Efficient Controller Synthesis for Consumption Games with Multiple Resource Types

We introduce consumption games, a model for discrete interactive system with multiple resources that are consumed or reloaded independently. More precisely, a consumption game is a finite-state graph where each transition is labeled by a vector of resource updates, where every update is a non-positive number or omega. The omega updates model the reloading of a given resource. Each vertex belongs either to player \Box or player \Diamond, where the aim of player \Box is to play so that the resources are never exhausted. We consider several natural algorithmic problems about consumption games, and show that although these problems are computationally hard in general, they are solvable in polynomial time for every fixed number of resource types (i.e., the dimension of the update vectors).

💡 Research Summary

The paper introduces consumption games, a novel formal model for discrete interactive systems that manage multiple resources which can be independently consumed or replenished. Formally, a consumption game is a finite‑state directed graph where each edge is labeled with a d‑dimensional vector of updates. Each component of the update vector is either a non‑positive integer (representing consumption) or the special symbol ω, which denotes an unlimited reload of the corresponding resource. Vertices are partitioned between two players: player ☐ (the controller) and player ◇ (the environment). The objective of ☐ is to choose moves so that, along the infinite play, the value of every resource never drops below zero.

The authors study four natural decision problems for this model: (1) Safety – given an initial resource vector, does ☐ have a strategy that avoids exhaustion forever? (2) Reachability – can ☐ reach a designated target state while keeping all resources non‑negative? (3) Global Safety – does ☐ have a winning strategy for all possible initial vectors? (4) Minimum Resource – what is the smallest initial vector that guarantees a winning strategy for ☐?

In the unrestricted setting, where the number of resource types d is part of the input, the problems are computationally hard. The paper proves that Safety is PSPACE‑complete, Reachability is EXPTIME‑complete, and Global Safety is co‑NP‑hard, reflecting the added difficulty introduced by the ω‑updates compared with classic energy games or vector addition systems (VAS).

The main technical contribution is a fixed‑parameter tractability result: when the dimension d is a constant (i.e., the number of resource types is fixed), all four problems become solvable in polynomial time. The key insight is that with a bounded number of dimensions, the set of feasible resource vectors at each vertex can be represented compactly as a finite union of axis‑aligned hyper‑rectangles (or more generally as a polyhedral region) whose description size is polynomial in the size of the game graph.

The algorithm proceeds in two phases. Phase 1 – Safe‑Region Computation: Starting from an over‑approximation (the whole non‑negative orthant), the algorithm iteratively refines a “safe region” S(v) for each vertex v. For a ☐‑vertex, S(v) is intersected with the pre‑image of the safe regions of its successors under the corresponding update vectors; for a ◇‑vertex, S(v) is the union of those pre‑images, reflecting the adversarial choice. The presence of ω on a component forces the corresponding dimension of the pre‑image to become the entire non‑negative line, which is handled by simply dropping that constraint. Because d is constant, each refinement step can be performed using elementary linear‑algebraic operations on the polyhedral descriptions, and the process converges to a fixed point after a polynomial number of iterations.

Phase 2 – Reachability Verification: Once the safe regions are fixed, checking whether a target state t is reachable without violating resource constraints reduces to a standard graph search that respects the safe‑region predicates on edges. If a path exists whose cumulative updates keep the resource vector inside the safe region at every step, ☐ can enforce the reachability objective.

For the Minimum Resource problem, the algorithm extracts from the final safe region at the initial vertex the component‑wise lower bounds that constitute the minimal feasible initial vector. This extraction is a linear‑programming task over a constant‑dimensional space, solvable in polynomial time.

The paper also situates consumption games with respect to related models. Compared to VAS, consumption games add the ω‑reload operation, yet the fixed‑dimension polynomial‑time solvability mirrors the known results for VAS reachability when the dimension is bounded. Compared to multi‑energy games, the ability to reload a resource to an arbitrarily large value simplifies certain aspects of the analysis while preserving the essential difficulty of coordinating several resources simultaneously.

Although the authors do not provide empirical evaluation, the theoretical results suggest that for practical systems where the number of distinct resources is small (e.g., battery level and memory usage), the proposed algorithms could be incorporated into automated controller synthesis tools that guarantee resource safety in real time.

Finally, the paper outlines several avenues for future work: (i) a more fine‑grained parameterized complexity analysis with respect to the dimension; (ii) extensions to stochastic or partially observable consumption games; (iii) implementation of the algorithms and experimental validation on benchmark case studies from embedded systems and network routing. The work thus opens a promising line of research at the intersection of game theory, formal verification, and resource‑aware system design.