Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
Conclusion
References

Independent set in P5-free graphs in polynomial time

Profile image of Martin VatshelleMartin Vatshelle

2014

https://doi.org/10.5555/2634074.2634117
visibility

…

description

17 pages

link

1 file

Abstract

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.

Related papers

An $O^*(1.0821^n)$-Time Algorithm for Computing Maximum Independent Set in Graphs with Bounded Degree 3
Davis Issac

2013

We give an $O^*(1.0821^n)$-time, polynomial space algorithm for computing Maximum Independent Set in graphs with bounded degree 3. This improves all the previous running time bounds known for the problem.

View PDFchevron_right
Confining sets and avoiding bottleneck cases: A simple maximum independent set algorithm in degree-3 graphs
Mingyu Xiao

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.

View PDFchevron_right
On unique independent sets in graphs
Jerzy Topp

Discrete Mathematics, 1994

A perfect independent set I of a graph G is defined to be an independent set with the property that any vertex not in I has at least two neighbors in I. For a nonnegative integer k, a subset I of the vertex set V (G) of a graph G is said to be k-independent, if I is independent and every independent subset where G[I, V (G) -I] is the bipartite graph obtained from G by deleting all edges which are not incident with vertices of I. It is easy to see that a set I is 0-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of G. In this paper we mainly investigate connections between perfect independent sets and k-independent as well as super k-independent sets for k = 0 and k = 1.

View PDFchevron_right
Independent Sets in Graphs without Subtrees with Many Leaves
Vladimir Alekseev

—A subtree of a graph is called inscribed if no three vertices of the subtree generate a triangle in the graph. We prove that, for fixed k, the independent set problem is solvable in polynomial time for each of the following classes of graphs: (1) graphs without subtrees with k leaves, (2) subcubic graphs without inscribed subtrees with k leaves, and (3) graphs with degree not exceeding k and lacking induced subtrees with four leaves.

View PDFchevron_right
Independence and Efficient Domination on P 6 -free Graphs
Emine Handan Tüzün

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2015

In the Maximum Weight Independent Set problem, the input is a graph G, every vertex has a non-negative integer weight, and the task is to find a set S of pairwise non-adjacent vertices, maximizing the total weight of the vertices in S. We give an n O(log 2 n) time algorithm for this problem on graphs excluding the path P 6 on 6 vertices as an induced subgraph. Currently, there is no constant k known for which Maximum Weight Independent Set on P k-free graphs becomes NP-complete, and our result implies that if such a k exists, then k > 6 unless all problems in NP can be decided in (quasi)polynomial time. Using the combinatorial tools that we develop for the above algorithm, we also give a polynomial-time algorithm for Maximum Weight Efficient Dominating Set on P 6-free graphs. In this problem, the input is a graph G, every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood, or to determine that no such set exists. Prior to our work, the class of P 6-free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of Maximum Weight Efficient Dominating Set was unknown.

View PDFchevron_right
Fast Algorithms for max independent set
Vangelis Paschos

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.

View PDFchevron_right
A Worst-Case Time Upper Bound for Counting the Number of Independent Sets
Guillermo De Ita

Lecture Notes in Computer Science, 2007

The problem of counting the number of independent sets of a graph G (denoted as NI(G)) is a classic #P-complete problem for graphs of degree 3 or higher. Exploiting the strong relation between NI(G) and Fibonacci numbers, we show that if the depth-first graph of G does not contain a pair of basic cycles with common edges, then NI(G) can be computed in linear time (in the size of the graph). This determines new classes of instances of graphs without restrictions on their degrees and where the number of independent sets is computed in polynomial time.

View PDFchevron_right
Further Improvement on Maximum Independent Set in Degree-4 Graphs
Mingyu Xiao

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.

View PDFchevron_right
Exact Algorithms for Maximum Independent Set
Mingyu Xiao

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.

View PDFchevron_right
Recognition and Optimization Algorithms for P5-Free Graphs
Victor Lepin

Symmetry, 2020

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique co...

View PDFchevron_right

References (49)

  1. V. Alekseev, The effect of local constraints on the complexity of determination of the graph independence number, Combinatorial-algebraic methods in applied mathematics, (1982), pp. 3-13. (in Russian).
  2. V. E. Alekseev, Polynomial algorithm for finding the largest independent sets in graphs without forks, Discrete Applied Mathematics, 135 (2004), pp. 3-16.
  3. V. E. Alekseev and V. V. Lozin, Augmenting graphs for independent sets, Discrete Applied Mathematics, 145 (2004), pp. 3 -10. Graph Optimization IV.
  4. S. Arora and B. Barak, Computational Complexity -A Modern Approach, Cambridge University Press, 2009.
  5. B. S. Baker, Approximation algorithms for np-complete problems on planar graphs, J. ACM, 41 (1994), pp. 153-180.
  6. E. Balas and C. S. Yu, On graphs with polynomially solvable maximal-weight clique problem, Networks, 19 (1989), pp. 247-253.
  7. J. R. S. Blair, P. Heggernes, and J. A. Telle, A practical algorithm for making filled graphs minimal, Theor. Comput. Sci., 250 (2001), pp. 125-141.
  8. J. R. S. Blair and B. W. Peyton, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computations, Springer, 1993, pp. 1-30. IMA Volumes in Mathematics and its Applications, Vol. 56.
  9. R. Boliac and V. V. Lozin, An augmenting graph approach to the stable set problem in P 5 -free graphs, Discrete Applied Mathematics, 131 (2003), pp. 567 -575.
  10. V. Bouchitté and I. Todinca, Treewidth and minimum fill-in: Grouping the minimal separators, SIAM J. Comput., 31 (2001), pp. 212-232.
  11. Listing all potential maximal cliques of a graph, Theor. Comput. Sci., 276 (2002), pp. 17-32.
  12. A. Brandstädt and R. Mosca, On the structure and stability number of P 5 -and co- chair-free graphs, Discrete Applied Mathematics, 132 (2003), pp. 47 -65. Stability in Graphs and Related Topics.
  13. A. Brandstädt, J. P. Spinrad, et al., Graph classes: a survey, no. 3, Siam, 1999.
  14. H. Broersma, T. Kloks, D. Kratsch, and H. Müller, Independent sets in asteroidal triple-free graphs, SIAM J. Discrete Math., 12 (1999), pp. 276-287.
  15. P. Buneman, A characterization of rigid circuit graphs, Discrete Math., 9 (1974), pp. 205- 212.
  16. B. N. Clark, C. J. Colbourn, and D. S. Johnson, Unit disk graphs, Discrete Math., 86 (1990), pp. 165 -177.
  17. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algo- rithms (3. ed.), MIT Press, 2009.
  18. D. Corneil, H. Lerchs, and L. Burlingham, Complement reducible graphs, Discrete Applied Mathematics, 3 (1981), pp. 163 -174.
  19. B. Courcelle, J. A. Makowsky, and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Syst., 33 (2000), pp. 125-150.
  20. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 25 (1961), pp. 71- 76.
  21. R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999.
  22. H. Flier, M. Mihalák, A. Schöbel, P. Widmayer, and A. Zych, Vertex disjoint paths for dispatching in railways, in Proceedings of ATMOS, 2010, pp. 61-73.
  23. F. V. Fomin and Y. Villanger, Finding induced subgraphs via minimal triangulations, in STACS, vol. 5 of LIPIcs, Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2010, pp. 383-394.
  24. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Series of Books in the Mathematical Sciences, W. H. Freeman and Co., 1979.
  25. F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory Ser. B, 16 (1974), pp. 47-56.
  26. M. U. Gerber and V. V. Lozin, On the stable set problem in special P 5 -free graphs, Discrete Applied Mathematics, 125 (2003), pp. 215-224.
  27. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
  28. M. Grötschel, L. Lovász, and A. Schrijver, The ellipsoid method and its conse- quences in combinatorial optimization., Combinatorica, 1 (1981), pp. 169-197.
  29. P. Heggernes, Minimal triangulations of graphs: A survey, Discrete Mathematics, 306 (2006), pp. 297-317.
  30. C.-W. Ho and R. Lee, Counting clique trees and computing perfect elimination schemes in parallel, Information Processing Letters, 31 (1989), pp. 61 -68.
  31. C. T. Hoàng, M. Kamiński, V. Lozin, J. Sawada, and X. Shu, Deciding k-colorability of P 5 -free graphs in polynomial time, Algorithmica, 57 (2010), pp. 74-81.
  32. R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strongly exponential complexity?, J. Comput. Syst. Sci., 63 (2001), pp. 512-530.
  33. R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, 1972, pp. 85-103.
  34. T. Kloks, D. Kratsch, and J. Spinrad, On treewidth and minimum fill-in of asteroidal triple-free graphs, Theor. Comput. Sci., 175 (1997), pp. 309-335.
  35. R. J. Lipton and R. E. Tarjan, Applications of a planar separator theorem, SIAM J. Comput., 9 (1980), pp. 615-627.
  36. D. Lokshtanov, D. Marx, and S. Saurabh, Lower bounds based on the exponential time hypothesis, Bulletin of the EATCS, 105 (2011), pp. 41-72.
  37. V. V. Lozin and M. Milanic, A polynomial algorithm to find an independent set of maximum weight in a fork-free graph, in SODA, 2006, pp. 26-30.
  38. V. V. Lozin and R. Mosca, Independent sets in extensions of 2k 2 -free graphs, Discrete Applied Mathematics, 146 (2005), pp. 74 -80.
  39. F. Maffray and G. Morel, On 3-colorable P 5 -free graphs, SIAM Journal on Discrete Mathematics, 26 (2012), pp. 1682-1708.
  40. G. J. Minty, On maximal independent sets of vertices in claw-free graphs, Journal of Combinatorial Theory, Series B, 28 (1980), pp. 284 -304.
  41. R. Mosca, Some observations on maximum weight stable sets in certain P 5 -free graphs, European Journal of Operational Research, 184 (2008), pp. 849 -859.
  42. R. Mosca, Some results on stable sets for k-colorable P 6 -free graphs and generalizations., Discrete Mathematics & Theoretical Computer Science, 14 (2012), pp. 37-56.
  43. A. Parra and P. Scheffler, Characterizations and algorithmic applications of chordal graph embeddings, Discrete Applied Mathematics, 79 (1997), pp. 171-188.
  44. S. Poljak, A note on stable sets and colorings of graphs, Commentationes Mathematicae Universitatis Carolinae, 15 (1974), pp. 307 -309.
  45. B. Randerath and I. Schiermeyer, On maximum independent sets in P 5 -free graphs, Discrete Applied Mathematics, 158 (2010), pp. 1041-1044.
  46. D. J. Rose, R. E. Tarjan, and G. S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput., 5 (1976), pp. 266-283.
  47. N. Sbihi, Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile, Discrete Mathematics, 29 (1980), pp. 53 -76.
  48. S. M. Takeshi Fukuda, Yasuhiko Morimoto and T. Tokuyama, Data mining with optimized two-dimensional association rules, ACM Transactions on Database Systems, 26 (2001), pp. 179-213.
  49. J. R. Walter, Representations of rigid cycle graphs, PhD thesis, Wayne State University, 1972.

Related papers

On maximum independent sets in P5-free graphs
Ingo Schiermeyer

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.

View PDFchevron_right
The complexity of the exact weighted independent set problem
Jérôme Monnot

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.

View PDFchevron_right
The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes
Jérôme Monnot

Electronic Notes in Discrete Mathematics, 2009

The exact weighted independent set (EWIS) problem consists in determining whether a given vertex-weighted graph contains an independent set of given weight. This problem is a generalization of two well-known problems, the NP-complete subset sum problem and the strongly NP-hard maximum weight independent set (MWIS) problem. Since the MWIS problem is polynomially solvable for some special graph classes, it is interesting to determine the complexity of this more general EWIS problem for such graph classes.

View PDFchevron_right
On the complexity of the independent set problem in triangle graphs
Jacek Blazewicz

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.

View PDFchevron_right
A Polynomial time Algorithm for the Maximum Weight Independent Set Problem on Outerstring Graphs
Martin Vatshelle

In the classic Maximum Weight Independent Set problem we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P 6-free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5-free graphs of Lokshtanov et al. [14], and the quasipolynomial-time algorithm on P 6-free graphs of Lokshtanov et al. [13]. The main technical contribution leading to our main result is enumeration of a polynomial-size family F of vertex subsets with the following property: for every maximal independent set I in the graph, F contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I.

View PDFchevron_right
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved