\) found in \MPS{}:
+  \begin{equation}
+    PlasmidnessScore = P
+    \cstlabel{mpsprime:cst:fixed_plasmidness_score} % chktex 25
+  \end{equation}
+\end{definition}
+
+\begin{definition}{\MPS{}' MILP objective function}{milp:mpsprime:objective}
+  The objective function aims to maximize the coverage flow prior to the GC score:
+  \begin{equation}
+    \max ~ |\Fragments{}| CoverageFlow + GCScore
+    \objlabel{mpsprime:obj:max_coverage_flow_then_gc_probability_score} % chktex 25 % chktex 35
+  \end{equation}
+\end{definition}

prelude.tex

@@ -18,7 +18,15 @@ \subsection*{Aims}\label{sec:prelude:aims}
 \end{itemize}
 
 \begin{newfeatbox}
-  Done since last times:
+  \textbf{2025--03--24} Done since last times:
+
+  \begin{CheckList}{Task}
+    \Task{done}{Propose a refinement of the \MPS{} problem (\Cref{sec:method:mps:refinement})}
+  \end{CheckList}
+\end{newfeatbox}
+
+\begin{newfeatbox}
+  \textbf{2025--03--17} Done since last times:
 
   \begin{CheckList}{Task}
     \Task{done}{Model a connected-flow in MILP to avoid iteratively resolving the problem until the solution subgraph has only one connected component. This answers one point of the PlasBin-flow discussion (\Cref{sec:method:mcf})}