Local Search Algorithms in Action: Max Cut Partition and Optimization
Explore local search algorithms applied to the Max Cut Partition problem, with insights on local changes, local vs. global optimization, and analysis of time and quality parameters for achieving local optima. Learn how to improve cuts iteratively and understand the nuances of obtaining optimal solutions in NP-hard scenarios.
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Local Search (part 1) Haim Kaplan and Uri Zwick Algorithms in Action Tel Aviv University Last updated: March 21 2016 1
Max Cut Partition ? = (?,?)into 2 parts ?,? ? such that the number of edges ?,? ,? ?,? ? ? is maximized ? ? = ? ? 3
Few comments This problem is NP-hard, best approximation guarantee is 0.878, NP-hard to get more than 0.91 The analogous problem of finding a minimum cut is easier (how ?) 4
The idea Start with some cut and try to improve it by making local changes Stop when there is no local change that improves the cut 5
Local change ? ? 6
Local change ? ? 7
Local change ? ? 8
Local change ? ? 9
Local change ? ? 10
Local change ? ? 11
Local vs. global OPT When we stop we have a cut that we cannot improve by a local change (local opt) This need not be the largest cut (global opt) 12
Local OPT ? ? B A C D 13
Global OPT ? ? A B C D 14
Analysis There are two main parameters of interest: Time: How long it takes to get to a local optimum ? Quality of the local optimum ? 15
Quality of local opt ? = ? ? ? v Can this be the situation at a local optimum ? 16
Quality of local opt ? = ? ? ? v ? ? ? ?(?) ? ? = #edges incident to v that cross the cut ? ? = #edges incident to v that do not cross the cut 17
Quality of local opt ? = ? ? ? v ? ? ? ? ? ? ?,? ? ?? ?,? ? ? ?,? ? 2 ?,? 2 ?,? 18
Quality of local opt ? = ? ? ? v ? ? ? ? ? ? ?,? ? ?? ?,? ? ? ?,? ? ?,? ?,? ??? 2 |?| ? ?,? ? ?,? 2 19
Steps to local opt ?( ? ) Each step the size of the cut increases by at least one 20
Weighted graphs Partition ? = (?,?)into 2 parts ?,? ? such that the ? ?,? ,? ?,? ? ? is maximized ? ? = ? ? 21
Weighted graphs Works the same: Make a swap if it increases the weight of the cut ??? 2 ? ? ?,? ? 2 ? ?,? ? = ?(?,?) 22
Steps to local opt We construct a example with a long running time The example uses two kinds of vertices 23
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 24
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 25
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 26
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 ? ?4 > ? ?2,? ?3 > ? ?1 ?2 ?3 ? ? ?2 + ? ?3 > ? ?1 + ? ?4 ?1 ?4 27
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 ? ?4 > ? ?2,? ?3 > ? ?1 ?2 ?3 ? ? ?2 + ? ?3 > ? ?1 + ? ?4 ?1 ?4 28
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 ? ?4 > ? ?2,? ?3 > ? ?1 ?2 ?3 ? ? ?2 + ? ?3 > ? ?1 + ? ?4 ?1 ?4 29
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 ? ?4 > ? ?2,? ?3 > ? ?1 ?2 ?3 ? ? ?2 + ? ?3 > ? ?1 + ? ?4 ?1 ?4 30
Steps to local opt ?2 ? ?3 > ? ?1 + ? ?2 + ? ?4 ? ?3 ?1 ?4 ? ?4 > ? ?2,? ?3 > ? ?1 ?2 ?3 ? ? ?2 + ? ?3 > ? ?1 + ? ?4 ?1 ?4 31
Steps to local opt ?0 ?1 Moral: 2 flips of ?0caused 4 flips of ?1 47
Steps to local opt ?0 ?1 ?2 48
Steps to local opt ?0 ?1 ?2 49
Steps to local opt ?0 ?1 ?2 50
