# Rendezvous on the Line with Different Speeds and Markers That Can Be Dropped at Chosen Time
## 5 Solution of the Problem with a Marker Held by the Slow Player  
The marker is held by one of the players and may be dropped off at any given time. Once dropped off, the marker is to be found by the other player when is passes at the location. Once found, the player stops following the original strategy and continues in  
Figure 4: Optimal strategy for \(v\in]0.619,0.990]\). the same direction until rendezvous occurs. We first assume that the marker is held by player I, who moves with the lowest speed bounded by \(v\leq 1\), and denote \(z\) as the dropping time. There are four possibilities, \(z\in[0,t_{1}],z\in[t_{1},t_{2}],z\in[t_{2},t_{3}]\) and \(z\in[t_{3},t_{4}]\), each one leading to a family of linear programs to solve. It happens that only the first case \(z\in[0,t_{1}]\) is relevant. For all other cases the optimal solutions do not make use of the marker and are given in Section 4. The strategy of player I is now given by \((a_{0},\ a_{1},\ a_{2},\ a_{3},\ a_{4})\) where \(a_{i}\) indicates whether \(v_{i}\) is positive or negative, i.e., the speed \(v_{i}\) are always assumed positive and the motions are depending on the product \(a_{i}\cdot v_{i}\). With respect to Section 4, the only novelty is the introduction of \(a_{0}\), see Equation (9) (compare with (7) with no marker). Agents of player II can find the marker or not. Hence, for each agent we must generate two linear programs each one assuming the agent finds the marker or not. Actually, for the first agent rendezvous (agent \((o_{1},d_{1})\)), we do not need to differentiate whether the marker is found or not, as the equations are the same. We use the new variable \(k_{1}\) to indicate that agent \((o_{2},d_{2})\) finds the marker \(k_{1}=1\) or not (\(k_{1}=0\)), in the interval \([0,t_{1}]\). Again if the marker is found by \((o_{2},d_{2})\) in the interval \([t_{1},t_{2}]\), the equations do not change. The variable \(k_{21},k_{22}\) indicate that agent \((o_{3},d_{3})\) finds the marker in the interval \([0,t_{1}]\) or \([t_{1},t_{2}]\), respectively. Finally, the variables \(k_{31},k_{32}\) and \(k_{33}\) indicate that agent \((o_{4},d_{4})\) finds the marker in the interval \([0,t_{1}]\), \([t_{1},t_{2}]\) or \([t_{2},t_{3}]\), respectively. This leads to the family of linear programs shown in Equations (15)-(17) (we denote \(\Delta t_{i}=(t_{i}-t_{i-1})\) and \(t_{iz}\) the time at which the marker is found by \((o_{i},b_{i})\)). \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\] \[o_{1}+d_{1}b_{1}t_{1}=a_{0}vz+a_{1}vt_{1}\] ( * ) \[k_{1}(o_{2}+d_{1}b_{2}t_{1}+d_{2}b_{2}\Delta t_{2})+\] \[(1-k_{1})(o_{2}+d_{1}b_{2}t_{1}+d_{1}b_{2}(t_{1}+\Delta t_{2}-t_{ 1z}))=a_{0}vz+a_{1}vt_{1}+a_{2}vt_{2}\] \[(1-k_{1})(o_{2}+d_{1}b_{2}t_{1z})=(1-k_{1})a_{0}vz\] ( ** ) \[k_{21}k_{22}(o_{3}+d_{1}b_{3}t_{1}+d_{2}b_{3}\Delta t_{2}+d_{3}b _{3}\Delta t_{3})+\] \[(1-k_{21})(o_{3}+d_{1}b_{3}t_{2z}+d_{1}b_{3}(t_{1}+\Delta t_{2}+ \Delta t_{3}-t_{2z}))+\] \[(1-k_{22})(o_{3}+d_{1}b_{3}t_{1}+d_{2}b_{3}t_{2}z+d_{2}b_{3}( \Delta t_{2}+\Delta t_{3}-t_{2z}))=\] \[a_{0}vz+a_{1}vt_{1}+a_{2}vt_{2}+a_{3}vt_{3}\] \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]0 ( *** ) \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]1 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]2 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]3 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]4 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]5 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]6 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]7 \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]8 ( *** ) \[\min_{\Delta_{i}}\ t_{1}+t_{2}+t_{3}+t_{4}\]9  
In this set of equations, the first one is the minimization problem to be solved. Notice that we minimize the sum of the rendezvous time while the number given in the Introduction and Results sections are the average of the rendezvous times. There are four sets of equations, (*), (**), (***) and (***). Equation (*) is the constraint that player I rendezvous with agent \((o_{1},b_{1})\) at time \(t_{1}\). Agent \((o_{1},b_{1})\) may find the marker before time \(t_{1}\).