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A faster approximate method to identify minimum dominating set
Abstract
The minimum dominating set (MDS) problem is known to be NP-hard. Compared with the currently fastest extract algorithm for MDS problem on graphs of n vertices using O(20.59n)time, which could merely tackle these problems with 200 vertices scale in 8 hours, the paper presents a faster approximate layer-method to address MDS problems in O(n2) time and the method is proved to be effective particularly for larger scale MDS. Furthermore, the programing phases are shown in detail.
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We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697n
, thus improving on the trivial \(\mathcal{O}(2^{n}/\sqrt{n})\) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an \(\mathcal{O}(1.7697^{n})\) listing algorithm.
Based on this result, we derive an \(\mathcal{O}(2.8805^{n})\) algorithm for the domatic number problem, and an \(\mathcal{O}(1.5780^{n})\) algorithm for the minimum-weight dominating set problem. Both algorithms improve over the previous algorithms.
A procedure is described for finding sets of key players in a social network. A key assumption is that the optimal selection of key players depends on what they are needed for. Accordingly, two generic goals are articulated, called KPP-POS and KPP-NEG. KPP- POS is defined as the identification of key players for the purpose of optimally diffusing something through the network by using the key players as seeds. KPP-NEG is defined as the identification of key players for the purpose of disrupting or fragmenting the network by removing the key nodes. It is found that off-the-shelf centrality measures are not optimal for solving either generic problem, and therefore new measures are presented.
We design fast exact algorithms for the problem of computing a minimum dominating set in undirected graphs. Since this problem is NP-hard, it comes with no big surprise that all our time complexities are exponential in the number eta of vertices. The contribution of this paper are 'nice' exponential time complexities that are bounded by functions of the form c(eta) with reasonably small constants c < 2: For arbitrary graphs we get a time complexity of 1.93782(eta). And for the special cases of split graphs, bipartite graphs, and graphs of maximum degree three, we reach time complexities of 1.41422(eta), 1.732061(eta), and 1.51433(eta), respectively.
An improved version of R. E. Tarjan and A. E. Trojanowski's (SIAM J. Comput. , vol 3, pp. 537-546, 1977) fast algorithm for finding a maximum independent set in a graph is presented. The algorithm has an upper bound of O(2**0 **. **3 **0 **4 **n ).
This paper provides a review on recent advances in the analysis and design of exact algorithms for domination in arbitrary graphs. The dominating set problem is one of the classical NP-complete problems and fits into broader class of domination problems. The authors provide the detailed algorithms, which have been very recently obtained, and illustrations for the domination problems in arbitrary graphs including minimum dominating set, maximum independent set, minimum independent dominating set, minimum connected dominating set and minimum weighted dominating set for the purpose of making the paper be self-contained and easy reading. Several techniques such as Branch & Reduce, Complexity analysis, Measure & Conquer, Memorization and so on are also discussed. After a long period of silence since Claude Berge formulated for the first time of modern conception of domination, the interest in design of exact exponential time algorithms has been growing significantly within the last five years. In addition to present a view of the latest results, the authors also hope more domestic research communities would pay attention to this rapidly developed field of research.
Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several new techniques are introduced, which lead to a simpler and improved algorithm of time complexity O(kn + 1.271k
k
2) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree.
An algorithm is presented which finds (the size of) a maximum independent set of an n vertex graph in time O(20.276n) improving on a previous bound of . The improvement comes principally from three sources: first, a modified recursive algorithm based on a more detailed study of the possible subgraphs around a chosen vertex; second, an improvement, not in the algorithm but in the time bound proved, by an argument about connected regular graphs; third, a time-space trade-off which can speed up recursive algorithms from a fairly wide class.
Closely related to the maximum clique problem, the Maximum Independent Set problem (MIS) was among the first problems proved NP-complete It is now known that the size of a maximum independent set cannot be approximated even within a factor of n 1-o(1) in polynomial time. Therefore it is important to find faster exact algorithms for MIS. Our algorithm produces subproblems that need not be induced subgraphs of G. One immediate advantage is that if G has a node of degree 2 we solve only one subproblem, whereas previous algorithms solved two. On the other hand, the resulting recursion may produce sub * problems containing nodes of arbitrarily high degree, even though the degree of G is bounded. By counting edges instead of vertices as our progress measure, we are nonetheless able to obtain good results for bounded degree graphs. We obtain the fastest MIS algorithms for sparse graphs and for graphs whose degree is bounded by 3 or by 4. We also obtain the fastest MIS algorithm for general graphs that runs in polynomial space, and the fastest MIS algorithm for general graphs that fits on one journal page. Our first simple algorithm finds maximum independent sets in time 2 0·114e for graphs with e edges, 2 0·171n for n-node graphs with maximum degree 3, 2 0·228n for n-node graphs with maximum degree 4. Our second simple algorithm finds maximum independent sets in time 2 0·303n for general graphs. Our third algorithm, which uses a complicated case analysis, finds maximum independent sets in time 2 0·290n for general graphs. All of our algorithms run in polynomial space. Space allows us to present only our first two algorithms.
Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times.
In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number.
This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.
We design fast exact algorithms for the problem of computing a minimum dominating set in undirected graphs. Since this problem
is NP-hard, it comes with no big surprise that all our time complexities are exponential in the number n of vertices. The contribution of this paper are ‘nice’ exponential time complexities that are bounded by functions of the
form c
n
with reasonably small constants c
n
. And for the special cases of split graphs, bipartite graphs, and graphs of maximum degree three, we reach time complexities
of 1.41422
n
, 1.73206
n
, and 1.51433
n
, respectively.
The currently (asymptotically) fastest algorithm for minimum dominating set on graphs of n nodes is the trivial �( 2n) algorithm which enumerates and checks all the subsets of nodes. In this paper we present a simple algorithm which solves this problem in O(1.81n) time. 2005 Published by Elsevier B.V.
We present an algorithm which finds a maximum independent set in an n-vertex graph in 0($2^{n/3}$) time. The algorithm can thus handle graphs roughly three times as large as could be analyzed using a naive algorithm.
The Vertex Cover problem asks, for input consisting of a graph G on n vertices, and a positive integer k, whether there is a set of k vertices such that every edge of G is incident with at least one of these vertices. We give an algorithm for this problem that runs in time O(kn + (1.324718)kk2). In particular, this gives an O((1.324718)nn2) algorithm to find the minimum vertex cover in the graph.
We consider worst case time bounds for NP-complete problems
including 3-coloring, 3-edge-coloring, and 3-list-coloring. Our
algorithms are based on a common generalization of these problems,
called symbol-system satisfiability or, briefly, SSS. 3-SAT is
equivalent to (2,3)-SSS while the other problems above are special cases
of (3,2)-SSS; there is also a natural duality transformation from
(a,b)-SSS to (b,a)-SSS. We give a fast algorithm for (3,2)-SSS and use
it to improve the time bounds for solving the other problems listed
above
A faster algorithm for finding a maximum independent set in a graph is presented. The algorithm is an improved version of the one by Tarjan and Trojanowski [7]. A technique to further accelerate this algorithm is also described.
We consider worst case time bounds for NP-complete problems including 3-SAT, 3coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a common generalization of these problems, called symbol-system satisfiability or, briefly, SSS [1]. 3-SAT is equivalent to (2,3)-SSS while the other problems above are special cases of (3,2)-SSS; there is also a natural duality transformation from (a; b)-SSS to (b; a)-SSS. We give a fast algorithm for (3,2)-SSS and use it to improve the time bounds for solving the other problems listed above. 1 Introduction There are many known NP-complete problems including such important graph theoretic problems as coloring and independent sets. Unless P=NP, we know that no polynomial time algorithm for these problems can exist, but that does not obviate the need to solve them as efficiently as possible, indeed the fact that these problems are hard makes efficient algorithms for them especially important. We are interested in this paper in...
We investigate the fixed-parameter complexity of WEIGHTED VERTEX COVER. Given a graph G = (V, E), a weight function ω: V → R+, and k ∈ R-. WEIGHTED VERTEX COVER WVC for short) asks for a vertex subset C ⊆ V of total weight at most k such that every edge of G has at least one endpoint in C. WVC and its natural variants are NP-complete. We observe that, when restricting the range of ω to positive integers, the so-called INTEGER-WVC can be solved as fast as unweighted VERTEX COVER. Our main result is that if the range of ω is restricted to positive reals ≥ 1, then socalled REAL-WVC can be solved in time O(1.3954k + k|V|). By way of contrast, unless P = NP, the problem is not fixed-parameter tractable if arbitrary weights > 0 are allowed. Using dynamic programming, at the expense of exponential memory use, we can improve the running time of REAL-WVC to O(1.3788k + k|V|). The same technique applied to a known algorithm yields the so far fastest algorithm for unweighted VERTEX COVER, running in time O(1.2832k k + k|V|).
We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3-SAT is equivalent to (2,3)-CSP while the other problems above are special cases of (3,2)-CSP. We give a fast algorithm for (3,2)-CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas.
Thomas Cvan Dijk. Inclusion/ Exclusion meets measure and conquer: Exact algorithms for counting doinating sets
- M Johan
- Jesper Van Rooij
- Nederlof