Smooth Routing in Decaying Trees

Abstract

Motivated by evacuation scenarios arising in extreme events such as flooding or forest fires, we study the problem of smoothly scheduling a set of paths in graphs where connections become impassable at some point in time. A schedule is smooth if no two paths meet on an edge and the number of paths simultaneously located at a vertex does not exceed its given capacity. We study the computational complexity of the problem when the underlying graph is a tree, in particular a star or a path. We prove that already in these settings, the problem is NP-hard even with further restrictions on the capacities or on the time when all connections ceased. We provide an integer linear program (ILP) to compute the latest possible time to evacuate. Using the ILP and its relaxation, we solve sets of artificial (where each underlying graph forms either a path or star) and semi-artificial instances (where the graphs are obtained from German cities along rivers), study the runtimes, and compare the results of the ILP with those of its relaxation.

Figures (13)

• Figure 1: Example instance of SRDG on a decaying path with 6 vertices (top) and a solution witnessing feasibility (bottom). The capacity of each vertex $v_i$ is encircled. (Top) On each connection $e$ we indicate its traversal time and deadline as $[\mathop{\mathrm{\theta}}\nolimits(e)\vert d(e)]$. Each of the paths $P_1,\dots,P_{10}$ is described by the vertically aligned source and sink and by an arc for its direction. (Bottom) For each path its timely location on any vertex or connections is drawn. E.g., the path $P_1$ (blue) starts at time step 1 from its source $v_1$, arrives at and departs from $v_2$ at time step 2, and so on. Red dashed lines/arrows indicate the deadline on the respective connection.
• Figure 2: Illustration of the temporal graph constructed by Klobas et al. KlobasMMNZ23, where the time label set of each edge (i.e., when it is present) is the union of the sets drawn below of it. Formally, they have no vertex capacities which translates to capacity one in our model. For the two color classes 1 and $k$, the corresponding paths are depicted in green/blue and orange/magenta, respectively.
• Figure 3: Red paths block all color classes, green none, and blue only the fist color class. Magenta and orange blocks indicate all possible starting times of the color-paths of color $1$ and $2$, respectively. Here, $A_1=L_1=\{2,7\}$, $A_2=\{5,9\}$, and $L_2=\{2,5,7,9\}$.
• Figure 4: Illustration to \ref{['const:nphard:stars:cap']}. For each edge $e$ we indicate the traversal time and deadline as $[\mathop{\mathrm{\theta}}\nolimits(e)\vert d(e)]$. Here, we picked $v$ and $e$ with $v\in e$ for illustration.
• Figure 5: Illustration to \ref{['constr:nphard:stars:cap:one']}. For each edge $e$ we indicate the traversal time and deadline as $[\mathop{\mathrm{\theta}}\nolimits(e)\vert d(e)]$ and short as $[[\mathop{\mathrm{\theta}}\nolimits(e)]]$ if $d(e)=\mathop{\mathrm{\theta}}\nolimits(e)+1$ (i.e., any path on such an edge has to depart on time step 1). Here, we picked $v$ and $e_i$ with $v\in e_i$ for illustration.

Theorems & Definitions (37)

• theorem 1
• lemma 1
• proof
• lemma 2
• proof
• proof : Proof of \ref{['thm:nphard:paths:capone']}
• theorem 2
• lemma 3
• proof
• lemma 4