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Approximation Results for the Optimum Cost Chromatic Partition Problem
 J. Algorithms
"... . In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation ..."
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. In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation results for the OCCP problem restricted to bipartite, chordal, comparability, interval, permutation, split and unimodular graphs. We prove that there exists no polynomial approximation algorithm with ratio O(jV j 0:5 ) for the OCCP problem restricted to bipartite and interval graphs, unless P = NP . Furthermore, we propose approximation algorithms with ratio O(jV j 0:5 ) for bipartite, interval and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O(jV j 1 ) for the OCCP problem restricted to split, chordal, permutation and comparability graphs, unless P = NP .
The Optimal Cost Chromatic Partition Problem for Trees and Interval Graphs
 GraphTheoretic Concepts in Computer Science (Cadenabbia, 1996), Lecture Notes in Computer Science
, 1996
"... In this paper we study the Optimal Cost Chromatic Partition (OCCP) problem for trees and interval graphs. The OCCP problem is the problem of coloring the nodes of a graph in such a way that adjacent nodes obtain different colors and that the total coloring costs are minimum. ..."
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In this paper we study the Optimal Cost Chromatic Partition (OCCP) problem for trees and interval graphs. The OCCP problem is the problem of coloring the nodes of a graph in such a way that adjacent nodes obtain different colors and that the total coloring costs are minimum.
A dichotomy for minimum cost graph homomorphisms
 European J. Combin
, 2007
"... For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost h ..."
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For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NPhard. This solves an open problem from an earlier paper. 1
A Provably Good Multilayer Topological Planar Routing Algorithm In IC Layout Designs
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1993
"... Given a number of routing layers, the multilayer topological planar routing problem is to choose a maximum (weighted) set of nets so that each net in the set can be topologically routed entirely in one of the given layers without crossing other nets. This problem has important application in the lay ..."
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Given a number of routing layers, the multilayer topological planar routing problem is to choose a maximum (weighted) set of nets so that each net in the set can be topologically routed entirely in one of the given layers without crossing other nets. This problem has important application in the layout design of multilayer IC technology, which has become available recently. In this paper, we present a provably good approximation algorithm for the multilayer topological planar routing problem. Our algorithm, called the iterativepeeling algorithm, finds a solution whose weight is guaranteed to be at least 1  e 1 ## ~ ~ 63.2% of the weight of an optimal solution. The algorithm works for multiterminal nets and arbitrary number of routing layers. For fixed number of routing layers, we have even tighter performance bounds. In particular, the performanceratio of the iterativepeeling algorithm is at least 75% for two layer routing, and is at least 70.4% for three layer routing. Experim...
Minimum Cost Homomorphisms to reflexive digraphs
 8th Latin American Theoretical Informatics (LATIN), Rio de Janeiro, Brazil
"... For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f(u)(u). For each fixed digraph H, the minimum cost hom ..."
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For digraphs G and H, a homomorphism of G to H is a mapping f: V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If moreover each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is u∈V (G) c f(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), is the following problem. Given an input digraph G, together with costs ci(u), u ∈ V (G), i ∈ V (H), and an integer k, decide if G admits a homomorphism to H of cost not exceeding k. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as chromatic partition optimization and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H (every vertex of H has a loop). It is known that the problem MinHOM(H) is polynomial time solvable if the digraph H has a MinMax ordering, i.e., if its vertices can be linearly ordered by < so that i < j, s < r and ir, js ∈ A(H) imply that is ∈ A(H) and jr ∈ A(H). We give a forbidden induced subgraph characterization of reflexive digraphs with a MinMax ordering; our characterization implies a polynomial time test for the existence of a MinMax ordering. Using this characterization, we show that for a reflexive digraph H which does not admit a MinMax ordering, the minimum cost homomorphism problem is NPcomplete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs. 1
Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs
"... Abstract. For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (D) c f(u)(u). For each fixed digraph H, we have the ..."
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Abstract. For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a kMinMax ordering of digraphs. Key words. homomorphisms, minimum cost homomorphisms, semicomplete bipartite digraphs
On the kLayer Planar Subset and Topological Via Minimization Problems
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1991
"... An important problem in performancedriven routing is the k layer planar subset problem which is to choose a maximum (weighted) subset of nets such that each net in the subset can be routed in one of k "preferred" layers. Related to the k layer planar subset problem is the k layer topol ..."
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An important problem in performancedriven routing is the k layer planar subset problem which is to choose a maximum (weighted) subset of nets such that each net in the subset can be routed in one of k "preferred" layers. Related to the k layer planar subset problem is the k layer topological viaminimization problem which is to determine the topology of each net using k routing layers such that a minimum number of vias is used. For the case k = 2, the topological via minimization problem has been studied by CAD researchers for a long time because of its practical and theoretical importance. In this paper, we show that both the general k layer planar subset problem and the k layer topological via minimization problem are NPcomplete. Moreover, we show that both problems can be solved in polynomial time when the routing regions are crossing channels. It can be shown that under a suitable assumption, all the channels for interblock connections in the general cell design style are ...
Minimum Cost “Homomorphisms to proper interval graphs and bigraphs
"... For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is ∑ u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost h ..."
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For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is ∑ u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NPhard. This solves an open problem from an earlier paper. Along the way, we prove a new characterization of the class of proper interval bigraphs. 1
Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops
"... For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomo ..."
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For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [12]. If each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [9], and a semicomplete multipartite digraph [11, 10]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [8].
An Analysis of Heuristics for Graph Planarization
 Journal of Information & Optimization Sciences
, 1997
"... We analyze several heuristics for graph planarization, i.e., deleting the minimum number of edges from a nonplanar graph to make it planar. The problem is NPhard, although some heuristics which perform well in practice have been reported. In particular, we compare the two principle methods, based o ..."
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We analyze several heuristics for graph planarization, i.e., deleting the minimum number of edges from a nonplanar graph to make it planar. The problem is NPhard, although some heuristics which perform well in practice have been reported. In particular, we compare the two principle methods, based on path addition and vertex addition, respectively, with a selective edge addition method, an incremental method, and a "cycle packing" approach. For the incremental, the path addition, and the edge addition methods, we prove theoretical worstcase performance bounds of 1=3. We also present an empirical analysis of the heuristics. Our results indicate that the "cyclepacking" method consistently yields the best solutions when applied to a large set of test graphs. 1