Primal Dual Method Lecture 20: March 28 primaldual restricted primal restricted dual y z found x, succeed! Construct a better dual

Primal Dual Program Primal ProgramDual Program If there is a feasible primal solution x and a feasible dual solution y, then both are optimal solutions. Primal-Dual Method : An algorithm to construct such a pair of solutions.

Optimality Condition Suppose there is a feasible primal solution x, and a feasible dual solution y. How do we check that they are optimal solutions? Avoid strict inequality

Optimality Condition

Complementary Slackness Conditions Primal complementary slackness condition: Dual complementary slackness condition:

Primal Dual Method Start from a feasible dual solution. Search for a feasible primal solution satisfying complementary slackness conditions. If not, improve the objective value of dual solution. repeat

Restricted Primal Formulate this as an LP itself! Given a feasible dual solution y, how do we search for a feasible primal solution x that satisfies complementary slackness conditions? Primal complementary slackness condition: Dual complementary slackness condition:

Restricted Primal If j not in J, then we need x(j) to be zero. If i not in I, then we need Restricted primal If zero, we are done

Restricted Dual Suppose the objective of the restricted primal is not zero, what do we do? Then we want to find a better dual solution. Restricted primalRestricted dual nonzero

Restricted Dual Restricted dual nonzero Dual Program Consider y+єz as the new dual solution. Still feasible Larger value

General Framework primaldual restricted primal restricted dual y z found x, succeed! Construct a better dual

Bipartite Matching Primal complementary slackness condition:

Start from a feasible vertex cover Find a perfect matching using tight edges non-zero Consider y- єz, a better dual Hungarian Method

Remarks It is not a polynomial time method. It reduces the weighted problem to the unweighted problem, so that the restricted primal linear program is easier to solve, and often there are combinatorial algorithms to solve it. Many combinatorial algorithms, like max-flow, matching, min-cost flow, shortest path, spanning tree, …, can be derived within this framework.

Approximation Algorithm How do we adapt the primal-dual method for approximation algorithms? We want to construct a primal feasible solution x and a dual feasible solution y so that cx and by are “close”. Avoid losing too much

Avoid strict inequality Approximate Optimality Condition

Approximate Complementary Slackness Condition Primal complementary slackness condition: Dual complementary slackness condition:Only a sufficient condition

Vertex Cover Primal complementary slackness condition: Dual complementary slackness condition:

Approximate Optimality Conditions Primal complementary slackness condition: Dual complementary slackness condition: Pick only vertices that go tight. Pick only edge with one vertex in the vertex cover. This would imply a 2-approximation. This is nothing! Just focus on this!

Algorithm Pick only vertices that go tight. Algorithm (2-approximation for vertex cover) Initially, x=0, y=0 When there is an uncovered edge Pick an uncovered edge, and raise y(e) until some vertices go tight. Add all tight vertices to the vertex cover. Output the vertex cover x. Familiar? This is the greedy matching 2-approximation when every vertex has the same cost.

Multicut Given k source-sink pairs {(s1,t1), (s2,t2),...,(sk,tk)}, a multicut is a set of edges whose removal disconnects each source-sink pair. The multicut problem asks for the minimum weight multicut. Today we only consider a tree! s1 s2 t2 t1

Hardness Why would this problem is hard? Vertex cover.

Primal Dual Programs Primal complementary slackness condition: Dual complementary slackness condition:

Approximate Optimality Conditions Primal complementary slackness condition: Dual complementary slackness condition: Only pick saturated edges. At most one edge can be picked from a path carrying nonzero flow. Relaxed dual complementary slackness condition: two

Algorithmic Idea Only pick saturated edges. At most two edges can be picked from a path carrying nonzero flow. We want to construct flow and cut so that: How to choose the flow? We should avoid sending flow along a long path…

Algorithm Multicut in Tree (2-approximation algorithm) Initially, f=0, D=0 Flow routing: Find a pair (si,ti) with lowest lowest common ancestor, Greedily send flow from si to ti. Add all the saturated edges to D. Let e1, e2,…, el be the ordered list of edges in D. Reverse delete: In reverse order, delete an edge e from D if D-e is still a multicut. Output the flow and the cut.

Reverse Delete At most two edges can be picked from a path carrying nonzero flow. Lemma. After reverse delete, the above condition is satisfied. siti v e’ e sj tj u there exists j where e is the only edge. j is processed earlier than i. i can send a flow, so e is added after e*. e* Idea?

Bonus Question Bouns Question 5 (50%): Show that a basic solution of the linear program for multicut in tree has an edge of value at least 1/2. Bouns Question 6 (50%): Show that a basic solution of the linear program for multiway cut has an edge of value at least 1/2.

Remarks Network design Facility location Online primal dual A unify primal-dual method for exact and approximation algorithm?