4 research outputs found
ARRIVAL: Next Stop in CLS
We study the computational complexity of ARRIVAL, a zero-player game on
-vertex switch graphs introduced by Dohrau, G\"{a}rtner, Kohler,
Matou\v{s}ek, and Welzl. They showed that the problem of deciding termination
of this game is contained in . Karthik C. S.
recently introduced a search variant of ARRIVAL and showed that it is in the
complexity class PLS. In this work, we significantly improve the known upper
bounds for both the decision and the search variants of ARRIVAL.
First, we resolve a question suggested by Dohrau et al. and show that the
decision variant of ARRIVAL is in . Second, we
prove that the search variant of ARRIVAL is contained in CLS. Third, we give a
randomized -time algorithm to solve both variants.
Our main technical contributions are (a) an efficiently verifiable
characterization of the unique witness for termination of the ARRIVAL game, and
(b) an efficient way of sampling from the state space of the game. We show that
the problem of finding the unique witness is contained in CLS, whereas it was
previously conjectured to be FPSPACE-complete. The efficient sampling procedure
yields the first algorithm for the problem that has expected runtime
with .Comment: 13 pages, 6 figure
The Classes PPA-k: Existence from Arguments Modulo k
The complexity classes PPA-, , have recently emerged as the main
candidates for capturing the complexity of important problems in fair division,
in particular Alon's Necklace-Splitting problem with thieves. Indeed, the
problem with two thieves has been shown complete for PPA = PPA-2. In this work,
we present structural results which provide a solid foundation for the further
study of these classes. Namely, we investigate the classes PPA- in terms of
(i) equivalent definitions, (ii) inner structure, (iii) relationship to each
other and to other TFNP classes, and (iv) closure under Turing reductions
Classification of Search Problems and Their Definability in Bounded Arithmetic
We present a new framework for the study of search problems and their definability in bounded arithmetic. We identify two notions of complexity of search problems: verification complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b i -definability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witness-oracle Turing machines. Based o