Combinatorial Optimization — Eureka, You Shrink!: Papers by Michael Jünger, Gerhard Reinelt, Giovanni Rinaldi

By Michael Jünger, Gerhard Reinelt, Giovanni Rinaldi

This e-book is devoted to Jack Edmonds in appreciation of his flooring breaking paintings that laid the principles for a wide number of next effects completed in combinatorial optimization.

The major half contains thirteen revised complete papers on present subject matters in combinatorial optimization, provided at Aussois 2001, the 5th Aussois Workshop on Combinatorial Optimization, March 5-9, 2001, and devoted to Jack Edmonds.

Additional highlights during this ebook are an account of an Aussois 2001 targeted consultation devoted to Jack Edmonds together with a speech given via William R. Pulleyblank in addition to newly typeset models of 3 up-to-now infrequently obtainable classical papers:

- Submodular features, Matroids, and sure Polyhedra
through Jack Edmonds

- Matching: A Well-Solved classification of Integer Linear Programs
through Jack Edmonds and Ellis L. Johnson

- Theoretical advancements in Algorithmic potency for community circulate Problems
through Jack Edmonds and Richard M. Karp.

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Extra resources for Combinatorial Optimization — Eureka, You Shrink!: Papers Dedicated to Jack Edmonds 5th International Workshop Aussois, France, March 5–9, 2001 Revised Papers

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4 Connected Matchings in Graphs with No Circuits on Four Nodes C4 is the circuit with four nodes. We consider graphs which do not have C4 as a subgraph. The Peterson graph is such a graph with a connected matching of size 5. The following theorem shows this is best possible. Theorem 1. The maximum size of a connected matching in a graph with no C4 as a subgraph is 5. Proof. Let G = (V, E) be a graph with no C4 as a subgraph. Suppose G contains a connected matching M of size 6, say M = {si ti : 1 ≤ i ≤ 6}.

If ti tj ∈ / E, then we have structure (A), and without loss of generality, i = 2 and j = 4. Then s2 , s4 , t4 , s5 is a C4 . Suppose there are two type 2 edges, say s5 ti and s5 tj . Then ti , s5 , tj , tk is a C4 , where k is the one of 2, 3, 4 different from i and j unless we are have structure (A) and {i, j} = {2, 3} or {i, j} = {3, 4}. So, without loss of generality, say s5 t2 and s5 t3 are type 2 edges, s5 t4 ∈ / E, and we have structure (A). If the edge joining s4 t4 and s5 t5 is a type 1 edge, s5 s4 , then s5 , s4 , t4 , t3 is a C4 .

If ti tj ∈ E, then s5 , si , ti , tj is a C4 . If ti tj ∈ / E, then we have structure (A), and without loss of generality, i = 2 and j = 4. Then s2 , s4 , t4 , s5 is a C4 . Suppose there are two type 2 edges, say s5 ti and s5 tj . Then ti , s5 , tj , tk is a C4 , where k is the one of 2, 3, 4 different from i and j unless we are have structure (A) and {i, j} = {2, 3} or {i, j} = {3, 4}. So, without loss of generality, say s5 t2 and s5 t3 are type 2 edges, s5 t4 ∈ / E, and we have structure (A). If the edge joining s4 t4 and s5 t5 is a type 1 edge, s5 s4 , then s5 , s4 , t4 , t3 is a C4 .

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