http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

The Another Solution Problem (ASP) of a problem Π is the following problem: for a given instance x of Π and a solution s to it, find a solution to x other than s. (The notion of ASP as a new class of problems was first introduced by Ueda and Nagao.) In this paper we consider n-ASP, the problem to find another solution when n solutions are given. In particular we consider ASP-completeness, the completeness with respect to the parsimonious reductions which allow polynomial-time transformation

/pdf/complexity-and-completeness-of-finding-another-solution-and-35honna3d2.pdf

Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

Publisher Summary This chapter discusses machine models and simulations. The existence of many different machine models does not necessarily mean that there are no uniform underlying notions of computational complexity. In the theory of computation, a large variety of models and formal calculi has been proposed to capture the notion of effective computation. This has not lead to a proliferation of computation theories, because of the basic fact that the proposed formalisms have all been shown to be equivalent in the following sense—each computation in one formalism can be simulated by a computation in the other formalism. From the equivalence, it can be concluded that if a problem is unsolvable in one particular model, then it is also unsolvable for all other formalized computing devices to which this particular model is related by mutual simulation. Consequently the notion of solvability does not really depend on the model used. The basic models of computation themselves remain on the shelf, to be used if and when needed.

Machine models and simulations

http://web.mit.edu/6.034/wwwbob/GamesSolved.pdf

Games solved: now and in the future

http://www.mathcs.emory.edu/~whalen/Hash/Hash_Articles/A%20Reliable%20Randomized%20Algorithm%20for%20the%20Closest-Pair%20Problem%20(DUPLICATE%20GROUPING).pdf

A Reliable Randomized Algorithm for the Closest-Pair Problem

There are a number of board games such as Checkers [2], Go [5], and Gobang [8], which are known to be PSPACE-hard This means that the problem to determine the player having a winning strategy in a given situation on an nA—n board of one of these games is as hard to solve as everything computable in polynomial space PSPACE-completeness has been previously proven for some combinatorial games played on graphs or by logical formulas [1, 9]

In this paper we will show that the same holds for the game of Hex The crucial point of the proof is to establish PSPACE-hardness for a generalization of Hex played on planar graphs This will be done by showing that the problem, whether a given quantified Boolean formula in conjunctive normal form is true, is polynomial time-reducible to the decision problem for generalized Hex In order to do this we will use methods, which were also used to prove PSPACE-completeness of planar Geography in [5] Therefore our proof is quite different from the proof provided by Even and Tarjan [1], who showed PSPACE-completeness of generalized Hex played on arbitrary graphs Since it is easy to see that the decision problem for Hex is in PS PACE, the decision problem for Hex is PSPACE-complete

Hex ist PSPACE-vollständig

Upper Bounds for Sorting Integers on Random Access Machines

For many games, the decision problem of whether a player in a given situation has a winning strategy has been shown to be PSPACE-complete. Following the PSPACE-completeness results of Even and Tarjan [1] for generalized Hex on graphs and of Schaefer [6] for a variety of combinatorial games, the decision problems were shown to be PSPACE-hard for generalizations of Go and Checkers. In this paper a corresponding theorem is proved for the board-game Gobang, a variant of Go. Since the decision problem for Gobang states-of-play itself lies in PSPACE, it can be shown that Gobang is in fact PSPACE-complete.

Gobang ist PSPACE-vollständig

We will construct a family of permutations having an entropy of lower order than their actual rearrangement-complexity. This result is a solution of an open problem that arose in the context of a result obtained by Stoβ [6], who showed that entropy yields a lower bound for complexity of permutations. Besides this we present some interesting results concerning the entropy of permutations defined by Stoβ [6].

A gap between the actual complexity of permutations and their entropy defined by Stoβ

Wir geben einen Sortieralgorithmus fur Registermaschinen an, wie sie in [1] definiert sind und die Additionen, modifizierte Subtraktionen, Multiplikationen und ganzzahlige Divisionen in einem Schritt durchfuhren konnen. Dieser Sortieralgorithmus sortiert n naturliche Zahlen x1,...,xn, die die Bedingung (*) erfullen, in uniformer Zeitkomplexitat von 0(n) +0(c).

Stefan Reisch

Papers

Hex ist PSPACE-vollständig

Upper Bounds for Sorting Integers on Random Access Machines

Gobang ist PSPACE-vollständig

A gap between the actual complexity of permutations and their entropy defined by Stoβ

Ein Sortierverfahren für Registermaschinen