The Schrijver system of the flow cone in series-parallel graphs

Abstract We represent a flow of a graph G = ( V , E ) as a couple ( C , e ) with C a circuit of G and e an edge of C , and its incidence vector is the 0 ∕ ± 1 vector χ C ∖ e − χ e . The flow cone of G is the cone generated by the flows of G and the unit vectors. When G has no K 5 -minor, this cone can be described by the system x ( M ) ≥ 0 for all multicuts M of G . We prove that this system is box-totally dual integral if and only if G is series–parallel. Then, we refine this result to provide the Schrijver system describing the flow cone in series–parallel graphs. This answers a question raised by Chervet et al., (2018).

[1]  Lei Tan,et al.  When Is the Matching Polytope Box-Totally Dual Integral? , 2017, Math. Oper. Res..

[2]  William J. Cook,et al.  On box totally dual integral polyhedra , 1986, Math. Program..

[3]  Alexander Schrijver,et al.  On total dual integrality , 1981 .

[4]  Denis Cornaz,et al.  Max-multiflow/min-multicut for G+H series-parallel , 2011, Discret. Math..

[5]  R. Duffin.  Topology of series-parallel networks , 1965 .

[6]  Xujin Chen,et al.  The box-TDI system associated with 2-edge connected spanning subgraphs , 2009, Discret. Appl. Math..

[7]  W. Cunningham,et al.  A primal algorithm for optimum matching , 1978 .

[8]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[9]  Patrick Chervet,et al.  Box-total dual integrality, box-integrality, and equimodular matrices , 2020, Math. Program..

[10]  William Cook.  A Minimal Totally Dual Integral Defining System for the b-Matching Polyhedron , 1983 .

[11]  Xujin Chen,et al.  A Characterization of Box-Mengerian Matroid Ports , 2008, Math. Oper. Res..

[12]  András Sebő,et al.  The Schrijver system of odd join polyhedra , 1988 .

[13]  William R. Pulleyblank,et al.  Linear Systems for Constrained Matching Problems , 1987, Math. Oper. Res..

[14]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[15]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[16]  Qiulan Zhao,et al.  On box-perfect graphs , 2016, J. Comb. Theory, Ser. B.

[17]  Patrick Chervet,et al.  Principally Box-integer Polyhedra and Equimodular Matrices , 2018, ArXiv.

[18]  William R. Pulleyblank.  Total dual integrality and b-matchings , 1981, Oper. Res. Lett..

[19]  Roland Grappe,et al.  Trader multiflow and box-TDI systems in series-parallel graphs , 2019, Discret. Optim..