2D-TUCKER Is PPAD-Complete
نویسنده
چکیده
Tucker’s lemma states that if we triangulate the unit disc centered at the origin and color the vertices with {1,−1, 2,−2} in an antipodal way (if |z| = 1, then the sum of the colors of z and −z is zero), then there must be an edge for which the sum of the colors of its endpoints is zero. But how hard is it to find such an edge? We show that if the triangulation is exponentially large and the coloring is determined by a deterministic Turing-machine, then this problem is PPAD-complete which implies that there is not too much hope for a polynomial algorithm.
منابع مشابه
2-D Tucker is PPA complete
The 2-D Tucker search problem is shown to be PPA-hard under many-one reductions; therefore it is complete for PPA. The same holds for k-D Tucker for all k ≥ 2. This corrects a claim in the literature that the Tucker search problem is in PPAD.
متن کاملUnderstanding PPA-Completeness
The search complexity classes PPA and PPAD were proposed by Papadimitriou twenty years ago for characterizing the computational difficulties of many interesting natural search problems. While many members in the complete class of PPAD, PPAD-complete, are established in the past twenty years, the understanding of the PPA-complete class falls far behind. We consider the problem of finding a fully...
متن کاملOn the Complexity of 2D Discrete Fixed Point Problem
While the 3-dimensional analogue of the Sperner problem in the plane was known to be PPAD-complete, the complexity of the 2DSPERNER itself is not known to be PPAD-complete or not. In this paper, we settle this open problem proposed by Papadimitriou [7] fifteen years ago. This also allows us to derive the computational complexity characterization of a discrete version of the 2-dimensional Brouwe...
متن کاملTree-like resolution complexity of two planar problems
We consider two CSP problems: the first CSP encodes 2D Sperner’s lemma for the standard triangulation of the right triangle on n2 small triangles; the second CSP encodes the fact that it is impossible to match cells of n× n square to arrows (two horizontal, two vertical and four diagonal) such that arrows in two cells with a common edge differ by at most 45◦, and all arrows on the boundary of t...
متن کاملOn the complexity of Sperner’s Lemma
We present several results on the complexity of various forms of Sperner’s Lemma. In the black-box model of computing, we exhibit a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is essentially linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get...
متن کامل