
Unique End of Potential Line
This paper studies the complexity of problems in PPAD ∩ PLS that have un...
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The Computational Complexity of Orientation Search Problems in CryoElectron Microscopy
In this report we study the problem of determining threedimensional ori...
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End of Potential Line
We introduce the problem EndOfPotentialLine and the corresponding comple...
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Nodimensional Tverberg Theorems and Algorithms
Tverberg's theorem is a classic result in discrete geometry. It states t...
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Tarski's Theorem, Supermodular Games, and the Complexity of Equilibria
The use of monotonicity and Tarski's theorem in existence proofs of equi...
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On the Complexity of Moduloq Arguments and the ChevalleyWarning Theorem
We study the search problem class PPA_q defined as a moduloq analog of ...
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Computational Intractability of Julia sets for real quadratic polynomials
We show that there exist real parameters c for which the Julia set J_c o...
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Computational Complexity of the αHamSandwich Problem
The classic HamSandwich theorem states that for any d measurable sets in ℝ^d, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and wellseparated, then for any given α_1, …, α_d ∈ [0, 1], there is a unique oriented hyperplane that cuts off a respective fraction α_1, …, α_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the αHamSandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^d3), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the HamSandwich problem, which is now known to be PPAcomplete (FilosRatsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new subclass of CLS (Continuous Local Search) called Unique EndofPotential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the αHamSandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first nontrivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the Pmatrix linear complementarity problem.
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