Totally Unimodular Congestion Games

Abstract

We investigate a new class of congestion games, called Totally Unimodular (TU) Congestion Games, where the players' strategies are binary vectors inside polyhedra defined by totally unimodular constraint matrices. Network congestion games belong to this class.In the symmetric case, when all players have the same strategy set, we design an algorithm that finds an optimal aggregated strategy and then decomposes it into the single players' strategies. This approach yields strongly polynomial-time algorithms to (i) find a pure Nash equilibrium, and (ii) compute a socially optimal state, if the delay functions are weakly convex. We also show how this technique can be extended to matroid congestion games.We then introduce some combinatorial TU congestion games, where the players' strategies are matchings, vertex covers, edge covers, and stable sets of a given bipartite graph. In the asymmetric case, we show that for these games (i) it is PLS-complete to find a pure Nash equilibrium even in case of linear delay functions, and (ii) it is NP-hard to compute a socially optimal state, even in case of weakly convex delay functions.

Computation and efficiency of potential function minimizers of combinatorial congestion games

“…Instead, we identify more general polyhedral properties of the describing systems that are sufficient to achieve (A) and (B). By doing so, we unify and extend the results in [16,24] to a much larger class of polytopal congestion games.…”

“…Our algorithm in (1) unifies and extends the aggregation/decomposition framework of [16]. In particular, the symmetric TUM congestion games and non-symmetric matroid congestion games (considered separately in [16]) fall into our class of polytopal congestion games satisfying IDP and box-TDI.…”

“…Our algorithm in (1) unifies and extends the aggregation/decomposition framework of [16]. In particular, the symmetric TUM congestion games and non-symmetric matroid congestion games (considered separately in [16]) fall into our class of polytopal congestion games satisfying IDP and box-TDI. Similarly, all combinatorial TUM congestion games (i.e., network, matching, edge cover, vertex cover and stable set congestion games) and their respective extensions to the maximum (or minimum) cardinality versions considered in [16] can be handled by our framework.…”

“…Only very recently, Del Pia et al [16] make further progress along these lines. The authors consider congestion games where the strategy sets of the players are given implicitly by a polyhedral description (see also [8]).…”

“…In order to tackle this question, we use a polyhedral approach similar to the ones used in [8,16]. But in contrast to these works, we do not restrict our attention to polyhedral descriptions arising from totally unimodular matrices only.…”