95 research outputs found

    Generalized Colorings of Graphs

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    A graph coloring is an assignment of labels called ā€œcolorsā€ to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k āˆ’ 1), conļ¬rming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We deļ¬ne the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive diļ¬€erent colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

    Computational Complexity of SAT, XSAT and NAE-SAT for linear and mixed Horn CNF formulas

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    The Boolean conjunctive normal form (CNF) satisfiability problem, called SAT for short, gets as input a CNF formula and has to decide whether this formula admits a satisfying truth assignment. As is well known, the remarkable result by S. Cook in 1971 established SAT as the first and genuine complete problem for the complexity class NP. In this thesis we consider SAT for a subclass of CNF, the so called Mixed Horn formula class (MHF). A formula F in MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set, can easily be encoded as MHF. Furthermore, we show that SAT remains NP-complete for some interesting subclasses of MHF. We also provide algorithms for some of these subclasses solving SAT in a better running time than O(2^0.5284n) which is the best bound for MHF so far. One of these subclasses consists of formulas, where the Horn part is negative monotone and the variable graph corresponding to the positive 2-CNF part P consists of disjoint triangles only. Regarding the other subclass consisting of certain k-uniform linear mixed Horn formulas, we provide an algorithm solving SAT in time O(k^(n/k)), for k>=4. Additionally, we consider mixed Horn formulas F in MHF for which holds: H is negative monotone, c=3. We also prove the NP-completeness of XSAT for CNF formulas which are l-regular meaning that every variable occurs exactly l times, where l>=3 is a fixed integer. On that basis, we can provide the NP-completeness of XSAT for the subclass of linear and l-regular formulas. This result is transferable to the monotone case. Moreover, we provide an algorithm solving XSAT for the subclass of monotone, linear and l-regular formulas faster than the so far best algorithm from J. M. Byskov et al. for CNF-XSAT with a running time of O(2^0.2325n). Using some connections to finite projective planes, we can also show that XSAT remains NP-complete for linear and l-regular formulas that in addition are l-uniform whenever l=q+1, where q is a prime power. Thus XSAT most likely is NP-complete for the other values of l>= 3, too. Apart from that, we are interested in exact linear formulas: Here each pair of distinct clauses has exactly one variable in common. We show that NAESAT is polynomial-time decidable restricted to exact linear formulas. Reinterpreting this result enables us to give a partial answer to a long-standing open question mentioned by T. Eiter: Classify the computational complexity of the symmetrical intersecting unsatisfiability problem (SIM-UNSAT). Then we show the NP-completeness of XSAT for monotone and exact linear formulas, which we can also establish for the subclass of formulas whose clauses have length at least k, k>=3. This is somehow surprising since both SAT and not-all-equal SAT are polynomial-time solvable for exact linear formulas. However, for k=3,4,5,6 we can show that XSAT is polynomial-time solvable for the k-uniform, monotone and exact linear formula class

    Reconstructing Geometric Structures from Combinatorial and Metric Information

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    In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable. Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets. Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding

    Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

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    Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable
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