Branch-And-Bound Strategies for Dynamic Programming
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Metaheuristics1
METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming. -
Lecture 4 Dynamic Programming
1/17 Lecture 4 Dynamic Programming Last update: Jan 19, 2021 References: Algorithms, Jeff Erickson, Chapter 3. Algorithms, Gopal Pandurangan, Chapter 6. Dynamic Programming 2/17 Backtracking is incredible powerful in solving all kinds of hard prob- lems, but it can often be very slow; usually exponential. Example: Fibonacci numbers is defined as recurrence: 0 if n = 0 Fn =8 1 if n = 1 > Fn 1 + Fn 2 otherwise < ¡ ¡ > A direct translation in:to recursive program to compute Fibonacci number is RecFib(n): if n=0 return 0 if n=1 return 1 return RecFib(n-1) + RecFib(n-2) Fibonacci Number 3/17 The recursive program has horrible time complexity. How bad? Let's try to compute. Denote T(n) as the time complexity of computing RecFib(n). Based on the recursion, we have the recurrence: T(n) = T(n 1) + T(n 2) + 1; T(0) = T(1) = 1 ¡ ¡ Solving this recurrence, we get p5 + 1 T(n) = O(n); = 1.618 2 So the RecFib(n) program runs at exponential time complexity. RecFib Recursion Tree 4/17 Intuitively, why RecFib() runs exponentially slow. Problem: redun- dant computation! How about memorize the intermediate computa- tion result to avoid recomputation? Fib: Memoization 5/17 To optimize the performance of RecFib, we can memorize the inter- mediate Fn into some kind of cache, and look it up when we need it again. MemFib(n): if n = 0 n = 1 retujrjn n if F[n] is undefined F[n] MemFib(n-1)+MemFib(n-2) retur n F[n] How much does it improve upon RecFib()? Assuming accessing F[n] takes constant time, then at most n additions will be performed (we never recompute). -
A Branch-And-Price Approach with Milp Formulation to Modularity Density Maximization on Graphs
A BRANCH-AND-PRICE APPROACH WITH MILP FORMULATION TO MODULARITY DENSITY MAXIMIZATION ON GRAPHS KEISUKE SATO Signalling and Transport Information Technology Division, Railway Technical Research Institute. 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, Japan YOICHI IZUNAGA Information Systems Research Division, The Institute of Behavioral Sciences. 2-9 Ichigayahonmura-cho, Shinjyuku-ku, Tokyo 162-0845, Japan Abstract. For clustering of an undirected graph, this paper presents an exact algorithm for the maximization of modularity density, a more complicated criterion to overcome drawbacks of the well-known modularity. The problem can be interpreted as the set-partitioning problem, which reminds us of its integer linear programming (ILP) formulation. We provide a branch-and-price framework for solving this ILP, or column generation combined with branch-and-bound. Above all, we formulate the column gen- eration subproblem to be solved repeatedly as a simpler mixed integer linear programming (MILP) problem. Acceleration tech- niques called the set-packing relaxation and the multiple-cutting- planes-at-a-time combined with the MILP formulation enable us to optimize the modularity density for famous test instances in- cluding ones with over 100 vertices in around four minutes by a PC. Our solution method is deterministic and the computation time is not affected by any stochastic behavior. For one of them, column generation at the root node of the branch-and-bound tree arXiv:1705.02961v3 [cs.SI] 27 Jun 2017 provides a fractional upper bound solution and our algorithm finds an integral optimal solution after branching. E-mail addresses: (Keisuke Sato) [email protected], (Yoichi Izunaga) [email protected]. -
Dynamic Programming Via Static Incrementalization 1 Introduction
Dynamic Programming via Static Incrementalization Yanhong A. Liu and Scott D. Stoller Abstract Dynamic programming is an imp ortant algorithm design technique. It is used for solving problems whose solutions involve recursively solving subproblems that share subsubproblems. While a straightforward recursive program solves common subsubproblems rep eatedly and of- ten takes exp onential time, a dynamic programming algorithm solves every subsubproblem just once, saves the result, reuses it when the subsubproblem is encountered again, and takes p oly- nomial time. This pap er describ es a systematic metho d for transforming programs written as straightforward recursions into programs that use dynamic programming. The metho d extends the original program to cache all p ossibly computed values, incrementalizes the extended pro- gram with resp ect to an input increment to use and maintain all cached results, prunes out cached results that are not used in the incremental computation, and uses the resulting in- cremental program to form an optimized new program. Incrementalization statically exploits semantics of b oth control structures and data structures and maintains as invariants equalities characterizing cached results. The principle underlying incrementalization is general for achiev- ing drastic program sp eedups. Compared with previous metho ds that p erform memoization or tabulation, the metho d based on incrementalization is more powerful and systematic. It has b een implemented and applied to numerous problems and succeeded on all of them. 1 Intro duction Dynamic programming is an imp ortant technique for designing ecient algorithms [2, 44 , 13 ]. It is used for problems whose solutions involve recursively solving subproblems that overlap. -
Heuristic Search Viewed As Path Finding in a Graph
ARTIFICIAL INTELLIGENCE 193 Heuristic Search Viewed as Path Finding in a Graph Ira Pohl IBM Thomas J. Watson Research Center, Yorktown Heights, New York Recommended by E. J. Sandewall ABSTRACT This paper presents a particular model of heuristic search as a path-finding problem in a directed graph. A class of graph-searching procedures is described which uses a heuristic function to guide search. Heuristic functions are estimates of the number of edges that remain to be traversed in reaching a goal node. A number of theoretical results for this model, and the intuition for these results, are presented. They relate the e])~ciency of search to the accuracy of the heuristic function. The results also explore efficiency as a consequence of the reliance or weight placed on the heuristics used. I. Introduction Heuristic search has been one of the important ideas to grow out of artificial intelligence research. It is an ill-defined concept, and has been used as an umbrella for many computational techniques which are hard to classify or analyze. This is beneficial in that it leaves the imagination unfettered to try any technique that works on a complex problem. However, leaving the con. cept vague has meant that the same ideas are rediscovered, often cloaked in other terminology rather than abstracting their essence and understanding the procedure more deeply. Often, analytical results lead to more emcient procedures. Such has been the case in sorting [I] and matrix multiplication [2], and the same is hoped for this development of heuristic search. This paper attempts to present an overview of recent developments in formally characterizing heuristic search. -
Automatic Code Generation Using Dynamic Programming Techniques
! Automatic Code Generation using Dynamic Programming Techniques MASTERARBEIT zur Erlangung des akademischen Grades Diplom-Ingenieur im Masterstudium INFORMATIK Eingereicht von: Igor Böhm, 0155477 Angefertigt am: Institut für System Software Betreuung: o.Univ.-Prof.Dipl.-Ing. Dr. Dr.h.c. Hanspeter Mössenböck Linz, Oktober 2007 Abstract Building compiler back ends from declarative specifications that map tree structured intermediate representations onto target machine code is the topic of this thesis. Although many tools and approaches have been devised to tackle the problem of automated code generation, there is still room for improvement. In this context we present Hburg, an implementation of a code generator generator that emits compiler back ends from concise tree pattern specifications written in our code generator description language. The language features attribute grammar style specifications and allows for great flexibility with respect to the placement of semantic actions. Our main contribution is to show that these language features can be integrated into automatically generated code generators that perform optimal instruction selection based on tree pattern matching combined with dynamic program- ming. In order to substantiate claims about the usefulness of our language we provide two complete examples that demonstrate how to specify code generators for Risc and Cisc architectures. Kurzfassung Diese Diplomarbeit beschreibt Hburg, ein Werkzeug das aus einer Spezi- fikation des abstrakten Syntaxbaums eines Programms und der Spezifika- tion der gewuns¨ chten Zielmaschine automatisch einen Codegenerator fur¨ diese Maschine erzeugt. Abbildungen zwischen abstrakten Syntaxb¨aumen und einer Zielmaschine werden durch Baummuster definiert. Fur¨ diesen Zweck haben wir eine deklarative Beschreibungssprache entwickelt, die es erm¨oglicht den Baummustern Attribute beizugeben, wodurch diese gleich- sam parametrisiert werden k¨onnen. -
Cooperative and Adaptive Algorithms Lecture 6 Allaa (Ella) Hilal, Spring 2017 May, 2017 1 Minute Quiz (Ungraded)
Cooperative and Adaptive Algorithms Lecture 6 Allaa (Ella) Hilal, Spring 2017 May, 2017 1 Minute Quiz (Ungraded) • Select if these statement are true (T) or false (F): Statement T/F Reason Uniform-cost search is a special case of Breadth- first search Breadth-first search, depth- first search and uniform- cost search are special cases of best- first search. A* is a special case of uniform-cost search. ECE457A, Dr. Allaa Hilal, Spring 2017 2 1 Minute Quiz (Ungraded) • Select if these statement are true (T) or false (F): Statement T/F Reason Uniform-cost search is a special case of Breadth- first F • Breadth- first search is a special case of Uniform- search cost search when all step costs are equal. Breadth-first search, depth- first search and uniform- T • Breadth-first search is best-first search with f(n) = cost search are special cases of best- first search. depth(n); • depth-first search is best-first search with f(n) = - depth(n); • uniform-cost search is best-first search with • f(n) = g(n). A* is a special case of uniform-cost search. F • Uniform-cost search is A* search with h(n) = 0. ECE457A, Dr. Allaa Hilal, Spring 2017 3 Informed Search Strategies Hill Climbing Search ECE457A, Dr. Allaa Hilal, Spring 2017 4 Hill Climbing Search • Tries to improve the efficiency of depth-first. • Informed depth-first algorithm. • An iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by incrementally changing a single element of the solution. • It sorts the successors of a node (according to their heuristic values) before adding them to the list to be expanded. -
Algorithmic System Design Under Consideration of Dynamic Processes
Algorithmic System Design under Consideration of Dynamic Processes Vom Fachbereich Maschinenbau an der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation vorgelegt von Dipl.-Phys. Lena Charlotte Altherr aus Karlsruhe Berichterstatter: Prof. Dr.-Ing. Peter F. Pelz Mitberichterstatter: Prof. Dr. rer. nat. Ulf Lorenz Tag der Einreichung: 19.04.2016 Tag der m¨undlichen Pr¨ufung: 24.05.2016 Darmstadt 2016 D 17 Vorwort des Herausgebers Kontext Die Produkt- und Systementwicklung hat die Aufgabe technische Systeme so zu gestalten, dass eine gewünschte Systemfunktion erfüllt wird. Mögliche System- funktionen sind z.B. Schwingungen zu dämpfen, Wasser in einem Siedlungsgebiet zu verteilen oder die Kühlung eines Rechenzentrums. Wir Ingenieure reduzieren dabei die Komplexität eines Systems, indem wir dieses gedanklich in überschaubare Baugruppen oder Komponenten zerlegen und diese für sich in Robustheit und Effizienz verbessern. In der Kriegsführung wurde dieses Prinzip bereits 500 v. Chr. als „Teile und herrsche Prinzip“ durch Meister Sun in seinem Buch „Die Kunst der Kriegsführung“ beschrieben. Das Denken in Schnitten ist wesentlich für das Verständnis von Systemen: „Das wichtigste Werkzeug des Ingenieurs ist die Schere“. Das Zusammenwirken der Komponenten führt anschließend zu der gewünschten Systemfunktion. Während die Funktion eines technischen Systems i.d.R. nicht verhan- delbar ist, ist jedoch verhandelbar mit welchem Aufwand diese erfüllt wird und mit welcher Verfügbarkeit sie gewährleistet wird. Aufwand und Verfügbarkeit sind dabei gegensätzlich. Der Aufwand bemisst z.B. die Emission von Kohlendioxid, den Energieverbrauch, den Materialverbrauch, … die „total cost of ownership“. Die Verfügbarkeit bemisst die Ausfallzeiten, Lebensdauer oder Laufleistung. Die Gesell- schaft stellt sich zunehmend die Frage, wie eine Funktion bei minimalem Aufwand und maximaler Verfügbarkeit realisiert werden kann. -
1 Introduction 2 Dijkstra's Algorithm
15-451/651: Design & Analysis of Algorithms September 3, 2019 Lecture #3: Dynamic Programming II last changed: August 30, 2019 In this lecture we continue our discussion of dynamic programming, focusing on using it for a variety of path-finding problems in graphs. Topics in this lecture include: • The Bellman-Ford algorithm for single-source (or single-sink) shortest paths. • Matrix-product algorithms for all-pairs shortest paths. • Algorithms for all-pairs shortest paths, including Floyd-Warshall and Johnson. • Dynamic programming for the Travelling Salesperson Problem (TSP). 1 Introduction As a reminder of basic terminology: a graph is a set of nodes or vertices, with edges between some of the nodes. We will use V to denote the set of vertices and E to denote the set of edges. If there is an edge between two vertices, we call them neighbors. The degree of a vertex is the number of neighbors it has. Unless otherwise specified, we will not allow self-loops or multi-edges (multiple edges between the same pair of nodes). As is standard with discussing graphs, we will use n = jV j, and m = jEj, and we will let V = f1; : : : ; ng. The above describes an undirected graph. In a directed graph, each edge now has a direction (and as we said earlier, we will sometimes call the edges in a directed graph arcs). For each node, we can now talk about out-neighbors (and out-degree) and in-neighbors (and in-degree). In a directed graph you may have both an edge from u to v and an edge from v to u. -
Backtracking / Branch-And-Bound
Backtracking / Branch-and-Bound Optimisation problems are problems that have several valid solutions; the challenge is to find an optimal solution. How optimal is defined, depends on the particular problem. Examples of optimisation problems are: Traveling Salesman Problem (TSP). We are given a set of n cities, with the distances between all cities. A traveling salesman, who is currently staying in one of the cities, wants to visit all other cities and then return to his starting point, and he is wondering how to do this. Any tour of all cities would be a valid solution to his problem, but our traveling salesman does not want to waste time: he wants to find a tour that visits all cities and has the smallest possible length of all such tours. So in this case, optimal means: having the smallest possible length. 1-Dimensional Clustering. We are given a sorted list x1; : : : ; xn of n numbers, and an integer k between 1 and n. The problem is to divide the numbers into k subsets of consecutive numbers (clusters) in the best possible way. A valid solution is now a division into k clusters, and an optimal solution is one that has the nicest clusters. We will define this problem more precisely later. Set Partition. We are given a set V of n objects, each having a certain cost, and we want to divide these objects among two people in the fairest possible way. In other words, we are looking for a subdivision of V into two subsets V1 and V2 such that X X cost(v) − cost(v) v2V1 v2V2 is as small as possible. -
Language and Compiler Support for Dynamic Code Generation by Massimiliano A
Language and Compiler Support for Dynamic Code Generation by Massimiliano A. Poletto S.B., Massachusetts Institute of Technology (1995) M.Eng., Massachusetts Institute of Technology (1995) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1999 © Massachusetts Institute of Technology 1999. All rights reserved. A u th or ............................................................................ Department of Electrical Engineering and Computer Science June 23, 1999 Certified by...............,. ...*V .,., . .* N . .. .*. *.* . -. *.... M. Frans Kaashoek Associate Pro essor of Electrical Engineering and Computer Science Thesis Supervisor A ccepted by ................ ..... ............ ............................. Arthur C. Smith Chairman, Departmental CommitteA on Graduate Students me 2 Language and Compiler Support for Dynamic Code Generation by Massimiliano A. Poletto Submitted to the Department of Electrical Engineering and Computer Science on June 23, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Dynamic code generation, also called run-time code generation or dynamic compilation, is the cre- ation of executable code for an application while that application is running. Dynamic compilation can significantly improve the performance of software by giving the compiler access to run-time infor- mation that is not available to a traditional static compiler. A well-designed programming interface to dynamic compilation can also simplify the creation of important classes of computer programs. Until recently, however, no system combined efficient dynamic generation of high-performance code with a powerful and portable language interface. This thesis describes a system that meets these requirements, and discusses several applications of dynamic compilation. -
Bellman-Ford Algorithm
The many cases offinding shortest paths Dynamic programming Bellman-Ford algorithm We’ve already seen how to calculate the shortest path in an Tyler Moore unweighted graph (BFS traversal) We’ll now study how to compute the shortest path in different CSE 3353, SMU, Dallas, TX circumstances for weighted graphs Lecture 18 1 Single-source shortest path on a weighted DAG 2 Single-source shortest path on a weighted graph with nonnegative weights (Dijkstra’s algorithm) 3 Single-source shortest path on a weighted graph including negative weights (Bellman-Ford algorithm) Some slides created by or adapted from Dr. Kevin Wayne. For more information see http://www.cs.princeton.edu/~wayne/kleinberg-tardos. Some code reused from Python Algorithms by Magnus Lie Hetland. 2 / 13 �������������� Shortest path problem. Given a digraph ����������, with arbitrary edge 6. DYNAMIC PROGRAMMING II weights or costs ���, find cheapest path from node � to node �. ‣ sequence alignment ‣ Hirschberg's algorithm ‣ Bellman-Ford 1 -3 3 5 ‣ distance vector protocols 4 12 �������� 0 -5 ‣ negative cycles in a digraph 8 7 7 2 9 9 -1 1 11 ����������� 5 5 -3 13 4 10 6 ������������� ���������������������������������� 22 3 / 13 4 / 13 �������������������������������� ��������������� Dijkstra. Can fail if negative edge weights. Def. A negative cycle is a directed cycle such that the sum of its edge weights is negative. s 2 u 1 3 v -8 w 5 -3 -3 Reweighting. Adding a constant to every edge weight can fail. 4 -4 u 5 5 2 2 ���������������������� c(W) = ce < 0 t s e W 6 6 �∈ 3 3 0 v -3 w 23 24 5 / 13 6 / 13 ���������������������������������� ���������������������������������� Lemma 1.