Mathematical background

This page explains the mathematical framework used by ConstrainedShortestPath.jl, and its application to a few examples It's simplified version of the framework from Parmentier 2017, retricted to acyclic graphs.

Generalized constrained shortest path: problem formulation

Let $D = (V, A)$ be an acyclic digraph. Let $o, d$ two vertices, and $\mathcal{P}\in\mathcal{P}_{od}$ a set of $o-d$ path in $D$. We want to find a path $P^\star \in\mathcal{P}$ minimizing a given cost function $P\mapsto C(P)$:

\[\min_{P\in\mathcal{P}}C(P)\]

Setting

Before describing the details of the algorithms, we must first define some modeling elements specific to the treated problem.

We define the two following sets:
- Set of forward resources $Q^F$, provided with a partial order $\leq^F$.
- Set of backward resources $Q^B$, provided with a partial order $\leq^B$.
$(Q^F, \leq^F)$ and $(Q^B, \leq^B)$ should be lattices: a partially ordered set $(S,\leq)$ is a lattice if any pair $s, s'\in S$, admits a greatest lower bound $s\wedge s'$ (i.e. $\forall b\in S,\, (b \leq s \text{ and } b\leq s') \implies b\leq s\wedge s'$)
For every arc $a\in A$, we define two functions:
- A forward extension function : $f^F_a:Q^F \to Q^F$, increasing
- A backward extension function : $f^B_a : Q^B \to Q^B$, increasing
We also define:
- A cost function : $c:Q^F\times Q^B \to \mathbb{R}\cup\{+\infty\}$, increasing
- Resources corresponding to vertices $o$ an $d$: $q^F_o\in Q^F$, $q^B_d\in Q^B$

All these new theoretical elements should be connected to the initial problem formulation by following one of these (equivalent) properties:

For all $o-d$ path $P = (o=v_0, a_1, v_1, \dots, a_k, v_k=d)$, we have :
\[\forall i, i'\in [k],\, c(f^F_i \circ f^F_{i-1}\circ\dots\circ f^F_1(q^F_o), f^B_{i+1} \circ f^B_{i+2}\circ\dots\circ f^B_k(q^B_d)) = C(P)\in\mathbb{R}\cup\{+\infty\}\]
For all $o-d$ path $P$ and every decomposition of $P$ as $(R_1, R_2)$, we have $C(P)=c(q^F_{R_1}, q^B_{R_2})$, with:
- For all $o-v$ path $R_1 = (o=v_0, a_1, v_1, \dots, a_i, v_i=v)$:
\[q^F_{R_1} = f^F_{a_i} \circ f^F_{a_{i-1}}\circ\dots \circ f^F_{a_1}(q^F_o)\]
- For all $v-d$ path $R_2 = (v=v_0, a_1, v_1, \dots, a_j, v_j=d)$:
\[q^B_{R_2} = f^B_{a_1} \circ f^B_{a_{2}}\circ\dots \circ f^B_{a_j}(q^B_d)\]

For examples of resources sets, extension and cost functions, checkout the Examples section.

Algorithms

We first assume that we have accesss to a lower bound $b^B_v\in Q^B$ for every vertex $v$, such that $b^B_v\leq^B q^B_R$ for all $v-d$ path $R$.

Lemma 1 (Cut using bounds) : let $R_1$ an $o-v$ path. Then :
\[\forall P = (R_1, R_2)\in \mathcal{P}_{od},\, c(q^F_{R_1}, b^B_v)\leq C(P)\]
Lemma 2 (Dominance) : if $q^F_{R_1} \leq^F q^F_{R'_1}$, then for all $R_2$ :
\[c(q^F_{R_1}, q^B_{R_2})\leq c(q^F_{R'_1}, q^B_{R_2})\]
i.e. $C(P) \leq C(P')$ ($P=(R_1, R_2),\, P'=(R'_1, R_2)$)

Generalized $A^\star$

Initialization :
- $L \leftarrow \{\text{Empty path in }o\}$: partial paths to process
- $M_v \leftarrow \{ q_o^F\}$ if $v=o$, $\emptyset$ else
- $c^\star = +\infty$
- $P^\star = \text{None}$
While $L \neq \emptyset$
- Extract from $L$ an $o-v$ path $P$ minimizing $c(q^F_P, b^B_v)$
- Extend $P$ : For all $a\in \delta^+(v)$
  - $Q \leftarrow P + a$ ($w$ destination vertex of $q$)
  - $q^F_Q \leftarrow f^F_a(q^F_P)$
  - If $w = d$ and $c_Q < c^\star$
    - $c^\star \leftarrow c_q$
    - $P^\star \leftarrow Q$
  - Else, if $q^F_q$ is not dominated by any path in $M_w$ and $c(q^F_Q, b^B_w) < c^\star$
    - Add $q^F_Q$ to $M_w$
    - Remove from $M_w$ every path dominated by $Q$
    - Add $Q$ to $L$

Computing bounds

Generalized dynamic programming equation in $b^B$:

\[\boxed{\left\{\begin{aligned} & b_d = q_d\\ & b_v = \bigwedge_{\substack{a = (v, w)\\ a\in\delta^+(v)}} f^B_a(b_w) \end{aligned}\right.}\]

example : when $(f^F_a(x) = )f^B_a(x) = c_a + x$ and $\wedge = \min$ then $b_v = \min_a c_a + b_w$

$\implies$ Iterative algorithm along inverse topological order

Proposition : the solution is a lower bound (it's even the greater: $q_{b_v} = \bigwedge\limits_{p\in\mathcal{P_{vd}}}q^B_p$)

Examples

Shortest path

$q_o^F = 0,\, q^B_d = 0$
$f^F_a(q) = f^B_a(q) = c_a + q$
$c(q^F, q^B) = q^F + q^B$

Usual resource constrained shortest path

$q_o^F = 0,\, q^B_d = 0$
$f^F_a(q) = f^B_a(q) = c_a + q$
$c(q^F, q^B) = q^F + q^B + \mathbb{I}_{\{q^F + q^B \leq W\}}$

Stochastic routing

$c_p = \mathbb{E}[\sum_{v\in p}R_v] = \sum_{v\in p} \mathbb{E}[R_v] = \sum_{v\in p}\frac{1}{m}\sum_{j=1}^mR_v^j$ ($m$ scenarios)
$s_{a=(u,v)}$ : slack between $u$ and $v$
$a=(u,v) \rightarrow R_v^j = \max(R_u^j - s_{uv}, 0) + \varepsilon_v^j$
$q^F_u = \begin{bmatrix}C\\ R_u^1\\ \vdots \\ R_u^m\end{bmatrix}\in \mathbb{R}_+^{m+1}$, $q^B_v = \begin{bmatrix}g^1\\ \vdots \\ g^m\end{bmatrix}\in (\mathbb{R}_+^{\mathbb{R}_+})^m$
$q^F_o = 0,\, q^B_d = 0\mapsto 0$
$f^F_{a=(u, v)}(q_u^F) = \begin{bmatrix}C + \frac{1}{m}\sum_{j=1}^m R_v^j\\ R_v^1 = \max(R_u^1 - S_a, 0) + \varepsilon_v^1\\ \vdots \\ R_v^m = \max(R_u^m - S_a, 0) + \varepsilon_v^m\end{bmatrix}$
Let $r_{a=(u,v)}^j(R) = \max(R - S_a, 0) + \varepsilon_v^j$
$f^B_{a=(u,v)}(q_v^B) = \begin{bmatrix}R\mapsto r_a^1(R) + g^1(r_a^1(R)) \\ \vdots \\ R\mapsto r_a^m(R) + g^m(r_a^m(R))\end{bmatrix}$
$c\left(\begin{bmatrix}C\\ R_u^1\\ \vdots \\ R_u^m\end{bmatrix}, \begin{bmatrix}g^1\\ \vdots \\ g^m\end{bmatrix}\right) = C + \frac{1}{m}\sum_{j=1}^m g^j(R^j_u)$