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Basic Ideas

In this page, we describe the basic ideas of Combinatorial Optimization Problem (COP), Stochastic Local Search (SLS) algorithms, Fitness Landscape and Search Trajectory (FLST) visualization, the implementation of FLST visualization in Viz, and the Integrated White and Black Box Approach. This page is just a summary and written in an informal tone, targeted for wider audience. More details and formal presentations can be found in the main author future thesis (later) and in our recent papers.

Table of Contents

1. Combinatorial Optimization Problem (COP): definition, examples, and algorithms to attack these problems
2. Stochastic Local Search (SLS) Algorithm: definition, examples, SLS engineering, SLS design and tuning problem
3. Fitness Landscape and Search Trajectory (FLST) visualization: the SLS visualization ideas
4. Implementation of FLST visualization in SLS engineering suite: Viz
5. The Integrated White and Black Box Approach (IWBBA): combining the strength of both worlds
6. References

1. Combinatorial Optimization Problem (COP)

Combinatorial Optimization Problem (COP) is a class of problem where the solution(s) are combinatoric, e.g. a permutation, an assignment, etc. The term optimization implies that one is interested in finding the maximum (or minimum) of a set of feasible combinatorial solutions. In general, COPs are classified as NP-hard, which in loose term means that we will need enormous computation time in order to find the best solution for these problems.

Examples of COPs include the Traveling Salesman (TSP), Quadratic Assignment (QAP), Vehicle Routing (VRP), Knapsack, etc.

To attack these COPs, people devised various computer algorithms that can be classified into two major extremes: exact (complete search) algorithms versus non-exact (incomplete) algorithms. There are pros and cons between these two approaches which are summarized in the following table.

-

Exact / Complete

Non-Exact / Incomplete
Pros Optimal solution can be determined Fast (iterative), can be stopped anytime
Cons Slow (exponential time) Optimality is not guaranteed
Examples Branch and Bound Approximation, Metaheuristic/Stochastic Local Search

2. Stochastic Local Search (SLS) Algorithm

Stochastic Local Search (SLS), also known as Metaheuristic, is a non-exact algorithm that can be loosely described as follows:

 1. SLS() {
 2.   start from any solution of the COP instance;
 3.   while (not finished) {
 4.     locally modify (search around) the current solution with some heuristic/stochastic rule;
 5.     if (the newly found solution is better than the best solution so far)
 6.      update the best solution status;
 7.   }
 8.   return the best found solution that it manages to find throughout the search process;
 9. }

SLS algorithms have been shown to be quite successful in attacking various COPs. Examples of SLS algorithms include: Tabu Search (TS), Iterated Local Search (ILS), Simulated Annealing (SA), Ants Colony Optimization (ACO), Genetic Algorithms (GA), Variable Neighborhood Search (VNS), Guided Local Search (GLS), etc.

While the basic version of an SLS algorithm is relatively simple, once we want to make it performs 'well', we will need to design (choose appropriate SLS components and search strategies), implement the algorithm properly using the best data structures, tune the SLS parameters, and analyze its performance. This SLS engineering process is not a straightforward task and often consumes a lot of development time! This issue is even more important within people without strong background in local search techniques. This hinders the adoption of SLS methods to wider community.

With this motivation, the main author proposed "to study the ways to address this issue of designing an effective SLS algorithm and tuning the corresponding SLS implementation for various COPs" in his PhD thesis.

3. Fitness Landscape and Search Trajectory (FLST) visualization
This section has been presented in UIST 2006, SLS 2007, and CP 2007.

To address the SLS Design and Tuning Problem, one needs to analyze the SLS behavior (which is closely related to its performance). However, analyzing SLS behavior is difficult. For example, if we look at this RunLog file (a file that records the search information every iteration) alone, we probably will not get many information from it. Thus people devise SLS analysis techniques that can be classified as either statistical techniques or information visualization techniques. With such analysis, people hope to gain insights on how to further improve the SLS algorithms.

We propose FLST visualization as an extension cum combination of the existing Fitness Distance Correlation (FDC) and Run Time Distribution (RTD) analysis. The basic ingredients to form this visualization are the fitness landscape components (search space, distance function, and objective function) of the COP instance and the solutions visited by an individual SLS runs. Obviously, visualizing exponentially large fitness landscape is not trivial. We explain the FLST visualization ideas using the series of illustrations below:

Explanation

Illustration

We visualize fitness landscape of a COP instance like this mountain range picture:
*search space (very big), is visualized as a collection of a lot of points (solutions)
*distance function spatially separates one mountain (solution) and the other mountains
*objective function determines the height of each mountain (solution)
*global optima is the highest mountain (solution with the best objective value)
*local optima is high mountains but not the highest

Notes: This fitness landscape formulation was proposed by P. Merz in his PhD thesis. This definition is slightly different with the one in H.H. Hoos and T. Stuetzle's book: SLS: Foundations and Applications where the fitness landscape is defined as <search space, neighborhood function, and objective function>. We do not use neighborhood function as it will cause our FLST visualization to be unstable (changing every SLS iteration).

We visualize the search trajectory of an individual SLS run as a movement of that SLS on the fitness landscape of the COP instance being attacked. The movement can be due to a local move within a local neighborhood or a non local move due to strong diversification mechanism. The objective is to find the global optima: imagine that you are one tiny human in this mountain range and can only see the surroundings within radius 1 km (local view) and you need to navigate locally to find the highest mountain.

However, without any 'reference point', it is quite hard to describe/explain what is going on here (see the picture on the right and try your best to explain the movement)...

Notes: This search trajectory formulation is defined in SLS: Foundations and Applications.

Now suppose that we record the solution denoted by the yellow rectangle (see the picture on the right). We can now describe the same search trajectory above as follows:

"The search trajectory once hits the yellow rectangle solution, then it moves somewhere else, then after certain number iterations, it hits the yellow rectangle solution again. Is this a solution cycling phenomenon? Is the SLS trapped?".

See that now we can say more things with the existence of a reference point.

To build the FLST visualization, we gather a constant amount of high quality (usually local optima), diverse, frequently visited solutions in the fitness landscape using the SLS algorithms themselves! We know that we cannot expect to record all points (exponential space!) therefore we should expect to miss some good points (look at the small blue arrows pointing at the other two mountain peaks at the background). Nevertheless, if the anchor points collected are reasonably good, diverse, and -that is- the important ones, we can say that we have a reasonable approximation of the fitness landscape.

In order to collect these points, we run the SLS with different configurations, with longer run times, and let it loose. It will then sample various points in the search space. We then filter the interesting points which we called: the Anchor Points, abbreviated as APs.

We can also add quality information by labeling these anchor points with color+shape:

*black dot-very bad,
*yellow rectangle-bad,
*green triangle-medium,
*blue circle-good.

We use both color and shape as color alone is hard to be distinguished in black and white scientific papers! Remember that not all scientific publications out there are in color at this point of time.

So now, we have this Fitness Landscape visualization based on these 4 selected APs.

With these APs, we can now describe the search trajectory:

In the picture on the right, the pink search trajectory encounters solution cycling in yellow/black (poor) APs. It fails to reach the better green/blue (better) APs.

And for this picture on the right, the blue search trajectory performs a diversification strategy after hitting an AP (local optima). From this visualization, we see that it manages to reach the better green/blue APs and we can say that it performs better than the pink search trajectory above.

 

This is our proposed FLST visualization.
If you have any comments, please don't hesitate to email the main author:
stevenhalim at gmail dot com

With FLST visualization, one can answer these COP fitness landscape characteristics and SLS behaviors questions (not exhaustive):

The fitness landscape characteristics of the COP instance:
 1. Are the local optima clustered or spread?
2. Does the fitness landscape smooth or rugged?
3. How many objective value levels in the fitness landscape? is it discrete or continuous?
4. Are there infeasible regions and where are they located?

The search trajectory behaviors of the SLS algorithm on a particular COP instance/fitness landscape:
1. Does the SLS behave like as what we intended? How does it make progress? Does it quickly find the best known solution or wander around in other regions?
2. How good is the SLS in intensification and diversification?
3. Is there any sign of cycling behavior (search stagnation)?
4. Where in the fitness landscape does the SLS spend most of its time?
5. How far is the starting/initial/greedy solution w.r.t the global optima/best known solution?
6. How wide is the SLS coverage?
7. What is the effect of modifying a certain search parameter/component/strategy w.r.t the SLS behavior?
8. How do two different SLS algorithms (on the same COP fitness landscape) compare?

Answering these questions can give insights on designing better performing SLS algorithm to attack the COP at hand.

4. Implementation of FLST visualization in SLS engineering suite: Viz
 Parts of this section has been presented in UIST 2006 and updated in SLS 2007.

In section 3 above, we show the FLST visualization ideas. In this section, we explain how we implement those ideas in a concrete visualization tool Viz. Viz consists of two main programs: Viz Experiment Wizard (EW) which executes SLS algorithms and processes the RunLogs and Viz Single Instance Multiple Runs Analyzer (SIMRA) which displays FLST visualization and some other statistical information in a user friendly GUI.

1. Potential AP Selection phase:

Here, Viz EW runs the SLS algorithms (each with its selected configuration) on the same COP instance (must be on the same fitness landscape, otherwise the collected points will be irrelevant with each other) to collect potential local optima points visited by those SLS. Of course, the SLS must have been augmented with simple codes to record search information. We use some of the points found by the SLS algorithms themselves to visually analyze the SLS performance.

Alternatively, the user can choose to feed Viz EW with RunLog files generated by their SLS algorithms running in non Windows operating system. That's it, Viz EW can be set to work with RunLog files only. The non-Windows users do not need to rewrite or recompile their SLS code in Windows platform to use this tool. They just need to install Viz on another Windows PC. e.g. friends' Windows PC, school's Windows PC, or company's Windows PC, etc and process the RunLog files there. (however, to fully utilize the benefits of Viz EW, we suggest that you write a Windows based wrapper code that perform these steps: (1) login to other OS, (2) executes the SLS algorithm in other OS, and (3) returns the log files back to Viz EW).

FAQs:
Q: Why do you use SLS algorithms to sample points in the search space?
A: Obviously, we do not have the time to generate all points using exact search just to analyze non-exact search trajectories. The best way is to analyze points visited by the SLS algorithms itself. Ensure that your SLS algorithm does not stuck (simply add random restart) so that it samples enough points. However, even if your SLS algorithm does stuck, this FLST visualization will show it to you.

Q: Why do you use log file (offline visualization) instead of real-time visualization?
A: Offline visualization has more advantages. First, we can display the final AP set so that the visualization is more stable rather than updating the AP set on the fly. Second, by communicating via log files, one does not need to depend on any Operating System. Third, visualization takes some CPU power to be processed, we simply do not want our visualization algorithm compete with the SLS algorithm in terms of CPU resource.

Q: Wait, is it true that Viz can only analyze 1 (ONE) single COP instance at one time. Will that be myopic?
A: We agree that analyzing SLS algorithm must be done with respect to several instances so that our observations are not instance specific only. Viz is designed to analyze the details of various SLS runs on a single/1/one COP instance. However, you can still run your SLS algorithm on several training instances using Viz EW. You will obviously see different details from one instance with the other instances but you should have identify more or less some general characteristics (e.g. all training instances have "Big Valley" properties, etc).

2. AP Update phase:

In this phase, after the potential APs have been collected, Viz EW filters them and pick a small number of more diverse and high quality APs to form an updated AP set. The reason of this step is simply because we cannot draw too many points on the screen.

FAQs:
Q: How many points in the screen is 'too many'?
A: As soon as one needs to use zoom in/out, panning (scroll left/right/up/down), or image distortion (e.g. fisheye technique), then the points are too many. Zooming, panning, image distortion, etc decrease the capability of the user in finding information from visualized data. Therefore, the number of points is set to be reasonably small, which is currently set to be min(50,0.5*instance size n).

Q: But, the number of points in the screen can be 'enlarged' if you use larger monitor, why don't you assume that the end users are using large monitor?
A: At this point of time, we designed Viz to work under 800*600 monitor resolution. That gives us 480,000 pixels to play with. Soon, we will increase Viz default window size to 1024*768, which is about 786,432 pixels, around 1.5 more pixels to work with. However, we cannot keep enlarging Viz window size as currently the typical monitor resolution is around [1024*768 .. 1200*800]. We want Viz to work for this kind of monitor resolution and thus can be used by most people that own a computer today!

Q: How do you actually filter the points?
A: The actual strategy is a bit too technical and still under research. In general, we select AP points based on these criteria: (1). have high quality (randomly selected from the top K% of the collected points), (2). increase diversity of the AP set (if we already have good point X in the AP set, we do not include X' which is very similar to X), (3). have some importance in the SLS runs used to form the AP set (e.g. the best solution found by the SLS, the most frequently visited solution, etc).

3. AP Layout phase:

After the AP set is selected, Viz layouts the APs in the abstract 2-D space using a spring model (also called force-directed layout algorithms). We measure the distance (bond-number of different edges, Hamming-number of different bits, etc) between APs to set up the spring model. The spring system will try to stretch and shrink itself into its most natural state (minimal tension) and that make APs that are close to each other to appear close in the visualization and vice versa.

We found that presenting the fitness landscape information in spatial manner like this is quite natural and easy to comprehend.

FAQs:
Q: Why use spring model and not any other ways?
A: It is so far the most natural representation that we can think of. We are also aware of other graph drawing technique like constraint stress majorization algorithm and will explore other techniques soon.

Q: What if the distance metric for the solutions in my COP are not Hamming or bond distance?
A: We are in process of extending the built-in distance metric inside Viz. Please co-operate with us by e-mailing the main author: stevenhalim at gmail dot com these information: the COP that you are planning to attack and the algorithm to compute distance metric that you think will be suitable for your COP.

4. AP Labeling phase:

After the AP layout phase, we further enhance the fitness landscape presentation by labeling the APs using color and shape labels as shown in the legend below. Now with one glance, we know the rough distribution and quality of the anchor points.

FAQs:
Q: Do you have any particular reason why you choose these set of colors?
A: The color set is actually taken from the color set used in geographical map where blue represents deep oceans, green represents the forests, yellow represents the mountains, and white surroundings around black dot represents the snowy peaks.

Q: Why do you use double features: shape and color, for the labels?
A: While color is the best means to group these information, it is quite hard to see in black and white scientific papers. So we add the shape :)

5. Search Trajectory Layout phase:

Now that we have the abstract 2-D space of anchor points ready, we can visualize the movements of the current point/solution (which is actually an N-dimensional combinatorial solution) of the SLS run as it pass through the COP Fitness Landscape. We playback the individual SLS run by measuring the distance of the current solution of an SLS run with the APs. If it is close, we draw a circle (smaller circle: more similar, larger circle: more different). If the current solution is nowhere near any of the selected APs, we do not draw anything (no information can be gained from this iteration). The movement (appearance and disappearance) of these circles tells us about the SLS trajectory.

As an example, we explain the interpretation of the search trajectory visualization of the four runs above (see step 1). The interpretation depends on the position and quality of APs visited by the search trajectory. These are the possible interpretations:

Run 1: SLS visits medium quality (green triangle) APs and encounter solution cycling issue near AP C. Perhaps AP C is too attractive even though it is not the best quality (AP D has better quality as indicated by its blue circle lable).

Run 2: SLS visits bad region (yellow rectangle): AP B, then somewhere near AP C (here we assume that solution F in run 2 is close to AP C, just to illustrate the drawing of circle with larger diameter around AP C), and finally arrive at AP A (very bad/black dot). Perhaps this SLS is not doing a good intensification.

Run 3: SLS starts from a very bad AP A (black dot), gradually moves to medium AP C (green triangle), then to good AP D (blue circle). This SLS behavior is quite okay.

Run 4: The SLS trajectory is not close to any known AP. It is exploring somewhere else, but definitely not somewhere near these selected anchor points, it only hits AP A (very bad/black dot AP).

FAQs:
Q: Do you know any other ways to visualize search trajectory?
A: To date this is our best way to visualize search trajectory. We will keep exploring other alternatives.

Q: Do you realize that when two people look at the same FLST visualization, they may arrive at different interpretations?
A: We agree with that. There may be more than one way to interpret the visualization and two algorithm designer may design two different SLS algorithms based on the visualization. Eventually, we just want improved SLS algorithms. If both interpretations yield improvements, that is ok.

Q: My SLS is not a trajectory based visualization! I am using population based algorithm. So how to visualize the 'search trajectory' of a population?
A: To date, our best way to overcome this situation is by defining the search trajectory as the movement of the 'best solution within the population' from one generation to the next... We are exploring some visualization ideas to better visualize population based SLS algorithm.

 Q: How accurate is this FLST visualization?
A: The accuracy of FLST visualization is still a research topic. We know that the accuracy will depends on at least two things: (1). the number of anchor points that the user wants to display on the screen (more APs, more layout errors), (2). the way we select AP set (if we miss important APs, the FLST visualization will be less accurate).

Viz SIMRA, the program that displays this FLST visualization, has few more features that have not been elaborated. We invite the reader the visit our Viz version history to appreciate the development of Viz SIMRA user interface. The GUI are designed with end user in mind and it boasts the following features: coordinated visualizations from various angle, visual comparison, animated search playback, heavy usage of color and highlighting, multiple level of details, and including some built-in statistical analysis.

This concludes section 4.

5. The Integrated White and Black Box Approach (IWBBA)
This section has been presented in CP 2007.

FLST visualization can give us insights of the approximate fitness landscape structures and SLS behavior. However, while that may give you insights to design good performing SLS algorithm, it is still not enough if we want the 'best' performing SLS algorithm for our COP.

We suggest that one combines the strength of the white-box approach with black-box approach:
1. In white box approach, we analyze our SLS algorithm by means of statistic or information visualization techniques to gain insights on how to design better performing SLS algorithm that match the observed fitness landscape characteristics.
2. In black-box approach, we use a specialized tuning algorithm which will find the best configuration for our SLS algorithm given the configuration space and the training set of COP instances. Examples of such algorithms are: ParamILS (Hutter et al., 2007), F-Race (Birattari, 2004), CALIBRA (Adenso-Diaz and Laguna, 2006).

We invite the reader to check our results page where we apply this IWBBA using Viz on real COPs and real SLS algorithms.

6. References

These basic ideas are explained in more details in our publications (download the local copy of those papers here):

  1. S. Halim, R. Yap, H.C. Lau. An Integrated White+Black Box Approach for Designing and Tuning Stochastic Local Search. In Principles and Practice of Constraint Programming (CP 2007, Providence, Rhode Island, USA, September 23-27, 2007): 332-347

  2. S. Halim, R. Yap. Designing and Tuning SLS through Animation and Graphics: an Extended Walk-through. In Engineering Stochastic Local Search Workshop (SLS 2007, Brussels, Belgium, September 6-8, 2007): 16-30

  3. S. Halim, R. Yap, H.C. Lau. Viz: A Visual Analysis Suite for Explaining Local Search Behavior. In User Interface System and Technology (UIST 2006, Montreux, Switzerland, October 15-18, 2006): 57-66
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