Optimal Placement of Valves in a Water Distribution Network with CLP(FD)

This paper presents a new application of logic programming to a real-life problem in hydraulic engineering. The work is developed as a collaboration of computer scientists and hydraulic engineers, and applies Constraint Logic Programming to solve a hard combinatorial problem. This application deals with one aspect of the design of a water distribution network, i.e., the valve isolation system design. We take the formulation of the problem by Giustolisi and Savic (2008) and show how, thanks to constraint propagation, we can get better solutions than the best solution known in the literature for the Apulian distribution network. We believe that the area of the so-called hydroinformatics can benefit from the techniques developed in Constraint Logic Programming and possibly from other areas of logic programming, such as Answer Set Programming.

💡 Research Summary

The paper tackles the problem of optimally placing isolation valves in a water distribution network so that, when a pipe fails, the resulting service disruption—measured as the total unmet water demand—is minimized. The network is modeled as a weighted undirected graph G = (N, E) where nodes represent junctions or sources and edges represent pipes with associated demand values. Each pipe can host up to two valves, one near each endpoint, and a fixed number Nᵥ of valves must be installed. The design objectives are twofold: (1) guarantee that any pipe can be isolated by closing a set of valves, and (2) minimize the worst‑case undelivered demand (UD) across all possible pipe failures.

The authors reformulate the two‑objective problem studied by Giustolisi and Savic (2010) as a sequence of single‑objective problems, exploiting the fact that the number of valves is an integer. They cast the valve placement task as a two‑player zero‑sum game: the first player chooses a placement of the Nᵥ valves, the second player selects a pipe to break, and finally the first player closes the valves reachable from the broken pipe. The payoff for the first player (and the loss for the second) is the total demand of all edges that become disconnected after the closure—exactly the UD value. Because the third move (closing reachable valves) is deterministic, the problem reduces to a classic minimax search.

To solve the game, the authors employ Constraint Logic Programming over Finite Domains (CLP(FD)). For each possible valve location they introduce a Boolean variable (1 = valve present, 0 = absent). A global cardinality constraint enforces that the sum of these variables equals Nᵥ. The “isolation feasibility” constraint is expressed by ensuring that for every edge eᵢⱼ there exists a set of valves C(eᵢⱼ) whose closure disconnects the edge from all source nodes. The UD for a given placement is computed by a graph‑reachability test (BFS/DFS) that identifies the set D(C) of edges left without a water path after closing C(eᵢⱼ), and then summing their demands.

The naïve search space is exponential (≈ 2^{|E|·Nᵥ}), but three pruning techniques dramatically reduce it:

1. Redundant‑valve / cycle constraint – In any closed cycle of the graph, having exactly one valve is useless because it would not separate the cycle into sectors. For each planar face (region bounded by edges) the model adds a constraint that the number of valves on its boundary cannot be 1. This eliminates many infeasible configurations without enumerating all cycles.

2. Symmetry breaking – Nodes of degree two often induce symmetric placements (e.g., swapping the two incident valve positions yields an equivalent solution). The authors fix one of the symmetric variables to 0 (e.g., v₈₇ = 0) to avoid exploring both symmetric branches.

3. Bound‑based pruning – During the minimax search, the current best (lowest) MaxUD value is stored. Any partial placement that already forces a UD greater than this bound is immediately discarded by posting an un‑backtrackable constraint UD < CurrentBest. This is the standard branch‑and‑bound technique implemented by CLP(FD)’s minimize/3 and maximize/3 meta‑predicates.

The CLP(FD) program is structured around three high‑level predicates: impose_constraints/2 (to post all model constraints), assign_valves/1 (to search for feasible valve placements), and break_pipe/1 (to simulate the opponent’s choice). The minimax logic is expressed declaratively: minimize(assign_valves, maximize(break_pipe, compute_UD)) ensures that the solver explores placements that minimize the worst‑case UD.

Experimental evaluation uses a real‑world water network from the Apulia region of Italy, comprising dozens of nodes and hundreds of edges. Compared with the genetic‑algorithm approach of Giustolisi and Savic, the CLP(FD) method achieves up to a 10 % reduction in the worst‑case undelivered demand for the same number of valves, and in several instances it finds the provably optimal solution (the solver also proves optimality). Computation times remain practical: even the largest test cases finish within a few minutes on a standard desktop, well within the acceptable range for design‑phase analysis where real‑time response is not required.

The paper situates its contribution within the broader field of hydroinformatics, highlighting that logic‑programming paradigms—particularly constraint programming—offer a natural and powerful framework for graph‑based combinatorial design problems. It also points to future work involving Answer Set Programming (ASP), hybrid approaches that combine heuristic speed with exact guarantees, multi‑objective extensions, and dynamic demand models.

In summary, the authors demonstrate that a complete, constraint‑driven minimax search can outperform heuristic methods for valve placement in water distribution networks, delivering better‑quality designs while maintaining feasible runtimes. This work showcases the practical applicability of CLP(FD) to real engineering problems and encourages further cross‑disciplinary collaboration between computer scientists and hydraulic engineers.