Expressing combinatorial optimization problems by linear programs

Many combinatorial optimization problems call for the optimization of a linear function over a certain polytope. Typically, these polytopes have an exponential number of facets. We explore the problem of finding small linear programming formulations when one may use any new variables and constraints. We show that expressing the matching and the Traveling Salesman Problem by a symmetric linear program requires exponential size. We relate the minimum size needed by a LP to express a polytope to a combinatorial parameter, point out some connections with communication complexity theory, and examine the vertex packing polytope for some classes of graphs.

[1]  M. Padberg,et al.  Addendum: Optimization of a 532-city symmetric traveling salesman problem by branch and cut , 1990 .

[2]  Ronald L. Rardin,et al.  Polyhedral Characterization of Discrete Dynamic Programming , 1990, Oper. Res..

[3]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[4]  Christos H. Papadimitriou,et al.  The complexity of facets resolved , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[5]  László Lovász,et al.  Vertex Packing Algorithms , 1985, ICALP.

[6]  A. Razborov.  Lower bounds on monotone complexity of the logical permanent , 1985 .

[7]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[8]  V. Chvátal,et al.  Topics on perfect graphs , 1984 .

[9]  Alfred V. Aho,et al.  On notions of information transfer in VLSI circuits , 1983, STOC.

[10]  Egon Balas,et al.  The perfectly matchable subgraph polytope of a bipartite graph , 1983, Networks.

[11]  Mihalis Yannakakis,et al.  The complexity of facets (and some facets of complexity) , 1982, STOC '82.

[12]  Kurt Mehlhorn,et al.  Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract) , 1982, STOC '82.

[13]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[14]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[15]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[16]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[17]  Richard J. Lipton,et al.  Linear Programming is Log-Space Hard for P , 1979, Inf. Process. Lett..

[18]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[19]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[20]  Robert G. Jeroslow.  On defining sets of vertices of the hypercube by linear inequalities , 1975, Discret. Math..

[21]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[22]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[23]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .

[24]  J. P. Secrétan,et al.  Der Saccus endolymphaticus bei Entzündungsprozessen , 1944 .