32 research outputs found
A PCP Characterization of AM
We introduce a 2-round stochastic constraint-satisfaction problem, and show
that its approximation version is complete for (the promise version of) the
complexity class AM. This gives a `PCP characterization' of AM analogous to the
PCP Theorem for NP. Similar characterizations have been given for higher levels
of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the
result for AM might be of particular significance for attempts to derandomize
this class.
To test this notion, we pose some `Randomized Optimization Hypotheses'
related to our stochastic CSPs that (in light of our result) would imply
collapse results for AM. Unfortunately, the hypotheses appear over-strong, and
we present evidence against them. In the process we show that, if some language
in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there
exist hard-on-average optimization problems of a particularly elegant form.
All our proofs use a powerful form of PCPs known as Probabilistically
Checkable Proofs of Proximity, and demonstrate their versatility. We also use
known results on randomness-efficient soundness- and hardness-amplification. In
particular, we make essential use of the Impagliazzo-Wigderson generator; our
analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page
Mastermind is NP-Complete
In this paper we show that the Mastermind Satisfiability Problem (MSP) is
NP-complete. The Mastermind is a popular game which can be turned into a
logical puzzle called Mastermind Satisfiability Problem in a similar spirit to
the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its
intrinsic computational property that makes it challenging and interesting.
This serves as an addition to our knowledge about a host of other puzzles, such
as Minesweeper, Mah-Jongg, and the 15-puzzle
Trainyard is NP-Hard
Recently, due to the widespread diffusion of smart-phones, mobile puzzle
games have experienced a huge increase in their popularity. A successful puzzle
has to be both captivating and challenging, and it has been suggested that this
features are somehow related to their computational complexity \cite{Eppstein}.
Indeed, many puzzle games --such as Mah-Jongg, Sokoban, Candy Crush, and 2048,
to name a few-- are known to be NP-hard \cite{CondonFLS97,
culberson1999sokoban, GualaLN14, Mehta14a}. In this paper we consider
Trainyard: a popular mobile puzzle game whose goal is to get colored trains
from their initial stations to suitable destination stations. We prove that the
problem of determining whether there exists a solution to a given Trainyard
level is NP-hard. We also \href{http://trainyard.isnphard.com}{provide} an
implementation of our hardness reduction
The Complexity of Local Proof Search in Linear Logic (Extended Abstract)
AbstractProof search in linear logic is known to be difficult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicative-additive fragment of propositional linear logic, mall, is known to be PSPACE-complete, and the pure multiplicative fragment, mll, is known to be np-complete. However, this still leaves open the possibility that there might be proof search heuristics (perhaps involving randomization) that often lead to a proof if there is one, or always lead to something close to a proof. One approach to these problems is to study strategies for proof games. A class of linear logic proof games is developed, each with a numeric score that depends on the number of certain preferred axioms used in a complete or partial proof tree. Using recent techniques for proving lower bounds on optimization problems, the complexity of these games is analyzed for the fragment mll extended with additive constants and for the fragment MALL. It is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy
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COMPLEXITY&APPROXIMABILITY OF QUANTIFIED&STOCHASTIC CONSTRAINT SATISFACTION PROBLEMS
Let D be an arbitrary (not necessarily finite) nonempty set, let C be a finite set of constant symbols denoting arbitrary elements of D, and let S and T be an arbitrary finite set of finite-arity relations on D. We denote the problem of determining the satisfiability of finite conjunctions of relations in S applied to variables (to variables and symbols in C) by SAT(S) (by SATc(S).) Here, we study simultaneously the complexity of decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. We present simple yet general techniques to characterize simultaneously, the complexity or efficient approximability of a number of versions/variants of the problems SAT(S), Q-SAT(S), S-SAT(S),MAX-Q-SAT(S) etc., for many different such D,C ,S, T. These versions/variants include decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. Our unified approach is based on the following two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic represent ability. Some of the results extend the earlier results in [Pa85,LMP99,CF+93,CF+94O]u r techniques and results reported here also provide significant steps towards obtaining dichotomy theorems, for a number of the problems above, including the problems MAX-&-SAT( S), and MAX-S-SAT(S). The discovery of such dichotomy theorems, for unquantified formulas, has received significant recent attention in the literature [CF+93,CF+94,Cr95,KSW97
Solving Mahjong Solitaire boards with peeking
We first prove that solving Mahjong Solitaire boards with peeking is
NP-complete, even if one only allows isolated stacks of the forms /aab/ and
/abb/. We subsequently show that layouts of isolated stacks of heights one and
two can always be solved with peeking, and that doing so is in P, as well as
finding an optimal algorithm for such layouts without peeking.
Next, we describe a practical algorithm for solving Mahjong Solitaire boards
with peeking, which is simple and fast. The algorithm uses an effective pruning
criterion and a heuristic to find and prioritize critical groups. The ideas of
the algorithm can also be applied to solving Shisen-Sho with peeking.Comment: 10 page