Decomposition and optimization over cycles in binary matroids

M Grötschel, K Truemper
1989 Journal of combinatorial theory. Series B (Print)  
DECOMPOSITION AND OPTIMIZATION 307 where c E [WE is a given objective function and c(C) stands for the sum c escce. We call this problem the maximum weight cycle problem, or just the cycle problem of binary matroids. Clearly, (1.2) is equivalent to the linear program max{cTx/xEP(M)}, (1.3) since every optimal solution of (1.2) yields an optimal vertex solution of (1.3) and vice versa. Problem (1.2) includes, among other interesting combinatorial optimization problems, the max-cut problem in
more » ... hs (if M is the cographic matroid of a graph G, then the cycles of A4 are the cuts of G) and the Eulerian subgraph problem (if A4 is the graphic matroid of a graph G, then the cycles of M are the (not necessarily connected) Eulerian subgraphs of G). Since the max-cut problem is &"Y-hard, the maximum cycle problem (1.2) is NY-hard as well. We use matroidal and polyhedral k-sums, k = 2, 3, to obtain a complete description of P(M), in case A4 can be k-separated into M-, and Mz and complete descriptions of P(M,) and P(M,) are known. We also prove that particular matroidal k-sums correspond to polyhedral k-sums of a certain LP-relaxation of P(M). These composition results are then combined with the characterization of the Euler subgraph polytope by Edmonds and Johnson [7] and with two decomposition theorems by Seymour [15] and Wagner [21] to a direct proof that the aforementioned LP-relaxation is
doi:10.1016/0095-8956(89)90052-x fatcat:5dk7tccsengrvppgtiipir5d7q