Carmine Cerrone, Raffaele Cerulli, Bruce Golden GO IX Sirmione, Italy July

Outline  Motivation  The Minimum Label Spanning Tree (MLST) problem  Experimental justification  Introduction to Carousel Greedy  Details of Carousel Greedy  Computational experiments in combinatorial optimization  Computational experiments in statistics  Conclusions 2

Motivation  We seek a heuristic framework that generalizes and enhances greedy algorithms  We want a heuristic that is fast  It should outperform a greedy algorithm  It should be applicable to many greedy algorithms  It should be simpler than a metaheuristic  It should involve a small number of parameters 3

Motivation  The Minimum Label Spanning Tree (MLST) Problem  Communications network design  Edges may be of different types or media (e.g., fiber optics, cable, microwave, telephone lines, etc.)  Each edge type is denoted by a unique letter or color  Construct a spanning tree that minimizes the number of colors 4

Motivation  A Small Example c e a d e a b bb d ee b b b InputSolution

Description of MVCA 0. Input: G (V, E, L). 1. Let C { } be the set of used labels. 2. repeat 3. Let H be the subgraph of G restricted to V and edges with labels from C. 4. for all i L – C do 5.Determine the number of connected components when inserting all edges with label i in H. 6. end for 7. Choose label i with the smallest resulting number of components and do: C C {i}. 8. Until H is connected. 6

How MVCA Works 7 Solution c e a d e a b bb d ee b b b Input Intermediate Solution b bb 1 6

An Example to Motivate Carousel Greedy b d b d b  Apply MVCA: add a, b, and c to obtain {a,b,c}  Note that label a looked best, but now we can discard it 8 a a c a a c c

An Example to Motivate Carousel Greedy  {b, c} is a MLST of cardinality 2  MVCA chose a wrong label initially  Carousel Greedy will try to correct this 9 b c b c c b

Observations about MVCA and Greedy Algorithms in General  We can divide MVCA into three phases 1 iterations S  Phase I  The set of candidate labels is very large  Many labels yield similar results  It is not possible to learn much from previous label selections, because there haven’t been many 10

Observations about MVCA--continued  Phase II  The set of candidate labels has been reduced  It is possible to learn from previous label selections  The Phase 2 selections are “smarter”  Some labels, chosen in Phase I, no longer look so good  Phase III  The set of candidate labels required for feasibility is small  This phase is short since we are near the end 11

MCVA Phase Experiment  We generated 10,000 random labeled graphs with V = 85, E = 340, and L = 85  The optimal solution is 20 labels in all cases (Xiong, Golden, Wasil – 2005)  The x-axis indicates the percentage of the selections in MVCA that have been completed  The y-axis indicates the average percentage of labels that are in the optimal solution 12

MVCA Phase Experiment 13 start of MVCA end of MVCA

A Limitation of MVCA  We generated 1,300 random labeled graphs with V = 81, L = {30, 60, 90, …, 390} and E = 4 L  The optimal solution is 20 labels in all cases  The x-axis indicates the number of available labels, L  The y-axis indicates the average number of labels in the MVCA solution 14

A Limitation of MVCA 15  When L is small, MVCA performs poorly  Can a generalized greedy algorithm do better?

Questions to Address  Can we extend Phase II?  Can we improve performance when L is small?  Can we ensure reasonable running times?  Can we keep it simple? 16

Iterated Greedy: A Simple Generalized Greedy Algorithm (Ruiz, Stutzle – 2008)  Step 1. Apply greedy algorithm to obtain a feasible solution  Step 2. Destruction phase: Remove some elements from the current solution, leaving a partial solution  Step 3. Construction phase: Apply the greedy algorithm to the partial solution to obtain another feasible solution  Step 4. Repeat Steps 2 & 3 until a stopping condition is satisfied 17

Carousel Greedy Illustration for S = 5, α = 1, β = 40% MVCA Solution Carousel Start Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6 Final Step 18  The length of a carousel string is S (1- β ) and the number of passes is α

More on Carousel Greedy  We can represent this example by images of a carousel in motion  The carousel is divided into S wedges  The number of full turns or passes is α  The final step obtains a feasible solution  The carousel representation of this generalized greedy algorithm is shown on the next slide 19

The Carousel Greedy Algorithm 20 R R RRR R G G GG O O O O P P P B G R O P Start

Carousel Greedy Performs Well for Small L 21

Further Experiments with Carousel Greedy  We look at the MLST problem  We generate 20 random labeled graphs with V = 100, L = 100, and E = 400  We tested different values of α and β  The optimal solution is 25 labels in all cases  Average improvements and running times are presented next 22

Carousel Greedy Improvements over MVCA 38%47%41%47%52%55%44%55% 54%60% 66%67%62% 64% 55%71%73%72%75%74%71%72% 60%72% 75%74%75%73% 23 α = 1 α = 2 α = 3 α = 4α = 5 α = 6α = 7 α = 8 β = 20% β = 15% β = 10% β = 5%  For example, when α = 5 and β = 10%, Carousel Greedy reduces the gap between the MVCA solution and the optimal solution by 75%

Carousel Greedy Running Time 24 α = 1 α = 2 α = 3 α = 4 α = 5 α = 6 α = 7 α = 8 β = 20% β = 15% β = 10% β = 5%  Time is in milliseconds  With α = 1, we expect the running time to be about twice the MVCA running time (10 ms)

Estimating the Complexity of Carousel Greedy  Let O(I) be the computational complexity of the basic greedy algorithm  We can show that Carousel Greedy has an approximate computational cost of (α + 1) O(I)  Next, we summarize our extensive computational results 25

MLST Instances  We looked at 420 small instances with V = L ≤ 200  Pilot outperforms Carousel Greedy (CG) and Iterated Greedy (IG)  CG was faster than IG which was faster than Pilot  We looked at 400 large instances with V = L ≤ 1000  Pilot is too slow to be useful  CG clearly outperforms IG  IG takes about five times as long as CG 26

Other Computational Experiments  We also tested CG on the Minimum Vertex Cover and Maximum Independent Set problems  CG improves upon greedy to a substantial degree  In addition, we tested CG on the Minimum Weight Vertex Cover problem  CG outperforms greedy, SA, TS, and ACO  Finally, we tested CG against Stepwise Regression on four instances  CG reduced the standard error in each instance 27

Conclusions  We introduced Carousel Greedy--a new, generalized greedy algorithm  It is fast and widely applicable  It has two (easy to set) parameters  It has been tested on problems in combinatorial optimization and statistics  It outperforms greedy, Iterated Greedy, and Pilot  Further testing on Stepwise Regression is required  Buon Compleanno Grazia!!! 28