I want to understand the optimum algorithm for a tree decomposition of any graph. Is there any good sites where I can look up because I cannot find proper materials to understand the logic behind tree decomposition.
The PACE (Parameterized Algorithms and Computational Experiments Challenge) challenge is a competition for implementing fast algorithms (with a worst-case exponential running time). In 2016 and 2017, one of the challenges was to compute tree decompositions. See here for reports and (inside the reports) the links to the implementations of submitted solutions.
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So here is the problem:
Ignore problem 7, i blanked out irrelevant parts.
I already know that the answer to Problem 8 is 13 as stated in the picture.
But i dont know how to algorithmically come to this conclusion.
I know how to create a MST from a graph using Prims Algorithm, but I feel like there is a better way to quickly come up with an answer here.
As it says in this link :
This problem can be solved by many different algorithms. It is the topic of some very recent research. There are several "best" algorithms, depending on the assumptions you make:
A randomized algorithm can solve it in linear expected time.1
It can be solved in linear worst case time if the weights are small integers.2
Otherwise, the best solution is very close to linear but not exactly linear. The exact bound is O(m log beta(m,n)) where the beta function has a complicated definition: the smallest i such that log(log(log(...log(n)...))) is less than m/n, where the logs are nested i times.3
These algorithms are all quite complicated, and probably not that great in practice unless you're looking at really huge graphs. The book tries to keep things simpler, so it only describes one algorithm but doesn't do a very good job of it. I'll go through three simple classical algorithms (spending not so much time on each one).
So it is better to stick with Prim or Kruskal
Karger, Klein, and Tarjan, "A randomized linear-time algorithm to find minimum spanning trees", J. ACM, vol. 42, 1995, pp. 321-328.
Fredman and Willard, "Trans-dichotomous algorithms for minimum spanning trees and shortest paths", 31st IEEE Symp. Foundations of Comp. Sci., 1990, pp. 719--725.
Gabow, Galil, Spencer, and Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, vol. 6, 1986, pp. 109--122.
I have been looking at a lot of graph searches -both informed and uninformed. I have also looked at and coded some optimization problems such as hill-climbing. My question is, how can I relate the two types of algorithms?
To be a little more clear, here is an example:
Let's say I run a graph algorithm like Depth first Iterative Deepening. I run it for one depth and get a goal node, then I run it again at a different depth and find a goal node and so on. So now, I have a set of possible goal nodes. Could I run an optimization algorithm like hill climbing to find which one is "optimal" according to different limitations?
One of the notions you encounter in graph search algorithms is optimality. You should consider that here optimality is just about the algorithm and not about the goal. Meaning that, if an algorithm is optimal it should return the minimum cost solution.
Now, if you want to optimize something it's completely a different problem. In this case the quality of the solution is of most importance not the way you achieve it.
Algorithms like Genetic Algorithms, PSO, ... and many more optimization methods exists to address this issue.
BTW, sometimes we combine a graph search method like A* with an optimization algorithm to gain better results from the optimization algorithm.
Note: The term Graph Optimization is not related to the topic Graph Search that I think is your main topic according to your question.
I'm developing an application in which I have to face the travelling salesman problem. I did my own tries but the times I'm getting are really bad. I was searching some optimization solutions but I'm not getting anything clear.
Any tips to start a optimization of this process or algorythms? My current algorithm is the basic backtracking one.
My graph satisfy all the typical conditions in a TSP graph (no directional, simetric, conex)...
Thanks
If your metric satisfy the triangle inequality I can suggest you to look for the christofides algorithm. It has a guarantee to be within the optimum solution. IMO the difficult part about christofides algorithm is the perfect matching. If you don't care about a guarantee you can look for the google map tsp solver. It uses the Ant Colony Optimization for large route. If you want really fast solving and less accuracy you can look for a monster curve for example a hilbert curve or a moore curve.
I am new to the subject of algorithm design and graph theory. I am simulating large content based network consisting of thousands of routers. I am using "Learning by reverse path" for routing. Requested content names and contents are propagated in network using flooding. Routers check for matching names in routing tables and either reply back or populate routing table with unmatched requested content name and contents. Will using optimization algorithm like Ant colony optimization, hill climbing etc instead of learning by reverse path improve routing efficiency?
If your graph satisfy the triangle inequality i.e. is an euklidian space then I suggest you to use the christofides approximation algorithm because it has a guarantee to be 3/2 in the optimum. Other heuristic algorithm like ant colony optimization are very fast and efficient but not very safe. A good example for an ant colony optimization (and brute force and dp solution) is the google map tsp solver in javascript. I believe a space-filling-curve is a good approximation too and has a certain guarantee. You may look into a z-curve or a hilbert curve. You can find a good article about the hilbert curve at Nick spatial index quadtree hilbert curve blog or in the book Hacker's delight. I suggest to look into constructing of monotonic n-ary gray codes and hamiltonian path either.
There are many basic graph algorithms such as topological sort, strongly/weakly connected components, all-pairs/single-source shortest paths, reachability and so on. Incremental variants of these algorithms have a variety of important practical applications. By "incremental" I mean those graph algorithms that can compute small changes to their outputs given small changes (e.g. edge insertion and deletion) to the input graph without having to recompute everything. For example, a garbage collector accumulating the subgraph of heap allocated blocks reachable from the global roots. However, I do not recall seeing the subject of incremental graph algorithms discussed outside domain-specific literature (e.g. Richard Jones' new book on GC).
Where can I find information on incremental graph algorithms or, for that matter, incremental algorithms in general?
There's a survey article by Eppstein, Galil, and Italiano from 1999. They would describe what you're looking for as "fully dynamic algorithms"; "partially dynamic algorithms" are divided into "incremental algorithms", which allow only insertions, and "decremental algorithms", which allow only deletions.
Beyond that, I suspect you're going to have to read the research literature – there are only a handful of researchers who work on dynamic graph algorithms. You should be able to find articles by examining the papers that cite the survey.