Combinatorial Optimization Top Ten List Discrete Mathematics 2000 As Selected by William R
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A Lagrangian Decomposition Approach Combined with Metaheuristics for the Knapsack Constrained Maximum Spanning Tree Problem
MASTERARBEIT A Lagrangian Decomposition Approach Combined with Metaheuristics for the Knapsack Constrained Maximum Spanning Tree Problem ausgeführt am Institut für Computergraphik und Algorithmen der Technischen Universität Wien unter der Anleitung von Univ.-Prof. Dipl.-Ing. Dr.techn. Günther Raidl und Univ.-Ass. Dipl.-Ing. Dr.techn. Jakob Puchinger durch Sandro Pirkwieser, Bakk.techn. Matrikelnummer 0116200 Simmeringer Hauptstraße 50/30, A-1110 Wien Datum Unterschrift Abstract This master’s thesis deals with solving the Knapsack Constrained Maximum Spanning Tree (KCMST) problem, which is a so far less addressed NP-hard combinatorial optimization problem belonging to the area of network design. Thereby sought is a spanning tree whose profit is maximal, but at the same time its total weight must not exceed a specified value. For this purpose a Lagrangian decomposition approach, which is a special variant of La- grangian relaxation, is applied to derive upper bounds. In the course of the application the problem is split up in two subproblems, which are likewise to be maximized but easier to solve on its own. The subgradient optimization method as well as the Volume Algorithm are deployed to solve the Lagrangian dual problem. To derive according lower bounds, i.e. feasible solutions, a simple Lagrangian heuristic is applied which is strengthened by a problem specific local search. Furthermore an Evolutionary Algorithm is presented which uses a suitable encoding for the solutions and appropriate operators, whereas the latter are able to utilize heuristics based on defined edge-profits. It is shown that simple edge-profits, derived straightforward from the data given by an instance, are of no benefit. -
Chapter 8 Stochastic Gradient / Subgradient Methods
Chapter 8 Stochastic gradient / subgradient methods Contents (class version) 8.0 Introduction........................................ 8.2 8.1 The subgradient method................................. 8.5 Subgradients and subdifferentials................................. 8.5 Properties of subdifferentials.................................... 8.7 Convergence of the subgradient method.............................. 8.10 8.2 Example: Hinge loss with 1-norm regularizer for binary classifier design...... 8.17 8.3 Incremental (sub)gradient method............................ 8.19 Incremental (sub)gradient method................................. 8.21 8.4 Stochastic gradient (SG) method............................. 8.23 SG update.............................................. 8.23 Stochastic gradient algorithm: convergence analysis....................... 8.26 Variance reduction: overview................................... 8.33 Momentum............................................. 8.35 Adaptive step-sizes......................................... 8.37 8.5 Example: X-ray CT reconstruction........................... 8.41 8.1 © J. Fessler, April 12, 2020, 17:55 (class version) 8.2 8.6 Summary.......................................... 8.50 8.0 Introduction This chapter describes two families of algorithms: • subgradient methods • stochastic gradient methods aka stochastic gradient descent methods Often we turn to these methods as a “last resort,” for applications where none of the methods discussed previously are suitable. Many machine learning applications, -
Parity Systems and the Delta-Matroid Intersection Problem
Parity Systems and the Delta-Matroid Intersection Problem Andr´eBouchet ∗ and Bill Jackson † Submitted: February 16, 1998; Accepted: September 3, 1999. Abstract We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lov´asz to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two delta- matroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lov´aszconcerning ‘double flowers’ and ‘projections’ carry over to parity systems. 1 Introduction: the delta-matroid intersec- tion problem A delta-matroid is a pair (V, ) with a finite set V and a nonempty collection of subsets of V , called theBfeasible sets or bases, satisfying the following axiom:B ∗D´epartement d’informatique, Universit´edu Maine, 72017 Le Mans Cedex, France. [email protected] †Department of Mathematical and Computing Sciences, Goldsmiths’ College, London SE14 6NW, England. [email protected] 1 the electronic journal of combinatorics 7 (2000), #R14 2 1.1 For B1 and B2 in and v1 in B1∆B2, there is v2 in B1∆B2 such that B B1∆ v1, v2 belongs to . { } B Here P ∆Q = (P Q) (Q P ) is the symmetric difference of two subsets P and Q of V . If X\ is a∪ subset\ of V and if we set ∆X = B∆X : B , then we note that (V, ∆X) is a new delta-matroid.B The{ transformation∈ B} (V, ) (V, ∆X) is calledB a twisting. -
Oasics-SOSA-2019-14.Pdf (0.4
Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph Michal Kotrbčík Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Martin Škoviera1 Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Abstract The maximum genus γM (G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least γM (G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong’s combinatorial characterisation of maximum genus. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms → Graph algorithms analysis, Mathematics of computing → Graph algorithms, Mathematics of computing → Graphs and surfaces Keywords and phrases maximum genus, embedding, graph, greedy algorithm Digital Object Identifier 10.4230/OASIcs.SOSA.2019.14 Acknowledgements The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions. 1 Introduction One of the paradigms in topological graph theory is the study of all surface embeddings of a given graph. The maximum genus γM (G) parameter of a graph G is then the maximum integer g such that G has a cellular embedding in the orientable surface of genus g.A result of Duke [12] implies that a graph G has a cellular embedding in the orientable surface of genus g if and only if γ(G) ≤ g ≤ γM (G) where γ(G) denotes the (minimum) genus of G. -
A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs
A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs Parinya Chalermsook Aalto University, Espoo, Finland parinya.chalermsook@aalto.fi Andreas Schmid Max Planck Institute for Informatics, Saarbrücken, Germany [email protected] Sumedha Uniyal Aalto University, Espoo, Finland sumedha.uniyal@aalto.fi Abstract A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least 1 a 6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing “dense planar structures” inside any graph: (i) 1 A 6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum 1 number of triangular faces; this improves upon the previous 11 -approximation; (ii) An alternate (and 4 arguably more illustrative) proof of the 9 approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind. 2012 ACM Subject Classification Mathematics of computing → Graph theory Keywords and phrases Graph Drawing, Matroid Matching, Maximum Planar Subgraph, Local Search Algorithms Digital Object Identifier 10.4230/LIPIcs.STACS.2019.19 Related Version Full Version: https://arxiv.org/abs/1804.03485. Funding Parinya Chalermsook: Part of this work was done while PC and AS were visiting the Simons Institute for the Theory of Computing. -
Subgradient Method
Subgradient Method Ryan Tibshirani Convex Optimization 10-725/36-725 Last last time: gradient descent Consider the problem min f(x) x n for f convex and differentiable, dom(f) = R . Gradient descent: (0) n choose initial x 2 R , repeat (k) (k−1) (k−1) x = x − tk · rf(x ); k = 1; 2; 3;::: Step sizes tk chosen to be fixed and small, or by backtracking line search If rf Lipschitz, gradient descent has convergence rate O(1/) Downsides: • Requires f differentiable this lecture • Can be slow to converge next lecture 2 Subgradient method n Now consider f convex, with dom(f) = R , but not necessarily differentiable Subgradient method: like gradient descent, but replacing gradients with subgradients. I.e., initialize x(0), repeat (k) (k−1) (k−1) x = x − tk · g ; k = 1; 2; 3;::: where g(k−1) 2 @f(x(k−1)), any subgradient of f at x(k−1) Subgradient method is not necessarily a descent method, so we (k) (0) (k) keep track of best iterate xbest among x ; : : : x so far, i.e., f(x(k) ) = min f(x(i)) best i=0;:::k 3 Outline Today: • How to choose step sizes • Convergence analysis • Intersection of sets • Stochastic subgradient method 4 Step size choices • Fixed step sizes: tk = t all k = 1; 2; 3;::: • Diminishing step sizes: choose to meet conditions 1 1 X 2 X tk < 1; tk = 1; k=1 k=1 i.e., square summable but not summable Important that step sizes go to zero, but not too fast Other options too, but important difference to gradient descent: all step sizes options are pre-specified, not adaptively computed 5 Convergence analysis n Assume that f convex, dom(f) = R , and also that f is Lipschitz continuous with constant G > 0, i.e., jf(x) − f(y)j ≤ Gkx − yk2 for all x; y Theorem: For a fixed step size t, subgradient method satisfies lim f(x(k) ) ≤ f ? + G2t=2 k!1 best Theorem: For diminishing step sizes, subgradient method sat- isfies lim f(x(k) ) = f ? k!1 best 6 Basic bound, and convergence rate (0) ? Letting R = kx − x k2, after k steps, we have the basic bound R2 + G2 Pk t2 f(x(k) ) − f(x?) ≤ i=1 i best Pk 2 i=1 ti Previous theorems follow from this. -
An Efficient Solution Methodology for Mixed-Integer Programming Problems Arising in Power Systems" (2016)
University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 12-15-2016 An Efficient Solution Methodology for Mixed- Integer Programming Problems Arising in Power Systems Mikhail Bragin University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Bragin, Mikhail, "An Efficient Solution Methodology for Mixed-Integer Programming Problems Arising in Power Systems" (2016). Doctoral Dissertations. 1318. https://opencommons.uconn.edu/dissertations/1318 An Efficient Solution Methodology for Mixed-Integer Programming Problems Arising in Power Systems Mikhail Bragin, PhD University of Connecticut, 2016 For many important mixed-integer programming (MIP) problems, the goal is to obtain near- optimal solutions with quantifiable quality in a computationally efficient manner (within, e.g., 5, 10 or 20 minutes). A traditional method to solve such problems has been Lagrangian relaxation, but the method suffers from zigzagging of multipliers and slow convergence. When solving mixed-integer linear programming (MILP) problems, the recently adopted branch-and-cut may also suffer from slow convergence because when the convex hull of the problems has complicated facial structures, facet- defining cuts are typically difficult to obtain, and the method relies mostly on time-consuming branching operations. In this thesis, the novel Surrogate Lagrangian Relaxation method is developed and its convergence is proved to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations a stepsizing formula, that guarantees convergence without requiring the optimal dual value, has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. -
Subgradient Methods
Subgradient Methods Stephen Boyd and Almir Mutapcic Notes for EE364b, Stanford University, Winter 2006-07 April 13, 2008 Contents 1 Introduction 2 2 Basic subgradient method 2 2.1 Negativesubgradientupdate. .... 2 2.2 Stepsizerules................................... 3 2.3 Convergenceresults.............................. .. 4 3 Convergence proof 4 3.1 Assumptions.................................... 4 3.2 Somebasicinequalities . ... 5 3.3 Aboundonthesuboptimalitybound . ... 7 3.4 Astoppingcriterion.............................. .. 8 3.5 Numericalexample ................................ 8 4 Alternating projections 9 4.1 Optimal step size choice when f ⋆ isknown................... 9 4.2 Finding a point in the intersection of convex sets . ......... 11 4.3 Solving convex inequalities . ..... 14 4.4 Positive semidefinite matrix completion . ........ 15 5 Projected subgradient method 16 5.1 Numericalexample ................................ 18 6 Projected subgradient for dual problem 18 6.1 Numericalexample ................................ 20 7 Subgradient method for constrained optimization 21 7.1 Numericalexample ................................ 24 8 Speeding up subgradient methods 24 1 1 Introduction The subgradient method is a very simple algorithm for minimizing a nondifferentiable convex function. The method looks very much like the ordinary gradient method for differentiable functions, but with several notable exceptions: The subgradient method applies directly to nondifferentiable f. • The step lengths are not chosen via a line search, as in the ordinary gradient method. • In the most common cases, the step lengths are fixed ahead of time. Unlike the ordinary gradient method, the subgradient method is not a descent method; • the function value can (and often does) increase. The subgradient method is readily extended to handle problems with constraints. Subgradient methods can be much slower than interior-point methods (or Newton’s method in the unconstrained case). -
1 Base Polytopes
6.883 Learning with Combinatorial Structure Note for Lecture 8 Authors: Hongyi Zhang A bit of history in combinatorial optimization. Last time we talked about the Lovász extension, which plays an important role in the optimization of submodular functions. Actually, despite the name Lovász, Jack Edmonds is another important figure who made huge contribution to this concept, and submodular optimization in general. In DISCML 20111, there was a brief introduction to Jack Edmonds’ contributions to mathematics and computer science by Jeff Bilmes. 1 Base polytopes 1.1 Examples of base polytopes Recall that in the last lecture we defined the base polytope BF of a submodular function, and showed that there is a greedy algorithm to solve the optimization problem max y>x y2BF and the solution is closely related to Lovász extension. Now we shall look at several other examples where interesting concepts turn out to be the base polytopes of some submodu- lar (or supermodular) functions. 1.1.1 Probability simplex Let the ground set be V such that jVj = n. Define F (S) = minfjSj; 1g;S ⊆ V, in last lecture we already saw that F (S) is submodular. It is easy to check that BF is a probability sim- plex. In fact, by definition ( ) X X BF = y : yi = F (V) and yi ≤ F (S) 8S ⊆ V i2V i2S ( n ) X = y : yi = 1 and yi ≥ 0; 8i i=1 n which is a standard (n−1)-simplex in R . 1http://las.ethz.ch/discml/discml11.html NIPS Workshop on Discrete Optimization in Ma- chine Learning 2011 1 1.1.2 Permutahedron PjSj Let the ground set be V such that jVj = n. -
Algebraic Algorithms in Combinatorial Optimization
Algebraic Algorithms in Combinatorial Optimization CHEUNG, Ho Yee A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Computer Science and Engineering The Chinese University of Hong Kong September 2011 Thesis/Assessment Committee Professor Shengyu Zhang (Chair) Professor Lap Chi Lau (Thesis Supervisor) Professor Andrej Bogdanov (Committee Member) Professor Satoru Iwata (External Examiner) Abstract In this thesis we extend the recent algebraic approach to design fast algorithms for two problems in combinatorial optimization. First we study the linear matroid parity problem, a common generalization of graph matching and linear matroid intersection, that has applications in various areas. We show that Harvey's algo- rithm for linear matroid intersection can be easily generalized to linear matroid parity. This gives an algorithm that is faster and simpler than previous known al- gorithms. For some graph problems that can be reduced to linear matroid parity, we again show that Harvey's algorithm for graph matching can be generalized to these problems to give faster algorithms. While linear matroid parity and some of its applications are challenging generalizations, our results show that the al- gebraic algorithmic framework can be adapted nicely to give faster and simpler algorithms in more general settings. Then we study the all pairs edge connectivity problem for directed graphs, where we would like to compute minimum s-t cut value between all pairs of vertices. Using a combinatorial approach it is not known how to solve this problem faster than computing the minimum s-t cut value for each pair of vertices separately. -
Matroid Matching: the Power of Local Search
Matroid Matching: the Power of Local Search [Extended Abstract] Jon Lee Maxim Sviridenko Jan Vondrák IBM Research IBM Research IBM Research Yorktown Heights, NY Yorktown Heights, NY San Jose, CA [email protected] [email protected] [email protected] ABSTRACT can be easily transformed into an NP-completeness proof We consider the classical matroid matching problem. Un- for a concrete class of matroids (see [46]). An important weighted matroid matching for linear matroids was solved result of Lov´asz is that (unweighted) matroid matching can by Lov´asz, and the problem is known to be intractable for be solved in polynomial time for linear matroids (see [35]). general matroids. We present a PTAS for unweighted ma- There have been several attempts to generalize Lov´asz' re- troid matching for general matroids. In contrast, we show sult to the weighted case. Polynomial-time algorithms are that natural LP relaxations have an Ω(n) integrality gap known for some special cases (see [49]), but for general linear and moreover, Ω(n) rounds of the Sherali-Adams hierarchy matroids there is only a pseudopolynomial-time randomized are necessary to bring the gap down to a constant. exact algorithm (see [8, 40]). More generally, for any fixed k ≥ 2 and > 0, we obtain a In this paper, we revisit the matroid matching problem (k=2 + )-approximation for matroid matching in k-uniform for general matroids. Our main result is that while LP- hypergraphs, also known as the matroid k-parity problem. based approaches including the Sherali-Adams hierarchy fail As a consequence, we obtain a (k=2 + )-approximation for to provide any meaningful approximation, a simple local- the problem of finding the maximum-cardinality set in the search algorithm gives a PTAS (in the unweighted case). -
On the Links Between Probabilistic Graphical Models and Submodular Optimisation Senanayak Sesh Kumar Karri
On the Links between Probabilistic Graphical Models and Submodular Optimisation Senanayak Sesh Kumar Karri To cite this version: Senanayak Sesh Kumar Karri. On the Links between Probabilistic Graphical Models and Submodular Optimisation. Machine Learning [cs.LG]. Université Paris sciences et lettres, 2016. English. NNT : 2016PSLEE047. tel-01753810 HAL Id: tel-01753810 https://tel.archives-ouvertes.fr/tel-01753810 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE` DE DOCTORAT de l’Universite´ de recherche Paris Sciences Lettres PSL Research University Prepar´ ee´ a` l’Ecole´ normale superieure´ On the Links between Probabilistic Graphical Models and Submodular Optimisation Liens entre modeles` graphiques probabilistes et optimisation sous-modulaire Ecole´ doctorale n◦386 ECOLE´ DOCTORALE DE SCIENCES MATHEMATIQUES´ DE PARIS CENTRE Specialit´ e´ INFORMATIQUE COMPOSITION DU JURY : M Andreas Krause ETH Zurich, Rapporteur M Nikos Komodakis ENPC Paris, Rapporteur M Francis Bach Inria Paris, Directeur de these` Soutenue par Senanayak Sesh Kumar KARRI le 27.09.2016 M Josef Sivic ENS Paris, Membre du Jury Dirigee´ par Francis BACH M Antonin Chambolle CMAP EP Paris, Membre du Jury M Guillaume Obozinski ENPC Paris, Membre du Jury RESEARCH UNIVERSITY PARIS ÉCOLENORMALE SUPÉRIEURE What is the purpose of life? Proof and Conjecture, ....