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Computational Complexity of Jumping Block Puzzles
Abstract
In combinatorial reconfiguration, the reconfiguration problems on a vertex subset (e.g., an independent set) are well investigated. In these problems, some tokens are placed on a subset of vertices of the graph, and there are three natural reconfiguration rules called “token sliding,” “token jumping,” and “token addition and removal”. In the context of computational complexity of puzzles, the sliding block puzzles play an important role. Depending on the rules and set of pieces, the sliding block puzzles characterize the computational complexity classes including P, NP, and PSPACE. The sliding block puzzles correspond to the token sliding model in the context of combinatorial reconfiguration. On the other hand, a relatively new notion of jumping block puzzles is proposed in puzzle society. This is the counterpart to the token jumping model of the combinatorial reconfiguration problems in the context of block puzzles. We investigate several variants of jumping block puzzles and determine their computational complexities.
Since the 1930s, mathematicians and computer scientists have been interested in computation. While mathematicians investigate recursion theory, computer scientists investigate computational complexity based on Turing machine model to understand what a computation is. Beside them, there is another approach of research on computation, which is the investigation of puzzles and games. Once we regard the rules used in puzzles and games as the set of basic operations of computation, we can perform some computation by solving puzzles and playing games. In fact, research on puzzles and games from the viewpoint of theoretical computer science has continued without any break in the history of theoretical computer science. Sometimes the research on computational complexity classes has proceeded by understanding the tons of puzzles. The wide collection of complete problems for a specific computational complexity class shares a common property, which gives us a deep understanding of the class. In this survey paper, we give a brief history of research on computational complexities of puzzles and games with related results and trends in theoretical computer science.
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when k is at most a constant.
Given a 3-colorable graph G together with two proper vertex 3-colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|2) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|2) recoloring steps. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 69-82, 2011 © 2011 Wiley Periodicals, Inc.
This paper deals with a popular puzzle known as Hi-Q. The puzzle is generalized: the board is extended to the size n × n, an initial position of the puzzle is given, and a place is given on which only one token is finally placed. The complexity of the generalized Hi-Q is proved NP-complete.
Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.
The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the puzzle to an board with squares, we can study the computational complexity of problems related to the puzzle; in particular, we consider the problem of determining whether a given end configuration can be reached from a given start configuration via at most a given number of moves. This problem was shown NP-complete in Ratner and Warmuth (1990). We provide an alternative simpler proof of this fact by reduction from the rectilinear Steiner tree problem.
The reconfiguration graph of the k-colourings of a graph G contains as its vertex set the proper vertex k-colourings of G, and two colourings are joined by an edge in the reconfiguration graph if they differ in colour on just one vertex of G. We prove that for a graph G on n vertices that is chordal or chordal bipartite, if G is k-colourable, then the reconfiguration graph of its ℓ-colourings, for ℓ⩾k+1ℓ⩾k+1, is connected and has diameter O(n2)O(n2). We show that this bound is asymptotically tight up to a constant factor.
We study problems of reconfigurability of independent sets in graphs. We consider three different models (token jumping, token sliding, and token addition and removal) and analyze relationships between them. We prove that independent set reconfigurability in perfect graphs (under any of the three models) generalizes the shortest path reconfigurability problem in general graphs and is therefore PSPACE-complete. On the positive side, we give polynomial results for even-hole-free graphs and P4P4-free graphs.
We present a nondeterministic model of computation based on reversing edge directions in weighted directed graphs with minimum in-flow constraints on vertices. Deciding whether this simple graph model can be manipulated in order to reverse the direction of a particular edge is shown to be PSPACE-complete by a reduction from Quantified Boolean Formulas. We prove this result in a variety of special cases including planar graphs and highly restricted vertex configurations, some of which correspond to a kind of passive constraint logic. Our framework is inspired by (and indeed a generalization of) the “Generalized Rush Hour Logic” developed by Flake and Baum [Theoret. Comput. Sci. 270(1–2) (2002) 895].We illustrate the importance of our model of computation by giving simple reductions to show that several motion-planning problems are PSPACE-hard. Our main result along these lines is that classic unrestricted sliding-block puzzles are PSPACE-hard, even if the pieces are restricted to be all dominoes (1×2 blocks) and the goal is simply to move a particular piece. No prior complexity results were known about these puzzles. This result can be seen as a strengthening of the existing result that the restricted Rush HourTM puzzles are PSPACE-complete [Theoret. Comput. Sci. 270(1–2) (2002) 895], of which we also give a simpler proof. We also greatly strengthen the conditions for the PSPACE-hardness of the Warehouseman's Problem [Int. J. Robot. Res. 3(4) (1984) 76], a classic motion-planning problem. Finally, we strengthen the existing result that the pushing-blocks puzzle Sokoban is PSPACE-complete [In: Proc. Internat. Conf. on Fun with Algorithms, Elba, Italy, June 1998, pp. 65–76.], by showing that it is PSPACE-complete even if no barriers are allowed.
The 8-puzzle and the 15-puzzle have been used for many years as a domain for testing heuristic search techniques. From experience it is known that these puzzles are “difficult” and therefore useful for testing search techniques. In this paper we give strong evidence that these puzzles are indeed good test problems. We extend the 8-puzzle and the 15-puzzle to an n×n board and show that finding a shortest solution for the extended puzzle is NP-hard and is thus believed to be computationally infeasible.We also sketch an approximation algorithm for transforming boards that is guaranteed to use no more than a constant times the minimum number of moves, where the constant is independent of the given boards and their side length n.The studied puzzles are instances of planar relocation problems where the reachability question is polynomial but efficient relocation is NP-hard. Such problems are natural robotics problems: A robot needs to efficiently relocate packages in the plane. Our research encourages the study of polynomial approximation algorithms for related robotics problems.
Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k=2, and decidable in polynomial time for k=3. Here we prove it is PSPACE-complete for all k≥4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4≤k≤6, and (ii) bipartite planar graphs and k=4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour α to β.We also exhibit, for every k≥4, a class of graphs {GN,k:N∈N∗}, together with two k-colourings for each GN,k, such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2N), whereas the size of GN is O(N2). This is in stark contrast to the k=3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4≤k≤6 planar graphs and (ii) for k=4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.
In this paper we describe a linear algorithm for embedding planar graphs in the rectilinear two-dimensional grid, where vertices are grid points and edges are noncrossing grid paths. The main feature of our algorithm is that each edge is guaranteed to have at most 2 bends (with the single exception of the octahedron for which 3 bends are needed). The total number of bends is at most2n + 4 if the graph is biconnected and at most(7/3)n in the general case. The area is(n + 1)2 in the worst case. This problem has several applications to VLSI circuit design, aesthetic layout of diagrams, computational geometry.
Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete, while some are also NP-hard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time.
We present an exact algorithm for a PSPACE-complete problem, denoted by CONNkSAT, which asks if the solution space for a given k-CNF formula is connected on the n-dimensional hypercube. The problem is known to be PSPACE-complete for k ≥ 3, and polynomial solvable for k ≤ 2 [6]. We show that CONN kSAT for k ≥ 3 is solvable in time for some constant ε
k
> 0, where ε
k
depends only on k, but not on n. This result is considered to be interesting due to the following fact shown by [5]: QBF-3-SAT, which is a typical PSPACE-complete problem, is not solvable in time O((2 − ε)n
) for any constant ε> 0, provided that the SAT problem (with no restriction to the clause length) is not solvable in time O((2 − ε)n
) for any constant ε> 0.
We study the following problem on reconfiguring shortest paths in graphs: Given two shortest s–t paths, what is the minimum number of steps required to transform one into the other, where each intermediate path must also be a shortest s–t path and must differ from the previous one by only one vertex. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length.
Packing trominoes is np-complete, #P-hard and ASP-complete
- T Horiyama
- T Ito
- K Nakatsuka
- A Suzuki
- R Uehara