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Iterative Rounding and Iterative Relaxation Some of the slides prepared by Lap Chi Lau Steiner Network undirected or directed • A graph G = (V,E); • A connectivity requirement r(u,v) for each pair of vertices u,v. Steiner Network • A subgraph H of G which has r(u,v) disjoint paths between each pair u,v. edge or vertex Examples of Steiner Network • Spanning tree: r(u,v) = 1 for all pairs of vertices. • Steiner tree: r(u,v) = 1 for all pair of required vertices. • Steiner forest: r(si,ti) = 1 for all source sink pairs. • k-edge-connected subgraph: r(u,v) = k for all pairs of vertices. Survivable Network Design Survivable network design: find a “good” Steiner network e.g. minimum Steiner tree, k-edge-connected subgraph Minimum cost Steiner network: Given a cost c(e) on each edge, find a Steiner network with minimum total cost. NP-complete… Linear Programming Relaxation at least r(u,v) edges crossing S S u v f(S) := max{ r(u,v) | S separates u and v}. • number of constraints is exponential. • can be handled by Ellipsoid algorithm • need a separation oracle Approximating Survivable network Design What does not seem to generalize … ? • primal-dual algorithm for constrained Steiner forest • only logarithmic approx. is known via this approach • one shot rounding as in • vertex cover - rounding up fractions ¸ ½ • set cover – randomized rounding • combinatorial algorithms • e.g., Steiner tree (reduction to spanning tree in the metric completion graph) New ideas are needed … Basic Solutions (1) A basic solution is the unique solution of m linearly independent “tight” constraints, where m is the number of variables in the LP. A basic solution cannot be represented as a convex sum of feasible solutions Basic Solutions (2) An edge of 1, pick it. An edge of 0, delete it. A basic solution is the unique solution of m linearly independent “tight” constraints, where m is the number of variables in the LP. Tight inequalities all come from the connectivity requirements. Petersen Graph: Non-Basic Feasible Solution ce 1, ruv 1. 15 edges with 0 < x(e) < 1 10 tight constraints (at use every edge to the the vertices) extent of 1/3. Basic Feasible Solution black edge – ¼ red edge – ½ Tight sets LP Solutions All 1/3 is a feasible solution. But not a basic feasible solution This is a basic feasible solution. Thick edges have value 1/2; Thin edges have value 1/4. Theorem [Jain 98] Every basic feasible solution has an edge of value at least 1/2 Corollary. There is a 2-approximation algorithm for survivable network design. Iterative Rounding Initialization: H = , f’ = f. While f’ ≠ 0 do: o Find a basic optimal solution, x, of the LP with function f’. o Add an edge with x(e) ≥ 1/2 into H. By Jain’s Theorem o Update f’: for every set S, set Output H. f’(S)=1 f’(S)=2 0.5 e The residual problem is feasible. 0.5 Corollary. This is a 2-approximation algorithm for the minimum cost survivable network problem. 0.5 0.5 Typical Rounding: Optimal Linear Suitable Integer Problem Fractional Programming Rounding Instance Solution Procedure Solution Solver Iterative Rounding: Integer Good Optimal Part Part Problem Linear Fractional Programming Instance Solver Solution Residual Too much Fractional Problem Laminar Basis tree representation any pair of sets in the basis are either disjoint or contained Weak Supermodularity A weakly supermodular function f satisfies: or Example: f(S) := max{ r(u,v) | S separates u and v}. Important: when f is updated by setting x(e)=1 for some edges e, it remains weakly supermodular Weakly supermodular functions are very useful for obtaining a laminar basis Obtaining a Laminar Basis Tight constraints: Uncrossing technique: A basic solution is defined by a laminar family of tight connectivity constraints. A[B A B AÅB Proof of Jain’s Theorem Theorem [Jain]. Every basic solution has an edge with value at least 1/3 •There are |L| constraints. •There are |E| variables. Assume every edge has value 0 < x(e) < 1/3. Prove that |E| > |L| by a counting argument. • At the beginning we give 2 tokens to each edge, 1 to each endpoint. • At the end we redistribute the tokens so that each member in the laminar family has at least 2 tokens, and there are still some tokens left. Then this would imply |E| > |L| Induction Basis Assume every edge has value 0 < x(e) < 1/3. 1/4 each leaf set has degree ¸ 4 1/4 otherwise 9e, x(e) · 1/3 each leaf has 4 tokens 1/4 1/4 Inductive Step (1) Induction Hypothesis: the root has 2 extra tokens (total of 4) +2 +2 +2 +2 +2 +2 +2 +2 Theorem [Jain]. Every basic solution has an edge with value at least 1/3 Inductive Step (2) root has two children – +2 each child can pass up 2 tokens root has 4 tokens +2 +2 root has one child and 2 new edges – child can pass up 2 tokens root has 4 tokens Inductive Step (3) root has one child root has no new edges but \delta(child) = \delta(root) linear dependence root has one child and only one new edge e (of the two in picture) e both root and child are tight f(root) = f(child) + x(e) or f(root) = f(child) - x(e) e but 0<x(e)<1 contradicting integrality of f Summary of Jain’s Algorithm Theorem [Jain]. Every basic solution has an edge with value at least 1/3 Theorem [Jain]. Every basic solution has an edge with value at least ½ Proof is more involved … Open Question: Combinatorial proof ? Survivable Network Design Survivable network design: find a “good” Steiner network e.g. minimum spanning tree, minimum Steiner tree Minimum cost Steiner network: Given a cost c(e) on each edge, find a Steiner network with minimum total cost. Minimum degree Steiner network: Find a Steiner network with minimum maximum degree. e.g. Hamiltonian path, Hamiltonian cycle The Problem Statement Goal: to find a good Steiner network w.r.t. to both criteria Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Without degree bounds, this is the minimum cost Steiner network problem. Without cost on edges, this is the minimum degree Steiner network problem. Ideal Approximation Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Let OPT(B) be the value of an optimal solution to this problem. Ideally, we would like to return a solution so that: SOL(B) ≤ c·OPT(B) However, it cannot be done for any polynomial factor, even for B=2, since this generalizes the minimum cost Hamiltonian path problem. Bicriteria Approximation Algorithms Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Let OPT(B) be the value of an optimal solution to this problem. A (c,f(B))-approximation algorithm if it returns a solution with maximum degree f(B) SOL(f(B)) ≤ c·OPT(B) e.g. f(B)=2B+1 This implies a c-approximation for minimum cost Steiner network, and an f(B)-approximation for minimum degree Steiner network. Minimum Bounded Degree Spanning Trees Theorem [Furer and Raghavachari ’92] Given k, there is a polynomial time algorithm which does the following: either the algorithm (i) finds a spanning tree with maximum degree at most k+1. (ii) shows that there is no spanning tree with maximum degree at most k. Theorem [Goemans 06]: Given k, there is a polynomial time algorithm that computes a spanning tree with cost at most OPT(k) and maximum degree at most k+2. (1,B+2) uncrossing! Results on Minimum Degree Survivable Networks Minimum cost Minimum degree Bicriteria Spanning tree 1 B+1 [FR] (1,B+2) [G] Steiner tree 1.55 [RZ] B+1 [FR] (2,2B+3) (O(logn),O(logn)B) Steiner forest 2 [AKR] 2B+3 ? (2,2B+3) ? k-ec subgraph 2 [KV] 2B+3 O(log n)·B [FMZ] (2,2B+3) ? Steiner network 2 [Jain] 2B+3 ? (2,2B+3) ? Theorem [Lau,Naor,Salavatipour,Singh, STOC 07]: There is a (2,2B+3)-approximation algorithm for the minimum bounded degree Steiner network problem. Corollary: There is a constant factor approximation algorithm for the Minimum Degree Steiner Network problem. Linear Programming Relaxation At least r(u,v) edges crossing S S u v f(S) := max{ r(u,v) | S separates u and v}. Nonuniform degree bounds First Try Observation: Half edges are good for degree bounds as well. Initialization: H = , f’ = f. Problem: A half edge While f’ ≠ 0 do: may not exist! o Find a basic optimal solution, x, of the LP with function f’. o Add an edge with x(e) ≥ 1/2 into H. o Update f’: for every set S, set o Update degree bounds: set Bv:=Bv-0.5 if e is incident on v. Output H. Bv=2 =1.5 0.5 e 0.5 The residual problem is feasible. 0.5 0.5 The Difference fractional values Uncrossing would just work fine. 0.25 0.25 But integrality is important in the counting argument. Bv=0.5 New Idea Idea: Relax the problem by removing the 0.25 0.25 degree constraint for v if v is of “low” degree. Bv=0.5 Effect: Only violates the degree bound by an additive constant. Lemma [LNSS]: If every vertex is of degree 5 when its degree constraint is present, then there is a half edge in a basic solution. Intuition: Removing a constraint decreases the number of linearly independent tight constraints, and makes the counting argument work. Counting Induction Hypothesis: The root has 2 extra tokens. +2 +2 +2 +2 +2 +2 +2 +3 By linear independence Theorem [LNSS]. Every basic solution has an edge with value at least 1/3 if every degree constraint has at least 5 edges. Iterative Relaxation Initialization: H = , f’ = f. While f’ ≠ 0 do: A multiplicative factor 2 o Find a basic optimal solution, x, of the LP with function f’. o (Rounding) Add an edge with x(e) ≥ 1/2 into H. o (Relaxing) Remove the degree constraint of v if v has degree ≤ 4 o Update the connectivity requirement f’ An additive constant +3 o Update degree bounds: set Bv:=Bv-1/2 if e is incident at v. Output H. Theorem: There is a (2,2B+3)-approximation algorithm for the minimum bounded degree Steiner network problem. Additive Approximation Theorem [Lau,Singh,STOC 08]: There is a (2,B+3)-approximation algorithm for the minimum bounded degree Steiner forest problem. Theorem [Lau,Singh,STOC 08]: There is a (2,B+O(rmax))-approximation algorithm for the minimum bounded degree Steiner network problem. Minimum Bounded Degree Spanning Trees Theorem [Singh,Lau,STOC 07] There is an (1,B+1)-approximation algorithm for the minimum bounded degree spanning tree problem. Improves on the (1,B+2)-approximation of Goemans 2006 Directed Connectivity (const, const)-approximation for certain directed Connectivity problems [LNSS STOC 07] [Bansal,Khandekar,Nagarajan,STOC 08] Exact Formulations This method can be applied to prove LP formulations are exact. •Spanning Tree •Arborescence •Matroid intersection •Perfect matching in general graphs •Rooted k-out-connected subgraphs •Submodular flows Limitation: not simple to prove the dual is integral. Approximation Algorithms Some NP-hard problems are variants of basic problems. • General assignment (bipartite matching) • Multicriteria spanning trees • Partial vertex cover • Prize collecting Steiner trees • Degree bounded matroids [Király,L,Singh,08] • Degree bounded submodular flows [Király,L,Singh,08] Open Problems • TSP, ATSP? • Other applications? • Combinatorial algorithms? • Connection to existing approaches?