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25
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 58 (15 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
A Simple and Fast Approach for Solving Problems on Planar Graphs
 In Proc. 21st STACS, volume 2996 of LNCS
, 2004
"... It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is th ..."
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Cited by 17 (2 self)
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It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is the number of vertices. However, the constants hidden in bigOh, usually are too large to claim the algorithms to be practical even on graphs of moderate size. Here we introduce a new algorithm design paradigm for solving problems on planar graphs. The paradigm is so simple that it can be explained in any textbook on graph algorithms: Compute tree or branch decomposition of a planar graph and do dynamic programming. Surprisingly such a simple approach provides faster algorithms for many problems. For example, Independent Set on planar graphs can be solved in time O(2 ) and Dominating Set in time O(2 ). In addition, significantly broader class of problems can be attacked by this method. Thus with our approach, Longest cycle on planar graphs is solved in time O(2 and Bisection is solved in time O(2 ). The proof of these results is based on complicated combinatorial arguments that make strong use of results derived by the Graph Minors Theory. In particular we prove that branchwidth of a planar graph is at most 2.122 # n. In addition we observe how a similar approach can be used for solving di#erent fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speedup for fundamental problems on planar graphs like Vertex Cover, (Weighted) Dominating Set, and many others.
Multicoloring trees
 Inform. and Comput
"... Scheduling jobs with pairwise conflicts is modeled by the graph multicoloring problem. It occurs in two versions: in the preemptive case, each vertex may get any set of colors, while in the nonpreemptive case, the set of colors assigned to each vertex has to be contiguous. We study these versions o ..."
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Cited by 12 (1 self)
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Scheduling jobs with pairwise conflicts is modeled by the graph multicoloring problem. It occurs in two versions: in the preemptive case, each vertex may get any set of colors, while in the nonpreemptive case, the set of colors assigned to each vertex has to be contiguous. We study these versions of the multicoloring problem on trees, under the sumofcompletiontimes objective. In particular, we give a quadratic algorithm for the nonpreemptive case, and a faster algorithm in the case that all job lengths are short, while we present a polynomialtime approximation scheme for the preemptive case. 1
Improved Bounds for Scheduling Conflicting Jobs with Minsum Criteria
"... We consider a general class of scheduling problems where a set of conflicting jobs needs to be scheduled (preemptively or nonpreemptively) on a set of machines so as to minimize the weighted sum of completion times. The conflicts among the jobs are formed as an arbitrary conflict graph. Building on ..."
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Cited by 7 (2 self)
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We consider a general class of scheduling problems where a set of conflicting jobs needs to be scheduled (preemptively or nonpreemptively) on a set of machines so as to minimize the weighted sum of completion times. The conflicts among the jobs are formed as an arbitrary conflict graph. Building on the framework of Queyranne and Sviridenko (J. of Scheduling, 5:287305, 2002), we present a general technique for reducing the weighted sum of completion times problem to the classical makespan minimization problem. Using this technique, we improve the best known results for scheduling conflicting jobs with minsum objective, on several fundamental classes of graphs, including line graphs, (k +1)claw free graphs and perfect graphs. In particular, we obtain the first constant factor approximation ratio for nonpreemptive scheduling on interval graphs. We also improve the results of Kim (SODA 2003, 97–98) for scheduling jobs on line graphs and for resourceconstrained scheduling.
GRAPH COLOURING PROBLEMS AND THEIR APPLICATIONS IN SCHEDULING
, 2003
"... Graph colouring and its generalizations are useful tools in modelling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applicatio ..."
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Cited by 7 (0 self)
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Graph colouring and its generalizations are useful tools in modelling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applications in scheduling.
Graph Coloring Problems and Their Applications in Scheduling
 IN PROC. JOHN VON NEUMANN PHD STUDENTS CONFERENCE
, 2004
"... Graph coloring and its generalizations are useful tools in modeling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph coloring, such as precoloring extension, list coloring, multicoloring, minimum sum coloring, and discuss their applications i ..."
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Cited by 6 (0 self)
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Graph coloring and its generalizations are useful tools in modeling a wide variety of scheduling and assignment problems. In this paper we review several variants of graph coloring, such as precoloring extension, list coloring, multicoloring, minimum sum coloring, and discuss their applications in scheduling.
A short proof of the NPcompleteness of minimum sum interval coloring
, 2004
"... In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive di#erent numbers and the sum of the numbers is minimized. ..."
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Cited by 5 (1 self)
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In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive di#erent numbers and the sum of the numbers is minimized.
Minimum sum multicoloring on the edges of trees
, 2003
"... The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color ..."
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Cited by 5 (1 self)
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The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper is a polynomialtime approximation scheme for minimum sum multicoloring the edges of trees. We also show that the problem is strongly NPhard for trees, even if every demand is at most 2.
Batch Coloring Flat Graphs and Thin
"... Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, namely, to partition the vertices to batches which are independent sets and assign to each b ..."
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Cited by 5 (0 self)
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Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, namely, to partition the vertices to batches which are independent sets and assign to each batch a contiguous set of colors, whose size is equal to the maximum color requirement of any vertex in this batch. When the objective is to minimize the sum of job completion times, we get the batch sum coloring problem; when we want to minimize the maximum completion time of any job (or, the makespan) we get the max coloring problem. Given the hardness of batch coloring on general graphs, already for the special case of unit color requirements (known as sum coloring and the classic graph coloring problem, respectively), it is natural to seek out classes of graphs where effective solutions can be obtained efficiently. In this paper we give the first polynomial time approximation schemes for batch sum coloring on several classes of “nonthick ” graphs that arise in applications. This includes paths, trees, partial ktrees, and planar graphs. Also, we give an improved O(n log n) exact algorithm for the maxcoloring problem on paths.
Minimum sum multicoloring on the edges of planar graphs and partial ktrees
 In Proc. 2nd Wshop. Approx. and Online Alg. (WAOA), volume 3351 of LNCS
, 2004
"... Abstract. The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the high ..."
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Cited by 4 (0 self)
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Abstract. The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper is a polynomial time approximation scheme for minimum sum multicoloring the edges of planar graphs and partial ktrees. 1