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19
Solving connectivity problems parameterized by treewidth in single exponential time (Extended Abstract)
, 2011
"... For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw |V | O(1) time algorithms, where tw is the treewidth of the input g ..."
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Cited by 34 (7 self)
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For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw |V | O(1) time algorithms, where tw is the treewidth of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best–known algorithms were naive dynamic programming schemes running in at least tw tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c tw |V | O(1) time Monte Carlo algorithms for most connectivity-type problems, including HAMIL-TONIAN PATH, STEINER TREE, FEEDBACK VERTEX SET and CONNECTED DOMINATING SET. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H-minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like CYCLE PACKING.
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal
, 2010
"... We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that fo ..."
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Cited by 19 (5 self)
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We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that for any ǫ> 0; • INDEPENDENT SET cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time, • DOMINATING SET cannot be solved in (3 − ǫ) tw(G) |V (G) | O(1) time, • MAX CUT cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time, • ODD CYCLE TRANSVERSAL cannot be solved in (3 − ǫ) tw(G) |V (G) | O(1) time, • For any q ≥ 3, q-COLORING cannot be solved in (q − ǫ) tw(G) |V (G) | O(1) time, • PARTITION INTO TRIANGLES cannot be solved in (2 − ǫ) tw(G) |V (G) | O(1) time. Our lower bounds match the running times for the best known algorithms for the problems, up to the ǫ in the base.
Fast Hamiltonicity checking via bases of perfect matchings
, 2014
"... For an even integer t ≥ 2, the Matching Connectivity matrix Ht is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph Kt on t vertices; an entry Ht[M1,M2] is 1 if M1 ∪M2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Ham ..."
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Cited by 12 (1 self)
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For an even integer t ≥ 2, the Matching Connectivity matrix Ht is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph Kt on t vertices; an entry Ht[M1,M2] is 1 if M1 ∪M2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Hamiltonicity problem, we present three results on the structure of Ht: We first show that Ht has rank at most 2t/2−1 over GF(2) via an appropriate factorization that explicitly provides families of matchings Xt forming bases for Ht. Second, we show how to quickly change representation between such bases. Third, we notice that the sets of matchings Xt induce permutation matrices within Ht. Subsequently, we use the factorization to obtain an 1.888nnO(1) time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Our algorithm as well counts the number of Hamiltonian cycles modulo two in directed bipartite or undirected graphs in the same time bound. Moreover, we use the fast basis change algorithm from the second result to present a Monte Carlo algorithm that given an undirected graph on n vertices along with a path decomposition of width at most pw decides Hamiltonicity in (2 + 2)pwnO(1) time. Finally, we use the third result to show that for every > 0 this cannot be improved to (2 + 2 − )pwnO(1) time unless the Strong Exponential Time Hypothesis fails, i.e., a faster algorithm for this problem would imply the breakthrough result of a (2 − )n time algorithm for CNF-Sat.
TRIMMED MOEBIUS INVERSION AND GRAPHS OF BOUNDED DEGREE
"... We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an n-ele ..."
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Cited by 8 (2 self)
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We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an n-element universe U and a family F of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from F in time within a polynomial factor (in n) of the number of supersets of the members of F. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree ∆. In particular, we show how to compute the Domatic Number in time within a polynomial factor of (2 ∆+1 − 2) n/(∆+1) and the Chromatic Number in time within a polynomial factor of (2 ∆+1 − ∆ − 1) n/(∆+1). For any constant ∆, these bounds are O ` (2 − ɛ) n ´ for ɛ> 0 independent of the number of vertices n.
A refined exact algorithm for edge dominating set
, 2011
"... We present an O ∗ (1.3160 n)-time algorithm for the edge dominating set problem in an n-vertex graph, which improves previous exact algorithms for this problem. The algorithm is analyzed by using the “Measure and Conquer method.” We design new branching rules based on conceptually simple local struc ..."
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Cited by 6 (3 self)
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We present an O ∗ (1.3160 n)-time algorithm for the edge dominating set problem in an n-vertex graph, which improves previous exact algorithms for this problem. The algorithm is analyzed by using the “Measure and Conquer method.” We design new branching rules based on conceptually simple local structures, called “clique-producing vertices/cycles,” which significantly simplify the algorithm and its running time analysis, attaining an improved time bound at the same time.
Parameterized Edge Dominating Set in Cubic Graphs
"... In this paper, we present an improved algorithm to decide whether a graph of maximum degree 3 has an edge dominating set of size k or not, which is based on enumerating vertex covers. We first enumerate vertex covers of size at most 2k and then construct an edge dominating set based on each vertex c ..."
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Cited by 5 (4 self)
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In this paper, we present an improved algorithm to decide whether a graph of maximum degree 3 has an edge dominating set of size k or not, which is based on enumerating vertex covers. We first enumerate vertex covers of size at most 2k and then construct an edge dominating set based on each vertex cover to find a satisfied edge dominating set. To enumerate vertex covers, we use a branch-and-reduce method, which will generate a search tree of size O(2.1479 k). Then we get the running time bound of the algorithm. Key words.
Bounded-degree techniques accelerate some parameterized graph algorithms
- FOMIN (EDS.), INT. WORKSHOP ON PARAMETERIZED AND EXACT COMP., IWPEC 2009, LNCS 5917
, 2009
"... Many parameterized algorithms for NP-hard graph problems are search tree algorithms with sophisticated local branching rules. But it has also been noticed that the global structure of input graphs can help improve the time bounds. Here we present some new results based on the global structure of bou ..."
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Cited by 3 (1 self)
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Many parameterized algorithms for NP-hard graph problems are search tree algorithms with sophisticated local branching rules. But it has also been noticed that the global structure of input graphs can help improve the time bounds. Here we present some new results based on the global structure of bounded-degree graphs after branching away the high-degree vertices. Some techniques and structural results are generic and should find more applications. First, we decompose a graph by branchings along a separator into cheap or small components where we can further branch separately. We apply this technique to accelerate the O∗(1.3803k) time algorithm for counting the vertex covers of size k (Mölle, Richter, and Rossmanith, 2006) to O∗(1.3740k). Next we give a complete characterization of graphs where every edge is in at most two conflict triples, i.e., triples of vertices with exactly two edges. This enables us to improve to O∗(1.47k) the previous O∗(1.53k) time algorithm (Gramm, Guo, Hüffner, Niedermeier, 2004) for Cluster Deletion. i.e., for deleting k edges
Improved Exact Algorithms for Counting 3- and 4-Colorings
"... Abstract. We introduce a generic algorithmic technique and apply it on decision and counting versions of graph coloring. Our approach is based on the following idea: either a graph has nice (from the algorithmic point of view) properties which allow a simple recursive procedure to find the solution ..."
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Cited by 3 (0 self)
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Abstract. We introduce a generic algorithmic technique and apply it on decision and counting versions of graph coloring. Our approach is based on the following idea: either a graph has nice (from the algorithmic point of view) properties which allow a simple recursive procedure to find the solution fast, or the pathwidth of the graph is small, which in turn can be used to find the solution by dynamic programming. By making use of this technique we obtain the fastest known exact algorithms – running in time O(1.7272 n) for deciding if a graph is 4-colorable and – running in time O(1.6262 n) and O(1.9464 n) for counting the number of k-colorings for k = 3 and 4 respectively. 1
Colorings With Few Colors: Counting, Enumeration and Combinatorial
"... Abstract. We provide exact algorithms for enumeration and counting problems on edge colorings and total colorings of graphs, if the number of (available) colors is fixed and small. For edge 3-colorings the following is achieved: there is a branching algorithm to enumerate all edge 3-colorings of a c ..."
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Cited by 2 (0 self)
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Abstract. We provide exact algorithms for enumeration and counting problems on edge colorings and total colorings of graphs, if the number of (available) colors is fixed and small. For edge 3-colorings the following is achieved: there is a branching algorithm to enumerate all edge 3-colorings of a connected cubic graph in time O ∗ (2 5n/8). This implies that the maximum number of edge 3-colorings in an n-vertex connected cubic graph is O ∗ (2 5n/8). Finally, the maximum number of edge 3-colorings in an n-vertex connected cubic graph is lower bounded by 12 n/10.Similar results are achieved for total 4-colorings of connected cubic graphs. We also present dynamic programming algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth. These algorithms can be used to obtain fast exact exponential time algorithms for counting edge k-colorings and total k-colorings on graphs, if k is small. 1
