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The Primal-Dual Method: Steiner Forest TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AA A A A AA A A

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Network Design Input: undirected graph G=(V,E) cost function: c:E ! R + connectivity requirements Output: minimum cost subgraph satisfying connectivity requirements Example: minimum spanning tree: minimum cost subgraph connecting all vertices

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Steiner Tree Input: set of terminals T = { t 1, …, t k } µ V Output: minimum cost subgraph connecting T NP-complete Algorithm: define complete graph H on T c(t i, t j ) = shortest path in G from t i to t j find a minimum spanning tree F in H replace each edge in F by the corresponding path in G (delete cycles)

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Steiner Tree: Analysis F * - optimal Steiner tree in G duplicate each edge in F * - all degrees are even open up F* into a (Euler) cycle Z c(Z) =2c(F*) Z can be transformed into a spanning tree in H: for any t,t 2 T: c Z (t,t ) ¸ c H (t,t ) Approximation factor = 2

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Constrained Steiner Forest Problem Requirement function f: S ! {0,1} 8 S ½ V f(S) =1 if there is a set T i split between S and V-S f(S) =0 otherwise Note: f(S)=f(V-S), f is a function on cuts Input: undirected graph G=(V,E) cost function: c:E ! R + disjoint sets T 1, …, T k ½ V Output: minimum cost subgraph H such that each set T i belongs to a connected component of H S

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Examples of Requirement Functions Steiner tree: for any cut S separating terminals, f(S)=1 shortest path between s and t: for any cut S separating s and t, f(S)=1 minimum weight perfect matching: f(S) =1 iff |S| is odd (At least one vertex is matched outside S)

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δ(S) S Linear Programming Formulation Covering LP Packing LP

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Integrality Gap Example Minimum spanning tree: all cuts are covered Graph: cycle on n vertices, unit cost edges Optimal integral solution: cost is n-1 Optimal fractional solution: for each edge e, x(e)=1/2 all cuts are covered cost is n/2 Gap is 2-1/n

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Primal-Dual Algorithm y Ã 0 A Ã ; (x Ã 0) While A is not a feasible solution C - set of connected components S 2 (V,A) such that f(S)=1 increase y S for all S 2 C until dual constraint for new e 2 E becomes tight A Ã A [ {e} Remove unnecessary edges from A

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Primal-Dual Algorithm y Ã 0 A Ã ; (x Ã 0) While A is not a feasible solution C - set of connected components S 2 (V,A) such that f(S)=1 increase y S for all S 2 C until dual constraint for new e 2 E becomes tight A Ã A [ {e} Remove unnecessary edges from A A – final solution

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Demo of Primal-Dual Algorithm

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Analysis (1) Lemma: The primal solution generated by the algorithm satisfies the requirement constraints Proof: The `while loop continues till we get a feasible solution Lemma: The dual solution generated by the algorithm is feasible Proof: Whenever an edge becomes tight, no more cuts are packed into it

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Analysis (2) Why is it important that we delete redundant edges at the end? Otherwise, we cannot charge tight primal edges to a single dual variable (the ball) – too many such edges! Goal: charge the increase in primal cost in each iteration to the increase in dual cost

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Analysis (3):

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Analysis (4):

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Summary: key ideas synchronizing dual update: all dual variables (of active sets) increase together complementary slackness: dual variable > 0 ! tight primal constraint primal variable > 0 ! tight dual constraint the primal-dual algorithm relaxes comp. slackness: if x e >0 (i.e., x e =1 ), then If y S >0, not necessarily ratio of primal and dual increase · 2 on average, not per constraint

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