On Fine-Grained Exact Computation in Regular Graphs

Encyclopedia of Algorithms, 2008

We show that the maximum independent set problem on an n-vertex graph can be solved in 1.1996 n n O(1) time and polynomial space, which even is faster than Robson's 1.2109 n n O(1) -time exponentialspace algorithm published in 1986. We also obtain improved algorithms for MIS in graphs with maximum degree 6 and 7, which run in time of 1.1893 n n O(1) and 1.1970 n n O(1) , respectively. Our algorithms are obtained by using fast algorithms for MIS in low-degree graphs in a hierarchical way and making a careful analyses on the structure of boundeddegree graphs.

Lecture Notes in Computer Science, 2011

We present a simple algorithm for the maximum independent set problem in an n-vertex graph with degree bounded by 4, which runs in O * (1.1455 n ) time and improves all previous algorithms for this problem. In this paper, we use the "Measure and Conquer method" to analyze the running time bound, and use some good reduction and branching rules with a new idea on setting weights to obtain the improved time bound.

Discrete Applied Mathematics, 2010

The complexity status of the Maximum Independent Set Problem (MIS) for the family of P 5-free graphs is unknown. Although for many subclasses of P 5-free graphs MIS can be solved in polynomial time, only exponential time MIS-algorithms for general graphs are known so far. In this note we present the first algorithm to solve MIS for P 5-free graphs in subexponential time.

Theoretical Computer Science, 2013

We present an O * (1.0836 n )-time algorithm for finding a maximum independent set in an n-vertex graph with degree bounded by 3, which improves all previous running time bounds for this problem. Our approach has the following two features. Without increasing the number of reduction/branching rules to get an improved time bound, we first successfully extract the essence from the previously known reduction rules such as domination, which can be used to get simple algorithms. More formally, we introduce a procedure for computing "confining sets," which unifies several known reducible subgraphs and covers new reducible subgraphs. Second we identify those instances that generate the worst recurrence among all recurrences of our branching rules as "bottleneck instances" and prove that bottleneck instances cannot appear consecutively after each branching operation.

Algorithmica, 2012

max independent set is a paradigmatic problem in theoretical computer science and numerous studies tackle its resolution by exact algorithms with non-trivial worst-case complexity, both in the general case and in the case of bounded degree graphs. Here, we improve several of these results. We first get an algorithm in O * (1.0854 n ) for graphs of average degree 3 using a case analysis based upon a detailed case analysis using several reduction rules. Then we propose a generic method showing how improvement of the worst-case complexity for max independent set in graphs of average degree d entails improvement of it in any graph of average degree greater than d and, based upon it, we tackle max independent set by improving its complexity in graphs of average degree 4, 5 and 6. Finally, we combine this method with measure and conquer techniques to get improved running times for general graphs. The best computation bounds obtained are O * (1.1571 n ), O * (1.1918 n ) and O * (1.2070 n ), for graphs of maximum degree 4, 5 and 6 respectively, and O * (1.2125 n ) for general graphs. These results improve upon the best known results for these cases.

Discrete Mathematics, 2011

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I, there is a vertex w ∈ I such that {u, v, w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPcomplete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P = NP.

Computing Research Repository, 2007

We develop an experimental algorithm for the exact solving of the maximum independent set problem. The algorithm consecutively finds the maximal independent sets of vertices in an arbitrary undirected graph such that the next such set contains more elements than the preceding one. For this purpose, we use a technique, developed by Ford and Fulkerson for the finite partially ordered sets, in particular, their method for partition of a poset into the minimum number of chains with finding the maximum antichain. In the process of solving, a special digraph is constructed, and a conjecture is formulated concerning properties of such digraph. This allows to offer of the solution algorithm. Its theoretical estimation of running time equals to is O(n 8 ), where n is the number of graph vertices. The offered algorithm was tested by a program on random graphs. The testing the confirms correctness of the algorithm.

In this paper, we introduce the exact weighted independent set problem (EWIS), the problem of determining whether a given weighted graph contains an independent set of a given weight. Our motivation comes from the related exact perfect matching problem, whose computational complexity is still unknown. We determine the complexities of the EWIS problem and its restricted version EWIS α (where the independent set is required to be of maximum size) for several graph classes. These problems are strongly NP-complete for cubic bipartite graphs; we also extend this result to a more general setting. On the positive side, we show that EWIS and EWIS α can be solved in pseudo-polynomial time for chordal graphs, AT-free graphs, distance-hereditary graphs, circle graphs, graphs of bounded clique-width, and several subclasses of P 5 -free and fork-free graphs. In particular, we show how modular decomposition can be applied to the exact weighted independent set problem.

Israel Journal of Mathematics, 1964

Lower and upper bounds for the maximal number of independent vertices in a regular graph are obtained, it is shown that the bounds are best possible. Some properties of regular graphs concerning the property .~ defined below are investigated.

2014

The Independent Set problem is NP-hard in general, however polynomial time algorithms exist for the problem on various specific graph classes. Over the last couple of decades there has been a long sequence of papers exploring the boundary between the NP-hard and polynomial time solvable cases. In particular the complexity of Independent Set on P5-free graphs has received significant attention, and there has been a long list of results showing that the problem becomes polynomial time solvable on sub-classes of P5-free graphs. In this paper we give the first polynomial time algorithm for Independent Set on P5-free graphs. Our algorithm also works for the Weighted Independent Set problem.