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Über dieses Buch

This book constitutes the thoroughly referred post-workshop proceedings of the 22nd International Workshop on Combinatorial Algorithms, IWOCA 2011, held in Victoria, BC, Canada, in July 2011. The 30 revised full papers presented were carefully reviewed and selected from a total of 71 submissions. A broad variety of topics in combinatorics and graph theory are addressed, such as combinatorics on words, string algorithms, codes, Venn diagrams, set partitions; Hamiltonian & Eulerian properties, graph drawing, colouring, dominating sets, spanning trees, and others.

Inhaltsverzeichnis

Frontmatter

Weighted Improper Colouring

Abstract
In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph G where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted t-improper k-colouring of G is a k-colouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed some threshold t. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted t-improper chromatic number defined as the minimum integer k such that G admits a weighted t-improper k-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number k of colours (frequencies) we want to determine the minimum real t such that G admits a weighted t-improper k-colouring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalisation of Lovász’s Theorem for the weighted t-improper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one where the weights are either 1 or M, for a sufficient big value M. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted t-improper chromatic number when G is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.
Julio Araujo, Jean-Claude Bermond, Frédéric Giroire, Frédéric Havet, Dorian Mazauric, Remigiusz Modrzejewski

Algorithmic Aspects of Dominator Colorings in Graphs

Abstract
In this paper we initiate a systematic study of a problem that has the flavor of two classical problems, namely Coloring and Domination, from the perspective of algorithms and complexity. A dominator coloring of a graph G is an assignment of colors to the vertices of G such that it is a proper coloring and every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G and is denoted by χ d (G). In the Dominator Coloring (DC) problem, a graph G and a positive integer k are given as input and the objective is to check whether χ d (G) ≤ k. We first show that unless P=NP, DC cannot be solved in polynomial time on bipartite, planar, or split graphs. This resolves an open problem posed by Chellali and Maffray [Dominator Colorings in Some Classes of Graphs, Graphs and Combinatorics, 2011] about the polynomial time solvability of DC on chordal graphs. We then complement these hardness results by showing that the problem is fixed parameter tractable (FPT) on chordal graphs and in graphs which exclude a fixed apex graph as a minor.
S. Arumugam, K. Raja Chandrasekar, Neeldhara Misra, Geevarghese Philip, Saket Saurabh

Parameterized Longest Previous Factor

Abstract
The longest previous factor (LPF) problem is defined for traditional strings exclusively from the constant alphabet Σ. A parameterized string (p-string) is a sophisticated string composed of symbols from a constant alphabet Σ and a parameter alphabet Π. We generalize the LPF problem to the parameterized longest previous factor (pLPF) problem defined for p-strings. Subsequently, we present a linear time solution to construct the pLPF array. Given our pLPF algorithm, we show how to construct the pLCP (parameterized longest common prefix) array in linear time. Our algorithm is further exploited to construct the standard LPF and LCP arrays all in linear time.
Richard Beal, Donald Adjeroh

p-Suffix Sorting as Arithmetic Coding

Abstract
The challenge of direct parameterized suffix sorting (p-suffix sorting) for a parameterized string (p-string) is the dynamic nature of parameterized suffixes (p-suffixes). In this work, we propose transformative approaches to direct p-suffix sorting by generating and sorting lexicographically numeric fingerprints and arithmetic codes that correspond to individual p-suffixes. Our algorithm to p-suffix sort via fingerprints is the first theoretical linear time algorithm for p-suffix sorting for non-binary parameter alphabets, which assumes that each code is represented by a practical integer. We eliminate the key problems of fingerprints by introducing an algorithm that exploits the ordering of arithmetic codes to sort p-suffixes in linear time on average.
Richard Beal, Donald Adjeroh

Periods in Partial Words: An Algorithm

Abstract
Partial words are finite sequences over a finite alphabet that may contain some holes. A variant of the celebrated Fine-Wilf theorem shows the existence of a bound L = L(h,p,q) such that if a partial word of length at least L with h holes has periods p and q, then it has period \(\gcd(p,q)\). In this paper, we associate a graph with each p- and q-periodic word, and study two types of vertex connectivity on such a graph: modified degree connectivity and r-set connectivity where \(r = q \bmod{p}\). As a result, we give an algorithm for computing L(h, p, q) in the general case.
Francine Blanchet-Sadri, Travis Mandel, Gautam Sisodia

The 1-Neighbour Knapsack Problem

Abstract
We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.
Glencora Borradaile, Brent Heeringa, Gordon Wilfong

A Golden Ratio Parameterized Algorithm for Cluster Editing

Abstract
The Cluster Editing problem asks to transform a graph by at most k edge modifications into a disjoint union of cliques. The problem is NP-complete, but several parameterized algorithms are known. We present a novel search tree algorithm for the problem, which improves running time from O*(1.76 k ) to O*(1.62 k ). In detail, we can show that we can always branch with branching vector (2,1) or better, resulting in the golden ratio as the base of the search tree size. Our algorithm uses a well-known transformation to the integer-weighted counterpart of the problem. To achieve our result, we combine three techniques: First, we show that zero-edges in the graph enforce structural features that allow us to branch more efficiently. Second, by repeatedly branching we can isolate vertices, releasing costs. Finally, we use a known characterization of graphs with few conflicts.
Sebastian Böcker

Stable Sets of Threshold-Based Cascades on the Erdős-Rényi Random Graphs

Abstract
Consider the following reversible cascade on the Erdős-Rényi random graph G(n,p). In round zero, a set of vertices, called the seeds, are active. For a given ρ ∈ ( 0,1 ], a non-isolated vertex is activated (resp., deactivated) in round t ∈ ℤ +  if the fraction f of its neighboring vertices that were active in round t − 1 satisfies f ≥ ρ (resp., f < ρ). An irreversible cascade is defined similarly except that active vertices cannot be deactivated. A set of vertices, S, is said to be stable if no vertex will ever change its state, from active to inactive or vice versa, once the set of active vertices equals S. For both the reversible and the irreversible cascades, we show that for any constant ε > 0, all p ∈ [ (1 + ε) (ln (e/ρ))/n,1 ] and with probability 1 − n − Ω(1), every stable set of G(n,p) has size O(⌈ρn⌉) or n − O(⌈ρn⌉).
Ching-Lueh Chang, Yuh-Dauh Lyuu

How Not to Characterize Planar-Emulable Graphs

Abstract
We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph)—a problem raised in Fellows’ thesis (1985) and conceptually related to the better known planar cover conjecture by Negami (1986). For over two decades, the planar emulator problem lived poorly in a shadow of Negami’s conjecture—which is still open—as the two were considered equivalent. But, in the end of 2008, a surprising construction by Rieck and Yamashita falsified the natural “planar emulator conjecture”, and thus opened a whole new research field. We present further results and constructions which show how far the planar-emulability concept is from planar-coverability, and that the traditional idea of likening it to projective embeddability is actually very out-of-place. We also present several positive partial characterizations of planar-emulable graphs.
Markus Chimani, Martin Derka, Petr Hliněný, Matěj Klusáček

Testing Monotone Read-Once Functions

Abstract
A checking test for a monotone read-once function f depending essentially on all its n variables is a set of vectors M distinguishing f from all other monotone read-once functions of the same variables. We describe an inductive procedure for obtaining individual lower and upper bounds on the minimal number of vectors T(f) in a checking test for any function f. The task of deriving the exact value of T(f) is reduced to a combinatorial optimization problem related to graph connectivity. We show that for almost all functions f expressible by read-once conjunctive or disjunctive normal forms, T(f) ~n / ln n. For several classes of functions our results give the exact value of T(f).
Dmitry V. Chistikov

Complexity of Cycle Transverse Matching Problems

Abstract
The stable transversal problem for a fixed graph H asks whether a graph contains a stable set that meets every induced copy of H in the graph. Stable transversal problems generalize several vertex partition problems and have been studied for various classes of graphs. Following a result of Farrugia, the stable transversal problem for each C with ℓ ≥ 3 is NP-complete. In this paper, we study an ‘edge version’ of these problems. Specifically, we investigate the problem of determining whether a graph contains a matching that meets every copy of H. We show that the problem for C 3 is polynomial and for each C with ℓ ≥ 4 is NP-complete. Our results imply that the stable transversal problem for each C with ℓ ≥ 4 remains NP-complete when it is restricted to line graphs. We show by contrast that the stable transversal problem for C 3, when restricted to line graphs, is polynomial.
Ross Churchley, Jing Huang, Xuding Zhu

Efficient Conditional Expectation Algorithms for Constructing Hash Families

Abstract
Greedy methods for solving set cover problems provide a guarantee on how close the solution is to optimal. Consequently they have been widely explored to solve set cover problems arising in the construction of various combinatorial arrays, such as covering arrays and hash families. In these applications, however, a naive set cover formulation lists a number of candidate sets that is exponential in the size of the array to be produced. Worse yet, even if candidate sets are not listed, finding the ‘best’ candidate set is NP-hard. In this paper, it is observed that one does not need a best candidate set to obtain the guarantee — an average candidate set will do. Finding an average candidate set can be accomplished using a technique employing the method of conditional expectations for a wide range of set cover problems arising in the construction of hash families. This yields a technique for constructing hash families, with a wide variety of properties, in time polynomial in the size of the array produced.
Charles J. Colbourn

2-Layer Right Angle Crossing Drawings

Abstract
A 2-layer drawing represents a bipartite graph so that the vertices of each partition set are points of a distinct horizontal line (called a layer) and the edges are straight-line segments. In this paper we study 2-layer drawings where all edge crossings form right angles. We characterize which graphs admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is \(\mathcal{NP}\)-complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings.
Emilio Di Giacomo, Walter Didimo, Peter Eades, Giuseppe Liotta

Hamiltonian Orthogeodesic Alternating Paths

Abstract
Given a set of red and blue points, an orthogeodesic alternating path is a path such that each edge is a geodesic orthogonal chain connecting points of different colour and no two edges cross. We consider the problem of deciding whether there exists a Hamiltonian orthogeodesic alternating path, i.e., an orthogeodesic alternating path visiting all points. We provide an O(n log2 n)-time algorithm for finding such a path if no two points are horizontally or vertically aligned. We show that the problem is NP-hard if bends must be at grid points. Nevertheless, we can approximate the maximum number of vertices of an orthogeodesic alternating path on the grid by roughly a factor of 3. Finally, we consider the problem of finding orthogeodesic alternating matchings, cycles, and trees.
Emilio Di Giacomo, Luca Grilli, Marcus Krug, Giuseppe Liotta, Ignaz Rutter

Ranking and Loopless Generation of k-ary Dyck Words in Cool-lex Order

Abstract
A binary string B of length n = k t is a k-ary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating k-ary Dyck words in cool-lex order: (1) The first requires two index variables and assumes k is a constant; (2) The second requires t index variables and works for any k. We also efficiently rank k-ary Dyck words in cool-lex order. Our results generalize the “coolCat” algorithm by Ruskey and Williams (Generating balanced parentheses and binary trees by prefix shifts in CATS 2008) and provide the first loopless and ranking applications of the general cool-lex Gray code by Ruskey, Sawada, and Williams (Binary bubble languages and cool-lex order under review).
Stephane Durocher, Pak Ching Li, Debajyoti Mondal, Aaron Williams

Two Constant-Factor-Optimal Realizations of Adaptive Heapsort

Abstract
In this paper we introduce two efficient priority queues. For both, insert requires O(1) amortized time and extract-min \(O(\lg n)\) worst-case time including at most \(\lg n + O(1)\) element comparisons, where n is the number of elements stored. One priority queue is based on a weak heap (array-based) and the other on a weak queue (pointer-based). In both, the main idea is to temporarily store the inserted elements in a buffer, and once it is full to move its elements to the main queue using an efficient bulk-insertion procedure. By employing the new priority queues in adaptive heapsort, we guarantee, for several measures of disorder, that the formula expressing the number of element comparisons performed by the algorithm is optimal up to the constant factor of the high-order term. We denote such performance as constant-factor optimality. Unlike some previous constant-factor-optimal adaptive sorting algorithms, adaptive heapsort relying on the developed priority queues is practically workable.
Stefan Edelkamp, Amr Elmasry, Jyrki Katajainen

A Unifying Property for Distribution-Sensitive Priority Queues

Abstract
We present a priority queue that supports the operations: insert in worst-case constant time, and delete, delete-min, find-min and decrease-key on an element x in worst-case \(O(\lg(\min\{w_x, q_x\}+2))\) time, where w x (respectively, q x ) is the number of elements that were accessed after (respectively, before) the last access of x and are still in the priority queue at the time when the corresponding operation is performed. Our priority queue then has both the working-set and the queueish properties; and, more strongly, it satisfies these properties in the worst-case sense. We also argue that these bounds are the best possible with respect to the considered measures. Moreover, we modify our priority queue to satisfy a new unifying property — the time-finger property — which encapsulates both the working-set and the queueish properties.
In addition, we prove that the working-set bound is asymptotically equivalent to the unified bound (which is the minimum per operation among the static-finger, static-optimality, and working-set bounds). This latter result is of tremendous interest by itself as it had gone unnoticed since the introduction of such bounds by Sleater and Tarjan [10].
Together, these results indicate that our priority queue also satisfies the static-finger, the static-optimality and the unified bounds.
Amr Elmasry, Arash Farzan, John Iacono

Enumerating Tatami Mat Arrangements of Square Grids

Abstract
We prove that the number of monomer-dimer tilings of an n×n square grid, with m < n monomers in which no four tiles meet at any point is m2 m  + (m + 1)2 m + 1, when m and n have the same parity. In addition, we present a new proof of the result that there are n2 n − 1 such tilings with n monomers, which divides the tilings into n classes of size 2 n − 1. The sum of these over all m ≤ n has the closed form 2 n − 1(3n − 4) + 2 and, curiously, this is equal to the sum of the squares of all parts in all compositions of n.
Alejandro Erickson, Mark Schurch

Quasi-Cyclic Codes over $\mathbb{F}_{13}$

Abstract
Let d q (n,k) be the maximum possible minimum Hamming distance of a linear [n,k] code over \(\mathbb{F}_{q}\). Tables of best known linear codes exist for all fields up to q = 9. In this paper, linear codes over \(\mathbb{F}_{13}\) are constructed for k up to 6. The codes constructed are from the class of quasi-cyclic codes. In addition, the minimum distance of the extended quadratic residue code of length 44 is determined.
T. Aaron Gulliver

Acyclic Colorings of Graph Subdivisions

Abstract
An acyclic coloring of a graph G is a coloring of the vertices of G, where no two adjacent vertices of G receive the same color and no cycle of G is bichromatic. An acyclic k-coloring of G is an acyclic coloring of G using at most k colors. In this paper we prove that any triangulated plane graph G with n vertices has a subdivision that is acyclically 4-colorable, where the number of division vertices is at most 2n − 6. We show that it is NP-complete to decide whether a graph with degree at most 7 is acyclically 4-colorable or not. Furthermore, we give some sufficient conditions on the number of division vertices for acyclic 3-coloring of subdivisions of partial k-trees and cubic graphs.
Debajyoti Mondal, Rahnuma Islam Nishat, Sue Whitesides, Md. Saidur Rahman

Kinetic Euclidean Minimum Spanning Tree in the Plane

Abstract
This paper presents the first kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of n moving points in 2-dimensional space. We build a KDS of size O(n) in O(nlogn) preprocessing time by which their EMST is maintained efficiently during the motion. In terms of the KDS performance parameters, our KDS is responsive, local, and compact.
Zahed Rahmati, Alireza Zarei

Generating All Simple Convexly-Drawable Polar Symmetric 6-Venn Diagrams

Abstract
An n-Venn diagram consists of n curves drawn in the plane in such a way that each of the 2 n possible intersections of the interiors and exteriors of the curves forms a connected non-empty region. A Venn diagram is convexly-drawable if it can be drawn with all curves convex and it is simple if at most two curves intersect at any point. A Venn diagram is called polar symmetric if its stereographic projection about the infinite outer face is isomorphic to the projection about the innermost face. We outline an algorithm that shows there are exactly 375 simple convexly drawable polar-symmetric 6-Venn diagrams.
Khalegh Mamakani, Wendy Myrvold, Frank Ruskey

The Rand and Block Distances of Pairs of Set Partitions

Abstract
The Rand distance of two set partitions is the number of pairs {x,y} such that there is a block in one partition containing both x and y, but x and y are in different blocks in the other partition. Let R(n,k) denote the number of distinct (unordered) pairs of partitions of n that have Rand distance k. For fixed k we prove that R(n,k) can be expressed as \(\sum_j c_{k,j} {n \choose j} B_{n-j}\) where c k,j is a non-negative integer and B n is a Bell number. For fixed k we prove that there is a constant K n such that \(R(n,{n \choose 2}-k)\) can be expressed as a polynomial of degree 2k in n for all n ≥ K n . This polynomial is explicitly determined for 0 ≤ k ≤ 3.
The block distance of two set partitions is the number of elements that are not in common blocks. We give formulae and asymptotics based on N(n), the number of pairs of partitions with no blocks in common. We develop an O(n) algorithm for computing the block distance.
Frank Ruskey, Jennifer Woodcock

On Minimizing the Number of Label Transitions around a Vertex of a Planar Graph

Abstract
We study the minimum number of label transitions around a given vertex v 0 in a planar multigraph G in which the edges incident with v 0 are labelled with integers 1, .…, l, where the minimum is taken over all embeddings of G in the plane. For a fixed number of labels, a linear-time FPT algorithm that (given the labels around v 0) computes the minimum number of label transitions around v 0 is presented. If the number of labels is unconstrained, then the problem of deciding whether the minimum number of label transitions is at most k is NP-complete.
Bojan Mohar, Petr Škoda

A New View on Rural Postman Based on Eulerian Extension and Matching

Abstract
We provide a new characterization of the NP-hard arc routing problem Rural Postman in terms of a constrained variant of minimum-weight perfect matching on bipartite graphs. To this end, we employ a parameterized equivalence between Rural Postman and Eulerian Extension, a natural arc addition problem in directed multigraphs. We indicate the NP-hardness of the introduced matching problem. In particular, we use it to make some partial progress towards answering the open question about the parameterized complexity of Rural Postman with respect to the number of weakly connected components in the graph induced by the required arcs. This is a more than thirty years open and long-time neglected question with significant practical relevance.
Manuel Sorge, René van Bevern, Rolf Niedermeier, Mathias Weller

Hamilton Cycles in Restricted Rotator Graphs

Abstract
The rotator graph has vertices labeled by the permutations of n in one line notation, and there is an arc from u to v if a prefix of u’s label can be rotated to obtain v’s label. In other words, it is the directed Cayley graph whose generators are \(\sigma_{k} := (1 \ 2 \ \cdots \ k)\) for 2 ≤ k ≤ n and these rotations are applied to the indices of a permutation. In a restricted rotator graph the allowable rotations are restricted from k ∈ {2,3,…,n} to k ∈ G for some smaller (finite) set G ⊆ {2,3,…,n}. We construct Hamilton cycles for G = {n−1,n} and G = {2,3,n}, and provide efficient iterative algorithms for generating them. Our results start with a Hamilton cycle in the rotator graph due to Corbett (IEEE Transactions on Parallel and Distributed Systems 3 (1992) 622–626) and are constructed entirely from two sequence operations we name ‘reusing’ and ‘recycling’.
Brett Stevens, Aaron Williams

Efficient Codon Optimization with Motif Engineering

Abstract
It is now common to add protein coding genes into cloning vectors for expression within non-native host organisms. Codon optimization supports translational efficiency of the desired protein product, by exchanging codons which are rarely found in the host organism with more frequently observed codons. Motif engineering, such as removal of restriction enzyme recognition sites or addition of immuno-stimulatory elements, is also often necessary. We present an algorithm for optimizing codon bias of a gene with respect to a well motivated measure of bias, while simultaneously performing motif engineering. The measure is the previously studied codon adaptation index, which favors the use, in the gene to be optimized, of the most abundant codons found in the host genome. We demonstrate the efficiency and effectiveness of our algorithm on the GENCODE dataset and provide a guarantee that the solution found is always optimal.
Anne Condon, Chris Thachuk

An Algorithm for Road Coloring

Abstract
A coloring of edges of a finite directed graph turns the graph into a finite-state automaton. The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph of uniform outdegree (constant outdegree of any vertex) is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word.
The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of the lengths of all its cycles is one. The problem posed in 1970 has evoked noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics as well as among the wide mathematical community.
A polynomial time algorithm of O(n 3) complexity in the worst case and quadratic in the majority of studied cases for the road coloring of the considered graph is presented below. The work is based on the recent positive solution of the road coloring problem. The algorithm was implemented in the freeware package TESTAS.
A. N. Trahtman

Complexity of the Cop and Robber Guarding Game

Abstract
The guarding game is a game in which several cops has to guard a region in a (directed or undirected) graph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, the robber on the remaining vertices (the robber-region). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to determine whether for a given graph and given number of cops the cops are able to prevent the robber from entering the guarded region. The problem is highly nontrivial even for very simple graphs. It is known that when the robber-region is a tree, the problem is NP-complete, and if the robber-region is a directed acyclic graph, the problem becomes PSPACE-complete [Fomin, Golovach, Hall, Mihalák, Vicari, Widmayer: How to Guard a Graph? Algorithmica, DOI: 10.1007/s00453-009-9382-4]. We solve the question asked by Fomin et al. in the previously mentioned paper and we show that if the graph is arbitrary (directed or undirected), the problem becomes E-complete.
Robert Šámal, Rudolf Stolař, Tomas Valla

Improved Steiner Tree Algorithms for Bounded Treewidth

Abstract
We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in \(\ensuremath{\mathcal{O}(B_{\ensuremath{\textit{tw}}+2}^2 \cdot \ensuremath{\textit{tw}}\ \cdot |V|)}\) time, where \(\ensuremath{\textit{tw}}\) is the graph’s treewidth and the Bell number B k is the number of partitions of a k-element set. This is a linear time algorithm for graphs with fixed treewidth and a polynomial algorithm for \(\ensuremath{\textit{tw}} = \ensuremath{\mathcal{O}(\log|V|/\log\log|V|)}\).
While being faster than the previously known algorithms, our thereby used coloring scheme can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems.
Markus Chimani, Petra Mutzel, Bernd Zey

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