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Independence Systems, Matroids, the Greedy Algorithm, and related Polyhedra

Independence Systems, Matroids, the Greedy Algorithm, and related Polyhedra

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Independence Systems, Matroids, the Greedy Algorithm, and related Polyhedra

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  1. Independence Systems, Matroids, the Greedy Algorithm, and related Polyhedra Martin Grötschel Summary of Chapter 4 of the classPolyhedral Combinatorics (ADM III)May 25, 2010 &June 1, 2010

  2. Matroids and Independence Systems Let E be a finite set, I a subset of the power set of E. The pair (E,I ) is called independence system on E if the following axioms are satisfied: (I.1) The empty set is in I. (I.2) If J is in I and I is a subset of J then I belongs to I. Let (E,I ) satisfy in addition: (I.3) If I and J are in I and if J is larger than I then there is an element j in J, j not in I, such that the union of I and j is in I. Then M=(E,I ) is called a matroid.

  3. Notation Let (E,I ) be an independence system. Every set in I is called independent. Every subset of E not in I is called dependent. For every subset F of E, a basis of F is a subset of F that is independent and maximal with respect to this property. The rank r(F) of a subset F of E is the cardinality of a largest basis of F. Important property, submodularity: The lower rank of F is the cardinality of a smallest basis of F.

  4. The Largest Independent Set Problem Problem: Let (E,I ) be an independence system with weights on the elements of E. Find an independent set of largest weight. We may assume w.l.o.g. that all weights are nonnegative (or even positive), since deleting an element with nonpositive weight from an optimum solution, will not decrease the value of the solution.

  5. The Greedy Algorithm Let (E,I ) be an independence system with weights c(e) on the elements of E. Find an independent set of largest weight. The Greedy Algorithm: 1. Sort the elements of E such that 2. Let 3. FOR i=1 TO n DO: 4. OUTPUT A key idea is to interprete the greedy solution as the solution of a linear program.

  6. The greedy algorithm works for matroids • Proof using axiom (I.3) on the blackboard. Martin Grötschel

  7. Polytopes and LPs Let M=(E,I ) be an independence system with weights c(e) on the elements of E.

  8. The Dual Greedy Algorithm Let (E,I ) be an independence system with weights c(e) for all e. After sorting the elements of E so that set Then is a feasible solution of the dual LP by construction. (integral if the weights are integral))

  9. Observation Let (E,I ) be an independence system with weights c(e) for all e. After sorting the elements of E so that We can express every greedy and optimum solution as follows:

  10. Rank Quotient Let (E,I ) be an independence system with weights c(e) for all e. The number q is between 0 and 1 and is called rank quotient of (E,I ). Observation: q = 1 iff (E,I ) is a matroid.

  11. The General Greedy Quality Guarantee a quality guarantee

  12. Consequences Let M=(E,I ) be an independence system with weights c(e) on the elements of E.

  13. Consequences Theorem. For every independence system (E,I with weights c(e) for all elements e of E, max cTx, xϵP(I) ≥ c(Iopt) ≥ c(Igreedy) ≥ q max cTx, xϵP(I) ≥ q c(Iopt), in other words, the greedy solution value is bounded from below by q times the maximum value of the LP relaxation and not only by q times the optimum value of the weighted independent set problem. Martin Grötschel

  14. More Proofs • Another proof of the completeness of the system of nonnegativity constraints and rank inequalities will be given in the class on the blackboard, see further slides. Martin Grötschel

  15. Completeness Proof of the Matroid Polytope Martin Grötschel

  16. Completeness Proof of the Matroid Polytope (continued) The proof above is from (GLS, pages 213-214), see http://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf Martin Grötschel

  17. Facets of the matroid polytope • It will also be shown on the blackboard that a rank inequality x(F) ≤ r(F)defines a facet of the matroid polytope if and only if the set F is closed and inseparable, seeMartin Grötschel, Facetten von Matroid-Polytopen, Operations Research Verfahren XXV, 1977, 306-313, downloadable fromhttp://www.zib.de/groetschel/pubnew/paper/groetschel1977d.pdf • A subset F of E is closed if r(F U {e})>r(F) for all e in E\F. • A subset F of E is separable if there exist two nonempty disjoint subsets F1 and F2 of F whose union is F and such that r(F)= r(F1)+ r(F2). Martin Grötschel

  18. The Forest Polytope Martin Grötschel

  19. A Partition Matroid Polytope Martin Grötschel

  20. The 1-Tree Polytope 1-trees come up as relaxations of the symmetric travelling salesman problem. Given a complete graph Kn=(V,E), V={1,2,…,n}. A 1-tree is the union of the edge set of a spanning tree of the complete graph on the node set {2,3,…,n} and two edges with endnode 1. Every 1-tree has n edges and contains exactly one cycle. This means that every travelling salesman tour is a 1-tree. The set of 1-trees is the set of bases of a matroid on E. A complete description of the convex hull of the incidence vectors of all 1-trees in Kn is given by: 0≤ xe ≤ 1 for all e in E x(E(W)) ≤ |W| - 1 for all node sets W in V not containing 1 x(δ(1)) = 2 x(E) = n Martin Grötschel

  21. The Branching and the Arborescence Polytope Martin Grötschel

  22. The Branching and the Arborescence Polytope Martin Grötschel

  23. The Matroid Intersection Polytope Martin Grötschel

  24. The Matroid Intersection Polytope Martin Grötschel

  25. The Matroid Intersection Polytope Martin Grötschel

  26. The Matroid Intersection Polytope The proof above is from (GLS, pages 214-216), see http://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf Martin Grötschel

  27. Submodular Functions and Polymatroids • The whole polyhedral and algorithmic theory developed so far can be generalized to submodular functions and polymatroids. • This is worked out in detail in Chapter 10 of (GLS), seehttp://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf Martin Grötschel

  28. Claude Berge (perfect graphs) andJack Edmonds (matching and matroids) Martin Grötschel

  29. Independence Systems, Matroids, the Greedy Algorithm and related Polyhedra Martin Grötschel Summary of Chapter 4 of the classPolyhedral Combinatorics (ADM III)May 18, 2010 The End