Chordal graphs are graphs with the property that each cycle of length greater than 3 has two non ... more Chordal graphs are graphs with the property that each cycle of length greater than 3 has two non consecutive vertices that are joined by an edge. An important subclass of chordal graphs are strongly chordal graphs \cite{Fa1}. Chordal graphs appear for example in the design of acyclic data base schemes \cite{BMFY}. In this paper we study the computational complexity (both sequential and parallel) of the maximum matching problem for chordal and strongly chordal graphs. We show that there is a linear time greedy algorithm for a maximum matching in a strongly chordal graph provided a strongly perfect elimination ordering is known. This algorithm can be also turned into a parallel algorithm. The technique used can be also extended for the multidimensional matching for chordal and strongly chordal graphs yielding the first polynomial time algorithms for these classes of graphs (the multidimensional matching is NP-complete in general).
We present efficient parallel algorithms for two hierarchical clustering heuristics. We point out... more We present efficient parallel algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solving some algorithmic problems in graphs, including split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of S. Cicerone and D. Di w x Stefano 7 that parity graphs are exactly those graphs that can be split decomposed into cliques and bipartite graphs.
Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vert... more Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vertices called termimds, and asked for a minimum weight set of edges that separates each terminal from all the others. when the number k of terminals is two, this is simply the min-cu~msx-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hsrd as sxrn as k = 3, but ctm be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hsrd however, if k is not tixad. We also describe a simple approximation slgorithnt for arbkmry graphs that is guaranteed to come within a fsctor of 2-2/k of the optimal cut weight.
The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degre... more The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are Pcomplete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the HIGH DEGREE SUEIGRAPH problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.
The paper considers the problem of computing a maximal independent set in hypergraphs (see [Karp,... more The paper considers the problem of computing a maximal independent set in hypergraphs (see [Karp, Ramachandran 88] and [Beame, Luby 89]). We present an efficient deterministic parallel algorithm for the case when the maximal cardinality of any hyperedge is 3. The algorithm works in $O(\log^4 n)$ parallel time with $O(n+m)$ processors on a CREW PRAM and is optimal up to a polylogarithmic factor.
... В 42 (1987), pp. 133-145 [F.Ï 86] Farber, M., and Jamison, RE, Convexity in Graphs and Hypcrg... more ... В 42 (1987), pp. 133-145 [F.Ï 86] Farber, M., and Jamison, RE, Convexity in Graphs and Hypcrgraphs, SIAM J. of Algebraic and Discrete Methods 7 (1986), pp. ... 161-165 [GJ 84] Goîumbic, MC, and Jamison, RE, The Edge Intersection Graphs of Pailis in a Tree, J. Combin. ...
Given a set of disjoint simple polygons σ 1 ,. .. , σ n , of total complexity N , consider a conv... more Given a set of disjoint simple polygons σ 1 ,. .. , σ n , of total complexity N , consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by their common convex hull. This process continues until no pair of polygons intersect. We show that this process has a unique output, which is a cover of the input polygons by a set of disjoint convex polygons, of total minimum area. Furthermore, we present a near linear time algorithm for computing this partition. The more general problem of covering a set of N segments (not necessarily disjoint) by min-area disjoint convex polygons can also be computed in near linear time. A similar result is already known, see the work by Barba et al. [BBBS13].
Chordal graphs became interesting as a generalization of interval graphs (see for example \cite{L... more Chordal graphs became interesting as a generalization of interval graphs (see for example \cite{LB}). We call a graph chordal if every cycle of length greater than three has a chord, i.e. an edge that joins two non consecutive vertices of the cycle. Note that interval graphs are not only chordal but strongly chordal as defined in \cite{Fa1}. Strongly chordal graphs are just those chordal graphs having a so called strongly perfect elimination ordering. In this paper we consider the sequential and parallel complexity of the maximum matching problem in chordal and strongly chordal graphs. Note that in general a linear time algorithm for perfect matching is not known. Here we shall show that, provided a strongly perfect elimination ordering is known, a maximum matching in a strongly chordal graph can be found in linear time by a simple greedy algorithm. This algorithm can be turned into a (non optimal) parallel algorithm.
The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degre... more The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are Pcomplete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the HIGH DEGREE SUEIGRAPH problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.
Page 1. Elias Dahlhaus and Johann A. Makowsky 287 ... To appear in the proceedings of the worksho... more Page 1. Elias Dahlhaus and Johann A. Makowsky 287 ... To appear in the proceedings of the workshop on "Logic and Computer Science: New Trends and Applications", Turin 1986. Gandy, RO 1980 Church's Thesis and Principles of Mechanism, The Kleene-Symposium, eds. ...
We prove that the problem to get an inclusion minimal elimination ordering can be solved in linea... more We prove that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs. The basic structure of the linear time algorithm is as follows. We select a vertex r as maximum and get a rst approximation of a minimal elimination ordering considering a vertex x as smaller than y if x has a larger distance than y from r. Using planarity, one can determine the ll-in edges joining two vertices of the same distance from r almost immediately. The algorithm determines an O(n)-representation of these ll-in edges. To determine the nal ll-in ordering, we use similar techniques as in the general parallel minimal elimination algorithm of 5].
Chordal graphs are graphs with the property that each cycle of length greater than 3 has two non ... more Chordal graphs are graphs with the property that each cycle of length greater than 3 has two non consecutive vertices that are joined by an edge. An important subclass of chordal graphs are strongly chordal graphs \cite{Fa1}. Chordal graphs appear for example in the design of acyclic data base schemes \cite{BMFY}. In this paper we study the computational complexity (both sequential and parallel) of the maximum matching problem for chordal and strongly chordal graphs. We show that there is a linear time greedy algorithm for a maximum matching in a strongly chordal graph provided a strongly perfect elimination ordering is known. This algorithm can be also turned into a parallel algorithm. The technique used can be also extended for the multidimensional matching for chordal and strongly chordal graphs yielding the first polynomial time algorithms for these classes of graphs (the multidimensional matching is NP-complete in general).
We present efficient parallel algorithms for two hierarchical clustering heuristics. We point out... more We present efficient parallel algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solving some algorithmic problems in graphs, including split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of S. Cicerone and D. Di w x Stefano 7 that parity graphs are exactly those graphs that can be split decomposed into cliques and bipartite graphs.
Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vert... more Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vertices called termimds, and asked for a minimum weight set of edges that separates each terminal from all the others. when the number k of terminals is two, this is simply the min-cu~msx-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hsrd as sxrn as k = 3, but ctm be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hsrd however, if k is not tixad. We also describe a simple approximation slgorithnt for arbkmry graphs that is guaranteed to come within a fsctor of 2-2/k of the optimal cut weight.
The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degre... more The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are Pcomplete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the HIGH DEGREE SUEIGRAPH problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.
The paper considers the problem of computing a maximal independent set in hypergraphs (see [Karp,... more The paper considers the problem of computing a maximal independent set in hypergraphs (see [Karp, Ramachandran 88] and [Beame, Luby 89]). We present an efficient deterministic parallel algorithm for the case when the maximal cardinality of any hyperedge is 3. The algorithm works in $O(\log^4 n)$ parallel time with $O(n+m)$ processors on a CREW PRAM and is optimal up to a polylogarithmic factor.
... В 42 (1987), pp. 133-145 [F.Ï 86] Farber, M., and Jamison, RE, Convexity in Graphs and Hypcrg... more ... В 42 (1987), pp. 133-145 [F.Ï 86] Farber, M., and Jamison, RE, Convexity in Graphs and Hypcrgraphs, SIAM J. of Algebraic and Discrete Methods 7 (1986), pp. ... 161-165 [GJ 84] Goîumbic, MC, and Jamison, RE, The Edge Intersection Graphs of Pailis in a Tree, J. Combin. ...
Given a set of disjoint simple polygons σ 1 ,. .. , σ n , of total complexity N , consider a conv... more Given a set of disjoint simple polygons σ 1 ,. .. , σ n , of total complexity N , consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by their common convex hull. This process continues until no pair of polygons intersect. We show that this process has a unique output, which is a cover of the input polygons by a set of disjoint convex polygons, of total minimum area. Furthermore, we present a near linear time algorithm for computing this partition. The more general problem of covering a set of N segments (not necessarily disjoint) by min-area disjoint convex polygons can also be computed in near linear time. A similar result is already known, see the work by Barba et al. [BBBS13].
Chordal graphs became interesting as a generalization of interval graphs (see for example \cite{L... more Chordal graphs became interesting as a generalization of interval graphs (see for example \cite{LB}). We call a graph chordal if every cycle of length greater than three has a chord, i.e. an edge that joins two non consecutive vertices of the cycle. Note that interval graphs are not only chordal but strongly chordal as defined in \cite{Fa1}. Strongly chordal graphs are just those chordal graphs having a so called strongly perfect elimination ordering. In this paper we consider the sequential and parallel complexity of the maximum matching problem in chordal and strongly chordal graphs. Note that in general a linear time algorithm for perfect matching is not known. Here we shall show that, provided a strongly perfect elimination ordering is known, a maximum matching in a strongly chordal graph can be found in linear time by a simple greedy algorithm. This algorithm can be turned into a (non optimal) parallel algorithm.
The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degre... more The HIGH DEGREE SUBGRAPH problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are Pcomplete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the HIGH DEGREE SUEIGRAPH problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.
Page 1. Elias Dahlhaus and Johann A. Makowsky 287 ... To appear in the proceedings of the worksho... more Page 1. Elias Dahlhaus and Johann A. Makowsky 287 ... To appear in the proceedings of the workshop on "Logic and Computer Science: New Trends and Applications", Turin 1986. Gandy, RO 1980 Church's Thesis and Principles of Mechanism, The Kleene-Symposium, eds. ...
We prove that the problem to get an inclusion minimal elimination ordering can be solved in linea... more We prove that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs. The basic structure of the linear time algorithm is as follows. We select a vertex r as maximum and get a rst approximation of a minimal elimination ordering considering a vertex x as smaller than y if x has a larger distance than y from r. Using planarity, one can determine the ll-in edges joining two vertices of the same distance from r almost immediately. The algorithm determines an O(n)-representation of these ll-in edges. To determine the nal ll-in ordering, we use similar techniques as in the general parallel minimal elimination algorithm of 5].
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