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Computation of equilibria for congestion games has been an important research subject. In many realistic scenarios, each strategy of congestion games is given by a combination of elements that satisfies certain constraints; such games are called combinatorial congestion games. For example, given a road network with some toll roads, each strategy of routing games is a path (a combination of edges) whose total toll satisfies a certain budget constraint. Generally, given a ground set of n elements, the set of all such strategies, called the strategy set, can be large exponentially in n, and it often has complicated structures; these issues make equilibrium computation very hard. In this paper, we propose a practical algorithm for such hard equilibrium computation problems. We use data structures, called zero-suppressed binary decision diagrams (ZDDs), to compactly represent strategy sets, and we develop a Frank–Wolfe-style iterative equilibrium computation algorithm whose per-iteration complexity is linear in the size of the ZDD representation. We prove that an ϵ-approximate Wardrop equilibrium can be computed in O(poly(n)/ϵ) iterations, and we improve the result to O(poly(n) log ϵ−1) for some special cases. Experiments confirm the practical utility of our method.

Practical Frank–Wolfe Method with Decision Diagrams for Computing Wardrop Equilibrium of Combinatorial Congestion Games