4 research outputs found
Parameterized pre-coloring extension and list coloring problems
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors
On Structural Parameterizations of Star Coloring
A Star Coloring of a graph G is a proper vertex coloring such that every path
on four vertices uses at least three distinct colors. The minimum number of
colors required for such a star coloring of G is called star chromatic number,
denoted by \chi_s(G). Given a graph G and a positive integer k, the STAR
COLORING PROBLEM asks whether has a star coloring using at most k colors.
This problem is NP-complete even on restricted graph classes such as bipartite
graphs.
In this paper, we initiate a study of STAR COLORING from the parameterized
complexity perspective. We show that STAR COLORING is fixed-parameter tractable
when parameterized by (a) neighborhood diversity, (b) twin-cover, and (c) the
combined parameters clique-width and the number of colors
Dominator Coloring and CD Coloring in Almost Cluster Graphs
In this paper, we study two popular variants of Graph Coloring -- Dominator
Coloring and CD Coloring. In both problems, we are given a graph and a
natural number as input and the goal is to properly color the vertices
with at most colors with specific constraints. In Dominator Coloring, we
require for each , a color such that dominates all vertices
colored . In CD Coloring, we require for each color , a
which dominates all vertices colored . These problems, defined due to their
applications in social and genetic networks, have been studied extensively in
the last 15 years. While it is known that both problems are fixed-parameter
tractable (FPT) when parameterized by where is the treewidth of
, we consider strictly structural parameterizations which naturally arise
out of the problems' applications.
We prove that Dominator Coloring is FPT when parameterized by the size of a
graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT
parameterized by CVD set size plus the number of remaining cliques. En route,
we design a simpler and faster FPT algorithms when the problems are
parameterized by the size of a graph's twin cover, a special CVD set. When the
parameter is the size of a graph's clique modulator, we design a randomized
single-exponential time algorithm for the problems. These algorithms use an
inclusion-exclusion based polynomial sieving technique and add to the growing
number of applications using this powerful algebraic technique.Comment: 29 pages, 3 figure
Parametrisierte Algorithmen für Ganzzahlige Lineare Programme und deren Anwendungen für Zuweisungsprobleme
This thesis is concerned with solving NP-hard problems. We consider two prominent strategies of coping with such computationally hard questions efficiently. The first approach aims to design approximation algorithms, that is, we are content to find good, but non-optimal solutions in polynomial time. The second strategy is called Fixed-Parameter Tractability (FPT) and considers parameters of the instance to capture the hardness of the problem and by that, obtain efficient algorithms with respect to the remaining input. This thesis employs both strategies jointly to develop efficient approximation and exact algorithms using parameterization and modeling the problem as structured integer linear programs (ILPs), which can be solved in FPT. In the first part of this work, we concentrate on these well-structured ILPs. On the one hand, we develop an efficient algorithm for block-structured integer linear programs called n-fold ILPs. On the other hand, we investigate the similarly block-structured 2-stage stochastic ILPs and prove conditional lower bounds regarding the running time of any algorithm solving them that match the best known upper bounds. We also prove the tightness of certain structural parameters called sensitivity and proximity for ILPs which arise from combinatorial questions such as allocation problems. The second part utilizes n-fold ILPs and structural properties to add to and improve upon known results for Scheduling and Bin Packing problems. We design exact FPT algorithms for the Scheduling With Clique Incompatibilities, Bin Packing, and Multiple Knapsack problems. Further, we provide constant-factor approximation algorithms and polynomial time approximation schemes (PTAS) for the Class Constraint Scheduling problems. Broadening our scope, we also investigate this problem and the closely related Cardinality Constraint Scheduling problem in the online setting and derive lower bounds for the approximation ratios as well as a PTAS for them. Altogether, this thesis contributes to the knowledge about structured ILPs, proves their limits and reaffirms their usefulness for a plethora of allocation problems. In doing so, various new and improved algorithms with respect to the running time or approximation quality emerge