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We present algorithms of two flavors—one rooted in constraint satisfaction problems (CSPs) and the other in learning dynamics—to compute pure-strategy Nash equilibrium (PSNE) in k-dimensional congestion games (k-DCGs) and their variants. The two algorithmic approaches are driven by whether or not a PSNE is guaranteed to exist. We first show that deciding the existence of a PSNE in a k-DCG is NP-complete even when players have binary and unit demand vectors. For general cost functions (potentially non-monotonic), we devise a new CSP-inspired algorithmic framework for PSNE computation, leading to algorithms that run in polynomial time under certain assumptions while offering exponential savings over standard CSP algorithms. We further refine these algorithms for variants of k-DCGs. Our experiments demonstrate the effectiveness of this new CSP framework for hard, non-monotonic k-DCGs. We then provide learning dynamics-based PSNE computation algorithms for linear and exponential cost functions. These algorithms run in polynomial time under certain assumptions. For general cost, we give a learning dynamics algorithm for an (α, β)-approximate PSNE (for certain α and β). Lastly, we also devise polynomial-time algorithms for structured demands and cost functions. 

We study pure-strategy Nash equilibrium (PSNE) computation in 𝑘-dimensional congestion games (𝑘-DCGs) where the weights or demands of the players are 𝑘-dimensional vectors. We first show that deciding the existence of a PSNE in a 𝑘-DCG is NP-complete even for games when players have binary and unit demand vectors. We then focus on computing PSNE for 𝑘-DCGs and their variants with general, linear, and exponential cost functions. For general cost functions (potentially non-monotonic), we provide the first configuration-space framework to find a PSNE if one exists. For linear and exponential cost functions, we provide potential function-based algorithms to find a PSNE. These algorithms run in polynomial time under certain assumptions. We also study structured demands and cost functions, giving polynomial-time algorithms to compute PSNE for several cases. For general cost functions, we give a constructive proof of existence for an (𝛼, 𝛽)-PSNE (for certain 𝛼 and 𝛽), where 𝛼 and 𝛽 are multiplicative and additive approximation factors, respectively.