583 research outputs found

    Computing Role Assignments of Proper Interval Graphs in Polynomial Time

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    A homomorphism from a graph G to a graph R is locally surjective if its restriction to the neighborhood of each vertex of G is surjective. Such a homomorphism is also called an R-role assignment of G. Role assignments have applications in distributed computing, social network theory, and topological graph theory. The Role Assignment problem has as input a pair of graphs (G,R) and asks whether G has an R-role assignment. This problem is NP-complete already on input pairs (G,R) where R is a path on three vertices. So far, the only known non-trivial tractable case consists of input pairs (G,R) where G is a tree. We present a polynomial time algorithm that solves Role Assignment on all input pairs (G,R) where G is a proper interval graph. Thus we identify the first graph class other than trees on which the problem is tractable. As a complementary result, we show that the problem is Graph Isomorphism-hard on chordal graphs, a superclass of proper interval graphs and trees

    Independent Set Reconfiguration in Cographs

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    We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets II and JJ of a graph GG, both of size at least kk, is it possible to transform II into JJ by adding and removing vertices one-by-one, while maintaining an independent set of size at least kk throughout? This problem is known to be PSPACE-hard in general. For the case that GG is a cograph (i.e. P4P_4-free graph) on nn vertices, we show that it can be solved in time O(n2)O(n^2), and that the length of a shortest reconfiguration sequence from II to JJ is bounded by 4n2k4n-2k, if such a sequence exists. More generally, we show that if XX is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in XX using disjoint union and complete join operations. Chordal graphs are given as an example of such a class XX

    Exploiting Chordality in Optimization Algorithms for Model Predictive Control

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    In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.Comment: arXiv admin note: text overlap with arXiv:1502.0638

    On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

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    We prove several results about the complexity of the role colouring problem. A role colouring of a graph GG is an assignment of colours to the vertices of GG such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with 1<k<n1< k <n colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as kk is either constant or has a constant difference with nn, the number of vertices in the tree. Finally, we prove that cographs are always kk-role-colourable for 1<kn1<k\leq n and construct such a colouring in polynomial time

    Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

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    An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

    Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree.

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    A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4 or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree
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