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23
Minimal triangulations of graphs: A survey
- DISCRETE MATHEMATICS
"... Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was ..."
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Cited by 35 (3 self)
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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.
Exact algorithms for treewidth and minimum fill-in
- In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci
, 2004
"... We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free g ..."
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Cited by 28 (17 self)
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We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectiv ..."
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Cited by 21 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
Treewidth Lower Bounds with Brambles
, 2005
"... In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm fo ..."
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Cited by 12 (3 self)
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In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms give often excellent lower bounds, in particular when applied to (close to) planar graphs.
Treewidth computation and extremal combinatorics
- PROCEEDINGS OF THE 35TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING, ICALP 2008, PART I
, 2008
"... For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n ` b+f b such vertex sub ..."
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Cited by 12 (8 self)
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For a given graph G and integers b, f ≥ 0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n ` b+f b such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G – compute the treewidth of G in time O(1.7549 n) by making use of exponential space and in time O(2.6151 n) and polynomial space; – decide in time O( ( 2n+k+1 3) k+1 · n 6) if the treewidth of G is at most k; – list all minimal separators of G in time O(1.6181 n) and all potential maximal cliques of G in time O(1.7549 n). This significantly improves previous algorithms for these problems.
A Note on Exact Algorithms for Vertex Ordering Problems on Graphs
, 2009
"... In this note, we give a proof that several vertex ordering problems can be solved in O ∗ (2n) time and O ∗ (2n) space, or in O ∗ (4n) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp [12] and Gurevich and Shelah [11]. We survey a nu ..."
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Cited by 10 (1 self)
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In this note, we give a proof that several vertex ordering problems can be solved in O ∗ (2n) time and O ∗ (2n) space, or in O ∗ (4n) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp [12] and Gurevich and Shelah [11]. We survey a number of vertex ordering problems to which the results apply.
Exact algorithms for graph homomorphisms
- Proceedings of FCT 2005, LNCS 3623, 2005
"... Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During the recent years the topic of exact (exponential-time) algorithms for NP-hard problems in genera ..."
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Cited by 5 (4 self)
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Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During the recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Neˇsetˇril, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is a cycle of length 5 is the first NP-hard case different from graph coloring. We show that, for a given graph G on n vertices and an odd integer k ≥ 5, whether G is homomorphic to a cycle of length k can be decided in time min { � � n n/2 O(1) n/k, 2} · n. We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: If H is of treewidth at most t, then whether G is homomorphic to H can be decided in time (2t + 1) n · nO(1).
Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions
, 2007
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Computing and exploiting tree-decomposition for (Max-)CSP
, 2005
"... Methods exploiting the tree-decomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate o ..."
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Cited by 2 (1 self)
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Methods exploiting the tree-decomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate optimal tree-decomposition before assessing their relevance for solving CSPs. Then, we propose and compare several strategies to achieve the best depth-first traversal of the associated cluster tree w.r.t. CSP solving. These strategies concern the choice of the root cluster (i.e. the first visited cluster) and the order according to which we visit the sons of a given cluster. Finally, we propose a new decomposition strategy and heuristics which both rely on probabilistic criteria and which significantly improve the runtime.
