800 research outputs found
Combinatorial simplex algorithms can solve mean payoff games
A combinatorial simplex algorithm is an instance of the simplex method in
which the pivoting depends on combinatorial data only. We show that any
algorithm of this kind admits a tropical analogue which can be used to solve
mean payoff games. Moreover, any combinatorial simplex algorithm with a
strongly polynomial complexity (the existence of such an algorithm is open)
would provide in this way a strongly polynomial algorithm solving mean payoff
games. Mean payoff games are known to be in NP and co-NP; whether they can be
solved in polynomial time is an open problem. Our algorithm relies on a
tropical implementation of the simplex method over a real closed field of Hahn
series. One of the key ingredients is a new scheme for symbolic perturbation
which allows us to lift an arbitrary mean payoff game instance into a
non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction
expanded, 18 pages, 3 figure
The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average
We introduce an algorithm which solves mean payoff games in polynomial time
on average, assuming the distribution of the games satisfies a flip invariance
property on the set of actions associated with every state. The algorithm is a
tropical analogue of the shadow-vertex simplex algorithm, which solves mean
payoff games via linear feasibility problems over the tropical semiring
. The key ingredient in our approach is
that the shadow-vertex pivoting rule can be transferred to tropical polyhedra,
and that its computation reduces to optimal assignment problems through
Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP
2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
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