Download presentation

Presentation is loading. Please wait.

Published byBrice Lindsey Modified over 2 years ago

1
Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem Yuval Rabani (Technion) Gabriel Scalosub (Tel Aviv University)

2
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 2 The Node-Cost Budget Problem Input: Undirected graph Cost function Profit function Budget Goal: Find a tree s.t. Budget constraint: is maximized cost=17 profit=14 cost=16 profit=16 B=16

3
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 3 Preliminaries An bicriteria approximate solution for the budget problem satisfies: WLOG, assume: The problem is rooted: Some predefined must be part of the solution

4
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 4 Our Results For any, a -approximation algorithm I.e., a tradeoff between the amount of budget violation, and the obtainable profit. The first result to reduce the budget violation below 2.

5
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 5 Previous Work Upper Bounds -approximation [Guha, Moss, Naor, Schieber 1999] -approximation [Moss, Rabani 2001] Lower Bounds Generalizes Budgeted Maximum Coverage Not approximable to within unless [Khuller, Moss, Naor 1999]

6
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 6 The Moss-Rabani Framework Solve an LP relaxation of the problem Use the solution to compute a polynomial set of trees Show there exists a tree which satisfies: Or Cheap, High profit Expensive, High profit-to-cost ratio In this case, we are done The Hard Part

7
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 7 Distance and Reachability Given two vertices, we let their distance be We say is reachable from with cost, if

8
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 8 The Trimming Lemma Assume: all vertices are reachable from the root with cost an -rooted tree satisfies then one can find an -rooted subtree such that Conclusion If all vertices are reachable from the root with cost then one can find a solution such that MR result:

9
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 9 An Intermediate Goal Some notation: - an optimal solution - a subtree of rooted at - the children of in Assume WLOG, Goal: the optimal profit, for the budget

10
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 10 A Structural Analysis of OPT Let be such that: Note that

11
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 11 A Structural Analysis of OPT (cont) Consider two instances: At least one of them has value is a feasible solution to rooted at

12
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 12 Algorithm Sketch Cand-1: MR solution with Enumerate over all Use the Trimming Lemma to approximate and Let, be the solutions obtained connect with Cand-2: Cand-3: Return: best of all rooted at

13
SWAT 2008 Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem 13 Summary and Open Questions This argument can be generalized by considering a partition into parts. Given any, taking gives a -approximate solution Can one do away completely with the budget violation? Logarithmic gap between upper and lower bound

14
Thank You!

Similar presentations

Presentation is loading. Please wait....

OK

Distributed Combinatorial Optimization

Distributed Combinatorial Optimization

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian villages our strength or weakness Esi ms ppt online Ppt on sports and games free download Ppt on intelligent traffic light control system Ppt on recycling of waste products Ppt on information security management system Ppt on stock market download Ppt on isobars and isotopes definition Ppt on role of water resources management Ppt on next generation 2-stroke engine diagram