deletions | additions       diff --git a/sectionReducing_Opti.tex b/sectionReducing_Opti.tex  index 51da9f6..2c3892d 100644  --- a/sectionReducing_Opti.tex  +++ b/sectionReducing_Opti.tex 
...
\section{Reducing MAPF to the CA Problem}  Different CA solvers provide different gurentees, guarantees,  e.g. social whelfer, welfare,  truthfulness, prioritized agents and more. Solving MAPF using a CA solver provides these guarntees guarantees  for MAPF. A solution for the MAPF problem is composed from a set of paths, one per agent. Each single agent path is composed from a set of coordinates, i.e. \{vertex, time step\} pairs. The MAPF problem can be easily reduced to CA when every coordinanet coordinate  is considerd considered  an item and every path a bondle. bundle.  CA is defined only on a finite set of items. Consequently only a finite number of coordinates can be taken into account. This calls for a bound on the number of time steps required until all agents reach their goal. It was proven that a solvable MAPF instance can be solved in less than $|V|^3$ time steps \cite{kornhauser1984coordinating}. Using this bound allows us to consider a finite number of items, $|V|^3 \times |V| = |V|^4$.   In CA each agent $k_i$, is required to provide a utility evaluation for any boundle $b$, this utility is used to detrmine the maximal price that agent $k_i$ is willing to pay for $b$.  CA is defined only on a finite set of items. Consequently only a finite number of coordinates can be taken into account. This calls for a bound on the number of time steps required until all agents reach thier goal. It was proven that a solvable MAPF instance can be solved in less than $|V|^3$ time steps \cite{kornhauser1984coordinating}