Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Constant Rank Bimatrix Games are PPAD-hard   The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-$0$, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank-$1$ games, leaving rank-2 and beyond unresolved.
  In this paper we show that NE computation in games with rank $\ge 3$, is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
  * An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
  * NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
  * Computing a symmetric NE of a symmetric bimatrix game with rank $\ge 6$ is
PPAD-hard.
  * Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
  The status of rank-$2$ games remains unresolved.

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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th