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By Martin Grötschel

The Sharpest lower is written in honor of Manfred Padberg, who has made basic contributions to either the theoretical and computational aspects of integer programming and combinatorial optimization. This amazing assortment provides fresh ends up in those components which are heavily attached to Padberg's learn. His deep dedication to the geometrical method of combinatorial optimization should be felt all through this quantity; his look for more and more higher and computationally effective slicing planes gave upward thrust to its name.

The peer-reviewed papers contained listed here are in response to invited lectures given at a workshop held in October 2001 to have fun Padberg's sixtieth birthday. Grouped by means of subject (packing, reliable units, and excellent graphs; polyhedral combinatorics; normal polytopes; semidefinite programming; computation), some of the papers got down to remedy demanding situations set forth in Padberg’s paintings. The publication additionally exhibits how Padberg's principles on slicing planes have encouraged glossy advertisement optimization software program. moreover, the amount encompasses a brief curriculum vitae, a private account of Padberg’s paintings through Laurence Wolsey, and an appendix with reflections from Egon Balas, Claude Berge, and Harold Kuhn.

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21] M. Grotschel, A. Martin, and R. Weismantel. Packing Steiner trees: A cutting plane algorithm and computational results. Mathematical Programming, 72:125–145, 1996. [22] M. Grotschel, A. Martin, and R. Weismantel. Packing Steiner trees: Further facets. European Journal on Combinatorics, 17:39-52, 1996. [23] M. Grotschel, A. Martin, and R. Weismantel. Packing Steiner trees: Polyhedral investigations. Mathematical Programming, 72:101–123, 1996. [24] M. Grotschel, A. Martin, and R. Weismantel.

3 each node of G belongs to three triangles. , T4 = {«•. 54, r4}). Consider in G — f u, w} a path P from s to t, with distinct nodes (but not necessarily a chordless path); let us prove first the following claim. Claim 2. If the subgraph ofG induced on P is bipartite, P is odd if and only if both triangles T), T\ are included in P U {u, w}. Proof. , 74) if the subgraph of G induced on P is bipartite. Assume now that F is even and that at least one of the two triangles Tj, F4 is included in F U {D, u>}; we can suppose that this triangle is T$ and that 53 appears before /j in the description of F from 5 to t.

4. Let G be a graph with clique number equal to three and with at least four edges; assume also that G contains a unique clique of size three {i>i, i>2, v$}. If all three graphs G — {i'i, ^2). , v$}, G — {1^2,^3} are connected, then G contains an odd hole. Proof. Let Q be a component of G — {v\, i>2, v$}. We may assume that Q is bipartite (else Q contains an odd hole and we are done); thus the set of nodes of Q splits into stable sets Si and S2. Since each G — {v,•, Vj} is connected, each of the three nodes vk must have a neighbor in Si US?.

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