[Paper Review] Stochastic Vehicle Routing with Recourse

Inge Li Goertz, Viswanath Nagarajan+1 more|arXiv (Cornell University)|Feb 26, 2012

Vehicle Routing Optimization Methods31 references5 citations

TL;DR

This paper presents an $O(\log^2 n \cdot \log(n\lambda))$-approximation algorithm for the Stochastic Vehicle Routing Problem with Recourse, where a fixed route is planned using demand distributions, and a more expensive recourse route is computed after demand realization. The approach reduces the problem to knapsack rank-function orienteering and uses sampling-based linear programming relaxation with randomized rounding to achieve the approximation guarantee.

ABSTRACT

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.

Motivation & Objective

• To address the two-stage stochastic vehicle routing problem where demands are random and recourse costs are inflated by a factor $\lambda$.
• To design a fixed-route strategy that minimizes the sum of fixed route cost and expected inflated recourse cost.
• To provide a polynomial-time approximation algorithm with provable performance guarantees under arbitrary demand distributions.
• To establish a tight approximation ratio by relating the problem to knapsack rank-function orienteering.
• To prove a Unique Games Conjecture-based $\omega(1)$ hardness of approximation, showing the algorithm's ratio is nearly optimal.

Proposed method

• The problem is reduced to a special case of submodular orienteering called knapsack rank-function orienteering.
• A sampling-based estimation technique is used to approximate the expected minimum spanning tree (MST) cost on random subsets of vertices with probabilistic presence.
• An integer program is formulated with variables for partitioning vertices into those served by the fixed route ($D_1$) and those requiring recourse ($D_2$).
• The IP uses $m = \text{poly}(n, \lambda)$ samples of demand realizations to estimate the expected recourse cost via empirical MST and flow values.
• The LP relaxation of the IP is solved, and a randomized rounding rule sets $D_1 = \{v : x_v > 1/2\}$ to define the fixed route.
• The fixed route is constructed as $O(\log(n\lambda)/\log\!\log(n\lambda))$ copies of an approximate VRP solution on $D_1$, with recourse computed greedily on remaining demands.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio for Stochastic VRP with Recourse under arbitrary demand distributions?
• RQ2 Can the problem be effectively reduced to a known combinatorial optimization problem with known approximation guarantees?
• RQ3 How can the expected recourse cost be estimated accurately enough to design a polynomial-time algorithm?
• RQ4 Is the proposed approximation ratio tight, or can stronger inapproximability bounds be proven?
• RQ5 Does the Unique Games Conjecture imply constant-factor inapproximability for Stochastic VRP with Recourse?

Key findings

• The paper achieves an $O(\log^2 n \cdot \log(n\lambda))$-approximation for Stochastic VRP with Recourse under arbitrary demand distributions.
• The algorithm's performance is derived by reducing the problem to knapsack rank-function orienteering and improving the approximation ratio for this subproblem.
• The expected recourse cost is estimated via $\text{poly}(n, \lambda)$ samples, ensuring concentration within a constant additive error.
• The LP relaxation with sampling-based estimation yields a solution that approximates the true objective within a constant factor with high probability.
• A Unique Games Conjecture-based $\omega(1)$ hardness result is established, showing that no constant-factor approximation exists unless the conjecture fails.
• The algorithm achieves a logarithmic approximation ratio on star-like metrics, matching the known inapproximability lower bound up to logarithmic factors.