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Extremely Complex 4Colored RectangleFree Grids: Solution of Open MultipleValued Problems
 IEEE 42ND INTERNATIONAL SYMPOSIUM ON MULTIPLEVALUED LOGIC (ISMVL 2012) MAY 1416
, 2012
"... This paper aims at the rectanglefree coloring of grids using four colors. It has been proven in a well developed theory that there is an upper bound of rectanglefree 4colorable grids as well as a lower bound of grids for which no rectanglefree color pattern of four colors exist. Between these ti ..."
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Cited by 7 (6 self)
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This paper aims at the rectanglefree coloring of grids using four colors. It has been proven in a well developed theory that there is an upper bound of rectanglefree 4colorable grids as well as a lower bound of grids for which no rectanglefree color pattern of four colors exist. Between these tight bounds the grids of the size 17×17, 17×18, 18×17, and 18×18 are located for which it is not known until now whether a rectanglefree coloring by four colors exists. We present in this paper an approach that solves all these open problems. From another point of view this paper aims at the solution of a multiplevalued problem having an extremely high complexity. There are 1.16798 ∗ 10 195 different grids of four colors. It must be detected whether at least one of this hardly imaginable large number of patterns satisfies strong additional conditions. In order to solve this highly complex problem, several approaches were taken into account to find out properties of the problem which finally allowed us to calculate the solution.
Ordering heuristics for parallel graph coloring
 In SPAA
, 2014
"... This paper introduces the largestlogdegreefirst (LLF) and smallestlogdegreelast (SLL) ordering heuristics for parallel greedy graphcoloring algorithms, which are inspired by the largestdegreefirst (LF) and smallestdegreelast (SL) serial heuristics, respectively. We show that although LF ..."
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Cited by 5 (1 self)
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This paper introduces the largestlogdegreefirst (LLF) and smallestlogdegreelast (SLL) ordering heuristics for parallel greedy graphcoloring algorithms, which are inspired by the largestdegreefirst (LF) and smallestdegreelast (SL) serial heuristics, respectively. We show that although LF and SL, in practice, generate colorings with relatively small numbers of colors, they are vulnerable to adversarial inputs for which any parallelization yields a poor parallel speedup. In contrast, LLF and SLL allow for provably good speedups on arbitrary inputs while, in practice, producing colorings of competitive quality to their serial analogs. We applied LLF and SLL to the parallel greedy coloring algorithm introduced by Jones and Plassmann, referred to here as JP. Jones and Plassman analyze the variant of JP that processes the vertices of a graph in a random order, and show that on an O(1)degree graph G = (V,E), this JPR variant has an expected parallel running time of O(lgV / lg lgV) in a PRAM model. We improve this bound to show, using workspan analysis, that JPR, augmented to handle arbitrarydegree graphs, colors a graph G = (V,E) with degree ∆ using Θ(V +E) work and O(lgV + lg ∆ ·min{√E,∆+ lg ∆ lgV / lg lgV}) expected span. We prove that JPLLF and JPSLL — JP using the LLF and SLL heuristics, respectively — execute with the same asymptotic work as JPR and only logarithmically more span while producing higherquality colorings than JPR in practice. We engineered an efficient implementation of JP for modern sharedmemory multicore computers and evaluated its performance on a machine with 12 Intel Corei7 (Nehalem) processor cores. Our implementation of JPLLF achieves a geometricmean speedup of 7.83 on eight realworld graphs and a geometricmean speedup of 8.08 on ten synthetic graphs, while our implementation using SLL achieves a geometricmean speedup of 5.36 on these realworld graphs and a geometricmean speedup of 7.02 on these synthetic graphs. Furthermore, on one processor, JPLLF is slightly faster than a wellengineered serial greedy algorithm using LF, and likewise, JPSLL is slightly faster than the greedy algorithm using SL.
Utilization of Permutation Classes for Solving Extremely Complex 4Colorable Rectanglefree Grids
 THE 2012 INTERNATIONAL CONFERENCE ON SYSTEMS AND INFORMATICS (ICSAI
, 2012
"... This paper aims at the rectanglefree coloring of grids using four colors. It has been proven that there are bounds for the size of rectanglefree fourcolorable grids outside of these values the grids cannot be colored. For the grids of the size 17×17, 17×18, 18×17, and 18×18 it is not yet known ..."
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This paper aims at the rectanglefree coloring of grids using four colors. It has been proven that there are bounds for the size of rectanglefree fourcolorable grids outside of these values the grids cannot be colored. For the grids of the size 17×17, 17×18, 18×17, and 18×18 it is not yet known whether rectanglefree colorings by four colors exist. A necessary condition for rectanglefree fourcolorable grids of the size 18×18 is that at least one fourth of the grid elements can be colored by a single color without violating the rectanglefree condition. This simplified task requires to evaluate 2 18∗18 = 2 324 = 3.41758 ∗10 97 assignments to the grid elements whether at least 18∗18/4 = 81 elements can be colored by the same color without violating the rectanglefree condition. We present in this paper an approach that allows to calculate such extremely large sets of assignments using the restricted memory space of a normal PC and a short period of time. The key to our solution is the utilization of permutation classes. However, even a single permutation class consists of such an extreme number of assignments that it can neither be stored nor enumerated. Hence, it is the main aim of this paper to find a way that allows us to decide whether two grid assignments are members of the same permutation class under these extreme conditions. On the basis of such an approach we are going to answer the question whether an extremely rare fourcolorable rectanglefree grid G18,18 can exist within the huge amount of 4 324 = 1.16798∗ 10 195 possible fourcolorings of such grids.
Solution of the Last Open FourColored Rectanglefree Grid an Extremely Complex MultipleValued Problem
"... Abstract—It is a challenge in the multivalued domain to solve problems that depend on a large number of variables, as large as possible. We selected for this paper the problem of rectanglefree colorings using four colors which could not be solved so far for the grids of the sizes 12 × 21 and 21 × ..."
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Abstract—It is a challenge in the multivalued domain to solve problems that depend on a large number of variables, as large as possible. We selected for this paper the problem of rectanglefree colorings using four colors which could not be solved so far for the grids of the sizes 12 × 21 and 21 × 12. This problem depends on 12∗21 = 252 fourvalued variables. It is the last of so far unsolved rectanglefree grid problems for four colors. This paper aims at the solution of a multivalued problem with an exceptionally high complexity. The search space for this finite problem is 4 252 which is approximately 5.2∗10 151. A similar coloring problem was solved for the grid of the size 18×18 that relies on the extreme search space of approximately 1.1 ∗ 10 195. The construction of a cyclically reusable solution for a single color simplifies this search space approximately to 3.4∗10 97. Unfortunately, such a restriction to a single color is not possible in the case of a grid of the size 12 × 21. Hence, the complexity which must be handled in maintainable time grows additionally by a factor of more than 10 54. Based on a very deep analysis of the properties of the problem we have constructed a strongly restricted SATmodel. This final model depends on 504 Boolean variables and 85.344 clauses. Using this SATinstance we could calculate not only one solution but 38,926 representatives of different permutation classes of fourcolored rectanglefree grids of the size 12×21. Keywordsfourvalued coloring, rectanglefree grid, Boolean equation, SATsolver, Latin square. I.
The solution of ultra large grid problems
 IN 21ST INTERNATIONAL WORKSHOP ON POSTBINARY USLI SYSTEMS
, 2012
"... SAT solvers have grown in power over the last decades. However, both certain intentions of their application and the complexity of the problems are still obstacles to their successful utilization. Here we are facing a problem where an extremely small fraction of solutions must be found in the unima ..."
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SAT solvers have grown in power over the last decades. However, both certain intentions of their application and the complexity of the problems are still obstacles to their successful utilization. Here we are facing a problem where an extremely small fraction of solutions must be found in the unimaginably large search space of more than 10 195. Up to now SAT solvers were able to find solutions for subproblems with the size of approximately 10 135. Hence, it was our challenge to bridge the gap of 10 60. In this paper we focus on a subproblem of the basic graph coloring problem. Due to the estimated complexity it seemed to be hopeless to find solutions. However, we enjoyed this extreme challenge and studied several approaches beyond the application of SAT solvers which allowed finally to solve the so far open graph coloring problem.
Solutions of Exceptionally Complex Boolean Problems
 Boolean Problems, Proceedings of the 10th International Workshops on Boolean Problems, 19.  21. September 2012, Freiberg University of Mining and Technology, Freiberg, 2012, ISBN 9783860124383
"... It is a challenge in the Boolean domain to solve problems that depend on a large number of variables, as large as possible. We selected for this paper the problem of rectanglefree colorings of grids using four colors which could not be solved so far for certain sizes of grids. It has been proven th ..."
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Cited by 1 (1 self)
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It is a challenge in the Boolean domain to solve problems that depend on a large number of variables, as large as possible. We selected for this paper the problem of rectanglefree colorings of grids using four colors which could not be solved so far for certain sizes of grids. It has been proven that there is an upper bound for the size of rectanglefree 4colorable grids as well as a lower bound for the size of grids for which no rectanglefree color patterns of four colors exists. There was a solution for grids of the size 16 × 16, and there was a proof that a grid of the size 19 × 19 is not rectanglefree 4colorable. The grids of the size 17 × 17, 17 × 18, 18 × 17, and 18 × 18 are located between these bounds until now it was not known whether a rectanglefree coloring by four colors exists or not. A binary model requires 648 Boolean variables which describe 4 324 = 1.16798 ∗ 10 195 color patterns, because two variables are needed for the coding of the four colors. It must be detected whether at least one of this hardly imaginable large number of patterns satisfies the requirements of the problem. In order to solve this highly complex problem, several approaches were taken into account to find out properties of the problem which finally allowed us to calculate the solution. Neither a search procedure executed on the most powerful computer nor a human being are able to solve such an extraordinarily complex task in an acceptable period of time. Human beings and computers must explore properties of the problem in a cooperative way. The utilization of problem properties within models and algorithms are the source of a successful solution. We see this solution as a very interesting cooperation between human beings and computers leading the way for future developments. 1
Search Space Restriction for Maximal RectangleFree Grids
"... This paper deals with a subtask of edge coloring of graphs: each edge of a complete graph must be colored with one of a restricted number of colors such that no cycle of four edges is colored by the same color [1]. We restrict ourselves to solutions for a single color and focus on bipartite graphs. ..."
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This paper deals with a subtask of edge coloring of graphs: each edge of a complete graph must be colored with one of a restricted number of colors such that no cycle of four edges is colored by the same color [1]. We restrict ourselves to solutions for a single color and focus on bipartite graphs. The maximal number of possible assignments of the selected color to the edges is a necessary condition for the minimal number of colors needed to color all edges of a complete graph. This problem is a challenge in the Boolean domain. In order to satisfy this condition a Boolean function f: B m∗n → B has to be found which is equal to 1 if the condition mentioned above will be satisfied. Within the set of all solution vectors of f = 1 such Boolean vectors must be selected which contain the largest number of values 1. The variables m and n specify numbers of vertices in the disjoint sets of the bipartite graph. We present in this paper several approaches to solve this problem for values of m and n as large as possible knowing that 2 m∗n Boolean vectors must be evaluated. 1
Artificial Intelligence and Creativity  Two Requirements to Solve an Extremely Complex Coloring Problem
"... The topic of this paper is the rectanglefree coloring of grids using four colors which is equivalent to the edge coloring of complete bipartite graphs without complete monochromatic subgraphs K2,2. So far unsolved are the grids of the sizes 17×17, 17×18, 18×17, and 18×18. The number of different 4 ..."
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The topic of this paper is the rectanglefree coloring of grids using four colors which is equivalent to the edge coloring of complete bipartite graphs without complete monochromatic subgraphs K2,2. So far unsolved are the grids of the sizes 17×17, 17×18, 18×17, and 18×18. The number of different 4color patterns of the grid 18×18 is equal to 4 324 ≈ 1.16798∗10 195. We summarize in this paper some basic approaches in order to gain the required knowledge. Three creative approaches are steps so solve the most complex grid of the size 18×18. Two advanced creative approaches reduce the required runtime to less than 12 percent.
Time and SpaceEfficient SelfStabilizing Algorithms
, 2012
"... In a distributed system error handling is inherently more difficult than in conventional systems that have a central control unit. To recover from an erroneous state the nodes have to cooperate and coordinate their actions based on local information only. Selfstabilization is a general approach to ..."
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In a distributed system error handling is inherently more difficult than in conventional systems that have a central control unit. To recover from an erroneous state the nodes have to cooperate and coordinate their actions based on local information only. Selfstabilization is a general approach to make a distributed system tolerate arbitrary transient faults by design. A selfstabilizing algorithm reaches a legitimate configuration in a finite number of steps by itself without any external intervention, regardless of the initial configuration. Furthermore, once having reached legitimacy this property is preserved. An important characteristic of an algorithm is its worstcase runtime and its memory requirements. This thesis presents new time and spaceefficient selfstabilizing algorithms for wellknown problems in algorithmic graph theory and provides new complexity analyses for existing algorithms. The main focus is on proof techniques used in the complexity analyses and the design of the algorithms. All algorithms presented in this thesis assume the most general concept with respect to concurrency.