Combinatorialalgorithms for Packings, Coverings and Tilings Of
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Solving Sudoku with Dancing Links
Solving Sudoku with Dancing Links Rob Beezer [email protected] Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington USA African Institute for Mathematical Sciences October 25, 2010 Available at http://buzzard.pugetsound.edu/talks.html Example: Combinatorial Enumeration Create all permutations of the set f0; 1; 2; 3g Simple example to demonstrate key ideas Creation, cardinality, existence? There are more efficient methods for this example Rob Beezer (U Puget Sound) Solving Sudoku with Dancing Links AIMS October 2010 2 / 37 Brute Force Backtracking BLACK = Forward BLUE = Solution RED = Backtrack root 0 1 2 0 1 3 3 0 2 1 0 2 3 3 0 3 1 0 3 1 0 2 0 0 1 2 2 0 1 3 0 2 1 3 0 2 3 0 3 1 3 0 3 3 1 0 2 1 0 0 0 1 2 0 1 0 2 1 0 2 0 3 1 0 3 1 0 2 0 0 1 2 3 0 0 2 0 0 3 0 1 0 2 2 0 1 0 1 2 0 2 0 2 2 0 3 0 3 2 root 1 0 2 0 1 0 0 1 0 2 0 0 2 0 3 0 0 3 2 0 1 1 0 2 3 0 1 0 1 3 0 2 0 2 3 0 3 0 3 2 1 0 1 0 2 . 0 1 1 0 1 3 0 0 2 1 0 2 3 0 0 3 1 0 3 2 1 1 0 0 . -
Representing Sudoku As an Exact Cover Problem and Using Algorithm X and Dancing Links to Solve the Represented Problem
Bhavana Yerraguntla Bhagya Dhome SUDOKU SOLVER Approach: Representing Sudoku as an exact cover problem and using Algorithm X and Dancing Links to solve the represented problem. Before we start, let us try to understand what Exact Cover is; Exact Cover: Given a set S and another set X where each element is a subset to S, select a set of subsets S* such that every element in X exist in exactly one of the selected sets. This selection of sets is said to be a cover of the set S. The exact cover problem is a decision problem to find an exact cover. E.g. Let S = {A, B, C, D, E, F} be a collection of subsets of a set X = {1, 2, 3, 4, 5, 6, 7} s t: A = {1, 4, 7} B = {1, 4} C = {4, 5, 7} D = {3, 5, 6} E = {2, 3, 6, 7} F = {2, 7} Then the set of subsets S* = {B, D, F} is an exact cover. Before we represent the Sudoku problem into an exact cover, let’s explore an example smaller. The basic approach towards solving an exact cover problem is simple 1) List the constraints that needs to be satisfied. (columns) 2) List the different ways to satisfy the constraints. (rows) Simple example: Imagine a game of Cricket (a batting innings), the batting order is almost filled with three batsmen A, B, C are yet to be placed and with three exact spots unfilled. One opener, one top order and one middle order. Each batsman must play in one order, and no two batsman must be in the same order. -
A Special Planar Satisfiability Problem and a Consequence of Its NP-Completeness
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector DISCRETE APPLIED MATHEMATICS ELSEVIER Discrete Applied Mathematics 52 (1994) 233-252 A special planar satisfiability problem and a consequence of its NP-completeness Jan Kratochvil Charles University, Prague. Czech Republic Received 18 April 1989; revised 13 October 1992 Abstract We introduce a weaker but still NP-complete satisfiability problem to prove NP-complete- ness of recognizing several classes of intersection graphs of geometric objects in the plane, including grid intersection graphs and graphs of boxicity two. 1. Introduction Intersection graphs of different types of geometric objects in the plane gained more attention in recent years, mainly in connection with fast development of computa- tional geometry and computer science. Just to mention the most frequently cited classes, these are interval graphs, circular arc graphs, circle graphs, permutation graphs, etc. If we consider only connected objects (more precisely arc-connected sets) the most general class of intersection graphs are string graphs (intersection graphs of curves in the plane) which were originally introduced by Sinden [16] in the connection with thin film RC-circuits. String graphs were then considered by several authors [4,6,7]. In a recent paper [S], I have shown that recognition of string graphs is NP-hard and in fact, the method developed in [S] is refined in this note to obtain other NP- completeness results. It is striking that so far no finite algorithm for string graph recognition is known. It seems that relatively simpler classes will arise if we consider straight-line segments instead of curves and furthermore, if these segments are allowed to follow only a bounded number of directions. -
Graphs with Small Intersection Dimension Patrick Lillis Advisor: Dr
Graphs With Small Intersection Dimension Patrick Lillis Advisor: Dr. R. Sritharan Abstract An X-Graph Split Graphs The boxicity of a graph G, denoted as box(G), We prove that the split graph of a is defined as the minimum integer k such that G is an intersection graph of axis-parallel k- convex graph has boxicity at most 2, dimensional boxes. We examine some known using intersecting chain graphs. A properties of graphs with respect to boxicity, chain graph is always an interval graph, so a 2 chain graph as well as show boxicity results pertaining to Add edge to get B several classes of graphs, including split representation is equivalent to a 2- graphs, X-graphs, and powers of trees. We dimensional box representation. also propose efficient algorithms to produce We then prove than any X-Graph is the intersection of 2 convex graphs, A the relevant k-dimensional representations. Add edge to get A and B (see left). As any convex graph Introduction has boxicity at most 2, any X-graph The graph classes we study all have low then has boxicity at most 4. bounds on boxicity (e.g. a tree has boxicity at most 2), or some result pertaining to small Powers of Trees (left) A tree T, with Δ ≤ 3 boxicity (e.g. it is NP-complete to determine if We find a constant bound on the boxicity (below) An embedding of a split graph has boxicity at most 3). We of powers of trees with Δ at most 3; any T in a revised perfect study specific subclasses of these graph even power of such a tree has binary tree T’. -
Geometric Representations of Graphs
' $ Geometric Representations of Graphs L. Sunil Chandran Assistant Professor Comp. Science and Automation Indian Institute of Science Bangalore- 560012. Email: [email protected] & 1 % ' $ • Conventionally graphs are represented as adjacency matrices, or adjacency lists. Algorithms are designed with such representations in mind usually. • It is better to look at the structure of graphs and find some representations that are suitable for designing algorithms- say for a class of problems. • Intersection graphs: The vertices correspond to the subsets of a set U. The vertices are made adjacent if and only if the corresponding subsets intersect. • We propose to use some nice geometric objects as the subsets- like spheres, cubes, boxes etc. Here U will be the set of points in a low dimensional space. & 2 % ' $ • There are many situations where an intersection graph of geometric objects arises naturally.... • Some times otherwise NP-hard algorithmic problems become polytime solvable if we have geometric representation of the graph in a space of low dimension. & 3 % ' $ Boxicity and Cubicity • Cubicity: Minimum dimension k such that G can be represented as the intersection graph of k-dimensional cubes. • Boxicity: Minimum dimension k such that G can be represented as the intersection graph of k-dimensional axis parallel boxes. • These concepts were introduced by F. S. Roberts, in 1969, motivated by some problems in ecology. • By the later part of eighties, the research in this area had diminished. & 4 % ' $ An Equivalent Combinatorial Problem • The boxicity(G) is the same as the minimum number k such that there exist interval graphs I1,I2,...,Ik such that G = I1 ∩ I2 ∩···∩ Ik. -
Bastian Michel, "Mathematics of NRC-Sudoku,"
Mathematics of NRC-Sudoku Bastian Michel December 5, 2007 Abstract In this article we give an overview of mathematical techniques used to count the number of validly completed 9 × 9 sudokus and the number of essentially different such, with respect to some symmetries. We answer the same questions for NRC-sudokus. Our main result is that there are 68239994 essentially different NRC-sudokus, a result that was unknown up to this day. In dit artikel geven we een overzicht van wiskundige technieken om het aantal geldig inge- vulde 9×9 sudokus en het aantal van essentieel verschillende zulke sudokus, onder een klasse van symmetrie¨en,te tellen. Wij geven antwoorden voor dezelfde vragen met betrekking tot NRC-sudoku's. Ons hoofdresultaat is dat er 68239994 essentieel verschillende NRC-sudoku's zijn, een resultaat dat tot op heden onbekend was. Dit artikel is ontstaan als Kleine Scriptie in het kader van de studie Wiskunde en Statistiek aan de Universiteit Utrecht. De begeleidende docent was dr. W. van der Kallen. Contents 1 Introduction 3 1.1 Mathematics of sudoku . .3 1.2 Aim of this paper . .4 1.3 Terminology . .4 1.4 Sudoku as a graph colouring problem . .5 1.5 Computerised solving by backtracking . .5 2 Ordinary sudoku 6 2.1 Symmetries . .6 2.2 How many different sudokus are there? . .7 2.3 Ad hoc counting by Felgenhauer and Jarvis . .7 2.4 Counting by band generators . .8 2.5 Essentially different sudokus . .9 3 NRC-sudokus 10 3.1 An initial observation concerning NRC-sudokus . 10 3.2 Valid transformations of NRC-sudokus . -
Box Representations of Embedded Graphs
Box representations of embedded graphs Louis Esperet CNRS, Laboratoire G-SCOP, Grenoble, France S´eminairede G´eom´etrieAlgorithmique et Combinatoire, Paris March 2017 Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. -
Posterboard Presentation
Dancing Links and Sudoku A Java Sudoku Solver By: Jonathan Chu Adviser: Mr. Feinberg Algorithm by: Dr. Donald Knuth Sudoku Sudoku is a logic puzzle. On a 9x9 grid with 3x3 regions, the digits 1-9 must be placed in each cell such that every row, column, and region contains only one instance of the digit. Placing the numbers is simply an exercise of logic and patience. Here is an example of a puzzle and its solution: Images from web Nikoli Sudoku is exactly a subset of a more general set of problems called Exact Cover, which is described on the left. Dr. Donald Knuth’s Dancing Links Algorithm solves an Exact Cover situation. The Exact Cover problem can be extended to a variety of applications that need to fill constraints. Sudoku is one such special case of the Exact Cover problem. I created a Java program that implements Dancing Links to solve Sudoku puzzles. Exact Cover Exact Cover describes problems in h A B C D E F G which a mtrix of 0’s and 1’s are given. Is there a set of rows that contain exactly one 1 in each column? The matrix below is an example given by Dr. Knuth in his paper. Rows 1, 4, and 5 are a solution set. 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 We can represent the matrix with toriodal doubly-linked lists as shown above. -
Arxiv:0910.0460V3 [Cs.DS] 3 Feb 2010 Ypsu Ntertclapcso Optrsine21 ( 2010 Science Computer of Aspects Theoretical on Symposium Matching
Symposium on Theoretical Aspects of Computer Science 2010 (Nancy, France), pp. 95-106 www.stacs-conf.org EXACT COVERS VIA DETERMINANTS ANDREAS BJORKLUND¨ E-mail address: [email protected] Abstract. Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-Dimensional Match- ing problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in O∗(2n(k−2)/k) time. The O∗() notation hides factors polynomial in n and k. The general Exact Cover by k-Sets problem asks the same when the partition constraint is dropped and arbitrary hyperedges of cardinality k are permitted. We show it can be ∗ n solved by a randomized polynomial space algorithm in O (ck ) time, where c3 = 1.496, c4 = 1.642, c5 = 1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k. They follow from the new observation that Lov´asz’ perfect matching detection via determinants (Lov´asz, 1979) admits an embedding in the recently proposed inclusion–exclusion counting scheme for set covers, despite its inability to count the perfect matchings. 1. Introduction The Exact Cover by k-Sets problem (XkC) and its constrained variant k-Dimensional Matching (kDM) are two well-known NP-hard problems. They ask, given a k-uniform hy- pergraph, if there is a subset of the hyperedges which cover the vertices without overlapping each other. -
Boxicity, Poset Dimension, and Excluded Minors
Boxicity, poset dimension, and excluded minors Louis Esperet∗ Veit Wiechert Laboratoire G-SCOP Institut f¨urMathematik CNRS, Univ. Grenoble Alpes Technische Universit¨atBerlin Grenoble, France Berlin, Germany [email protected] Submitted: Apr 12, 2018; Accepted: Nov 27, 2018; Published: Dec 21, 2018 c The authors. Abstract In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no Kt-minor can be represented as the intersection of O(t2 log t) interval graphs (improving the previous bound of 4 15 2 O(t )), and as the intersection of 2 t circular-arc graphs. Mathematics Subject Classifications: 05C15, 05C83, 06A07 1 Introduction The intersection G1 \···\ Gk of k graphs G1;:::;Gk defined on the same vertex set V , is the graph (V; E1 \:::\Ek), where Ei (1 6 i 6 k) denotes the edge set of Gi. The boxicity box(G) of a graph G, introduced by Roberts [19], is defined as the smallest k such that G is the intersection of k interval graphs. Scheinerman proved that outerplanar graphs have boxicity at most two [20] and Thomassen proved that planar graphs have boxicity at most three [24]. Outerplanar graphs have no K4-minor and planar graphs have no K5-minor, so a natural question is how these two results extend to graphs with no Kt-minor for t > 6. It was proved in [6] that if a graph has acyclic chromatic number at most k, then its boxicity is at most k(k − 1). -
Contact Representations of Planar Graphs with Cubes
Contact representations of planar graphs with cubes Stefan Felsner∗ Mathew C. Francis [email protected] [email protected] Technische Universit¨at Berlin Department of Applied Mathematics, Institut f¨ur Mathematik Charles University Strasse des 17. Juni 136 Malostransk´eN´am. 25, 10623 Berlin, Germany 118 00 Praha 1, Czech Republic Abstract. We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent. Moreover, when two cubes have a non-empty intersection, they just touch each other. This result is a strengthening of a result by Thomassen which states that every planar graph has such a representation using axis-parallel boxes. 1 Introduction An axis-aligned d-dimensional box, or d-box in short, is defined to be the Cartesian product of d closed intervals. The boxicity of a graph G is defined as the smallest d such that G can be represented as the intersection graph of d-boxes in IRd . A graph has boxicity at most one if and only if it is an interval graph. Roberts [12] showed that removing a perfect matching from a complete graph on 2n vertices yields a graph of boxicity n. The octahedron graph is the instance with n = 3 from this series of examples. This shows that the boxicity of planar graphs can be as large as 3. Thomassen [16] proved that the boxicity of planar graphs is at most 3. -
Boxicity and Cubicity of Product Graphs
Boxicity and Cubicity of Product Graphs L. Sunil Chandran1, Wilfried Imrich2, Rogers Mathew ∗3, and Deepak Rajendraprasad †1 1 Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India - 560012. {sunil, deepakr}@csa.iisc.ernet.in 2 Department Mathematics and Information Technology, Montanuniversit¨at Leoben, Austria. [email protected] 3 Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada - B3H 3J5. [email protected] Abstract The boxicity (cubicity) of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in Rk. In this article, we give estimates on the boxic- ity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of d, of the boxicity and the cubicity of the d-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the d-th Cartesian power of any given finite graph is in O(logd/loglogd) and Θ(d/log d), respectively. On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products. Keywords: intersection graphs, boxicity, cubicity, graph products, boolean lattice. 1 Introduction Throughout this discussion, a k-box is the Cartesian product of k closed intervals on the real line R, and a k-cube is the Cartesian product of k closed unit length intervals k arXiv:1305.5233v1 [math.CO] 22 May 2013 on R.