Abstract
We study the maximum node-weighted induced bipartite subgraph problem in planar graphs with maximum degree three. We show that this is polynomially solvable. It was shown in [6] that it is NP-complete if the maximum degree is four. We extend these ideas to the problem of balancing signed graphs.
We also consider maximum weighted induced acyclic subgraphs of planar directed graphs. If the maximum degree is three, it is easily shown that this is polynomially solvable. We show that for planar graphs with maximum degree four the same problem is NP-complete.
This work has been supported by project PICS05891, CNRS-IBM.
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References
Barahona, F.: Planar multicommodity flows, max cut and the Chinese Postman Problem. In: Polyhedral Combinatorics. DIMACS Series on Discrete Mathematics and Theoretical Computer Science No. 1, pp. 189–202. DIMACS, NJ (1990)
Barahona, F., Mahjoub, A.: Facets of the balanced (acyclic) induced subgraph polytope. Mathematical Programming 45, 21–33 (1989)
Barahona, F., Mahjoub, A.: Compositions of graphs and polyhedra I: Balanced induced subgraphs and acyclic subgraphs. SIAM Journal on Discrete Mathematics 7, 344–358 (1994)
Barahona, F., Maynard, R., Rammal, R., Uhry, J.: Morphology of ground states of two-dimensional frustration model. Journal of Physics A: Mathematical and General 15, 673 (1982)
Chen, R.-W., Kajitani, Y., Chan, S.-P.: A graph-theoretic via minimization algorithm for two-layer printed circuit boards. IEEE Transactions on Circuits and Systems 30, 284–299 (1983)
Choi, H.-A., Nakajima, K., Rim, C.S.: Graph bipartization and via minimization. SIAM Journal on Discrete Mathematics 2, 38–47 (1989)
Edmonds, J., Johnson, E.L.: Matching, euler tours and the chinese postman. Mathematical Programming 5, 88–124 (1973)
Even, G., Naor, J.S., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20, 151–174 (1998)
Festa, P., Pardalos, P.M., Resende, M.G.: Feedback set problems. In: Handbook of Combinatorial Optimization, pp. 209–258. Springer (1999)
Fouilhoux, P., Mahjoub, A.: Solving VLSI design and DNA sequencing problems using bipartization of graphs. Computational Optimization and Applications 51, 749–781 (2012)
Funke, M., Reinelt, G.: A polyhedral approach to the feedback vertex set problem. In: Integer Programming and Combinatorial Optimization, pp. 445–459. Springer (1996)
Gabow, H.N.: An efficient implementation of edmonds’ algorithm for maximum matching on graphs. Journal of the ACM (JACM) 23, 221–234 (1976)
Gabow, H.N.: Centroids, representations, and submodular flows. Journal of Algorithms 18, 586–628 (1995)
Gardarin, G., Spaccapietra, S.: Integrity of data bases: A general lockout algorithm with deadlock avoidance. In: IFIP Working Conference on Modelling in Data Base Management Systems, pp. 395–412 (1976)
Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness (1979)
Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi) cut theorems and their applications. SIAM Journal on Computing 25, 235–251 (1996)
Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 18, 37–59 (1998)
Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4, 221–225 (1975)
Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J., Bohlinger, J. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer US (1972)
Leiserson, C.E., Saxe, J.B.: Retiming synchronous circuitry. Algorithmica 6, 5–35 (1991)
Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11, 329–343 (1982)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36, 177–189 (1979)
Lucchesi, C., Younger, D.: A minimax theorem for directed graphs. J. London Math. Soc. 17(2), 369–374 (1978)
Lucchesi, C.L.: A minimax equality for directed graphs, PhD thesis, Thesis (Ph. D.)–University of Waterloo (1976)
Pinter, R.Y.: Optimal layer assignment for interconnect. Journal of VLSI and Computer Systems 1, 123–137 (1984)
Seymour, P.D.: On odd cuts and plane multicommodity flows. Proceedings of the London Mathematical Society 3, 178–192 (1981)
Silberschatz, A., Peterson, J.L., Galvin, P.B.: Operating system concepts. Addison-Wesley Longman Publishing Co., Inc. (1991)
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Baïou, M., Barahona, F. (2014). Maximum Weighted Induced Bipartite Subgraphs and Acyclic Subgraphs of Planar Cubic Graphs. In: Lee, J., Vygen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2014. Lecture Notes in Computer Science, vol 8494. Springer, Cham. https://doi.org/10.1007/978-3-319-07557-0_8
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