255,847 research outputs found

    Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms

    Get PDF
    Consider the following 2-respecting min-cut problem. Given a weighted graph GG and its spanning tree TT, find the minimum cut among the cuts that contain at most two edges in TT. This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs. * An O(mlog2nloglogn+nlog6n)O(m\frac{\log^2 n}{\log\log n} + n\log^6 n)-time sequential algorithm: This improves Karger's O(mlog3n)O(m \log^3 n) and O(m(log2n)log(n2/m)loglogn+nlog6n)O(m\frac{(\log^2 n)\log (n^2/m)}{\log\log n} + n\log^6 n) bounds when the input graph is not extremely sparse or dense. Improvements over Karger's bounds were previously known only under a rather strong assumption that the input graph is simple [Henzinger et al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel edges, our bound can be improved to O(mlog1.5nloglogn+nlog6n)O(m\frac{\log^{1.5} n}{\log\log n} + n\log^6 n). * An algorithm requiring O~(n)\tilde O(n) cut queries to compute the min-cut of a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18, who obtained a similar bound for simple graphs. * A streaming algorithm that requires O~(n)\tilde O(n) space and O(logn)O(\log n) passes to compute the min-cut: The only previous non-trivial exact min-cut algorithm in this setting is the 2-pass O~(n)\tilde O(n)-space algorithm on simple graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19). In contrast to Karger's 2-respecting min-cut algorithm which deploys sophisticated dynamic programming techniques, our approach exploits some cute structural properties so that it only needs to compute the values of O~(n)\tilde O(n) cuts corresponding to removing O~(n)\tilde O(n) pairs of tree edges, an operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2; (2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1

    Multiway Cut, Pairwise Realizable Distributions, and Descending Thresholds

    Full text link
    We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013]. We proceed in three steps. First, we analyze the rounding scheme of Buchbinder et al., 2013 and design a modification that improves the approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a tight example showing that this is the best approximation one can achieve with the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by exponential clocks, and (2) single-coordinate cuts with equal thresholds. Then, we prove that this factor can be improved by introducing a new rounding scheme: (3) single-coordinate cuts with descending thresholds. By combining these three schemes, we design an algorithm that achieves a factor of (10 + 4 sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that we are able to verify by hand. Finally, we show that by combining these three rounding schemes with the scheme of independent thresholds from Karger et al., 2004, the approximation factor can be further improved to 1.2965. This approximation factor has been verified only by computer.Comment: This is an updated version and is the full version of STOC 2014 pape

    Approximate Graph Coloring by Semidefinite Programming

    Full text link
    We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on nn vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

    New Approximability Results for Two-Dimensional Bin Packing

    Get PDF
    We study the two-dimensional bin packing problem: Given a list of n rectangles the objective is to find a feasible, i.e. axis-parallel and non-overlapping, packing of all rectangles into the minimum number of unit sized squares, also called bins. Our problem consists of two versions; in the first version it is not allowed to rotate the rectangles while in the other it is allowed to rotate the rectangles by 90∘, i.e. to exchange the widths and the heights. Two-dimensional bin packing is a generalization of its one-dimensional counterpart and is therefore strongly NP-hard. Furthermore Bansal et al. showed that even an APTAS is ruled out for this problem, unless P=NP. This lower bound of asymptotic approximability was improved by Chlebik and Chlebikova to values 1+1/3792 and 1+1/2196 for the version with and without rotations, respectively. On the positive side there is an asymptotic 1.69.. approximation by Caprara without rotations and an asymptotic 1.52... approximation by Bansal et al.for both versions. We give a new asymptotic upper bound for both versions of our problem: For any fixed ε and any instance that fits optimally into OPT bins, our algorithm computes a packing into (3/2+ε)⋅OPT+69 bins in the version without rotations and (3/2+ε)⋅OPT+39 bins in the version with rotations. The algorithm has polynomial running time in the input length. In our new technique we consider an optimal packing of the rectangles into the bins. We cut a small vertical or horizontal strip out of each bin and move the intersecting rectangles into additional bins. This enables us to either round the widths of all wide rectangles, or the heights of all long rectangles in this bin. After this step we round the other unrounded side of these rectangles and we achieve a solution with a simple structure and only few types of rectangles. Our algorithm initially rounds the instance and computes a solution that nearly matches the modified optimal solution

    Algorithms and algorithmic obstacles for probabilistic combinatorial structures

    Get PDF
    Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 209-214).We study efficient average-case (approximation) algorithms for combinatorial optimization problems, as well as explore the algorithmic obstacles for a variety of discrete optimization problems arising in the theory of random graphs, statistics and machine learning. In particular, we consider the average-case optimization for three NP-hard combinatorial optimization problems: Large Submatrix Selection, Maximum Cut (Max-Cut) of a graph and Matrix Completion. The Large Submatrix Selection problem is to find a k x k submatrix of an n x n matrix with i.i.d. standard Gaussian entries, which has the largest average entry. It was shown in [13] using non-constructive methods that the largest average value of a k x k submatrix is 2(1 + o(1) [square root] log n/k with high probability (w.h.p.) when k = O(log n/ log log n). We show that a natural greedy algorithm called Largest Average Submatrix LAS produces a submatrix with average value (1+ o(1)) [square root] 2 log n/k w.h.p. when k is constant and n grows, namely approximately [square root] 2 smaller. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a k x k matrix with asymptotically the same average value (1+o(1) [square root] 2log n/k w.h.p., for k = o(log n). Since the maximum clique problem is a special case of the largest submatrix problem and the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor [square root] 2 performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a k x k matrix with average value (1 + o[subscript]k(1) + o(1))(4/3) [square root] 2log n/k for k = o((log n)¹.⁵), that is, with asymptotic factor 4/3 when k grows. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically (1 + o(1))[alpha][square root] 2 log n/k for a fixed value [alpha] [epsilon] [1, fixed value a E [1, [square root]2]. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at [alpha]* [delta]= 5[square root]2/(3[square root]3) ~~ 1.3608.. [epsilon] [4/3, [square root]2]: when [alpha] [alpha]*, appropriately defined. We conjecture that OGP observed for [alpha] > [alpha]* also marks the onset of the algorithmic hardness - no polynomial time algorithm exists for finding matrices with average value at least (1+o(1)[alpha][square root]2log n/k, when [alpha] > [alpha]* and k is a growing function of n. Finding a maximum cut of a graph is a well-known canonical NP-hard problem. We consider the problem of estimating the size of a maximum cut in a random Erdős-Rényi graph on n nodes and [cn] edges. We establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523[square root]c,c/2 + 0.55909[square root]c] w.h.p. as n increases, for all sufficiently large c. We observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved multi-dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2 + 0.47523[square root]c. We also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n. Matrix Completion is the problem of reconstructing a rank-k n x n matrix M from a sampling of its entries. We propose a new matrix completion algorithm using a novel sampling scheme based on a union of independent sparse random regular bipartite graphs. We show that under a certain incoherence assumption on M and for the case when both the rank and the condition number of M are bounded, w.h.p. our algorithm recovers an [epsilon]-approximation of M in terms of the Frobenius norm using O(nlog² (1/[epsilon])) samples and in linear time O(nlog² (1/[epsilon])). This provides the best known bounds both on the sample complexity and computational cost for reconstructing (approximately) an unknown low-rank matrix. The novelty of our algorithm is two new steps of thresholding singular values and rescaling singular vectors in the application of the "vanilla" alternating minimization algorithm. The structure of sparse random regular graphs is used heavily for controlling the impact of these regularization steps.by Quan Li.Ph. D

    Le problème du postier chinois cumulatif

    Get PDF
    Résumé Le sujet de cette thèse est le problème du postier chinois cumulatif (PPCC). Dans ce problème, nous considérons l'importance du moment où une arête est traitée complètement. Cette façon de procéder introduit un caractère cumulatif et dynamique dans le coût réel des arêtes, ce qui a pour effet de changer la structure du problème du postier chinois. Nous démontrons que ce problème est fortement NP-difficile et réductible à une version du problème de voyageur de commerce cumulatif. Ce problème est, à notre connaissance, nouveau. Nous continuons ici l'étude entreprise dans notre mémoire de maîtrise. Notre but dans cette thèse est de résoudre exactement ce problème à l'aide des outils de la programmation linéaire en nombres entiers. Notre contribution est de plusieurs ordres. Premièrement, nous développons une vingtaine de modèles différents. Dans cette thèse, nous étudions les huit meilleurs et les comparons aussi bien empiriquement que théoriquement entre eux et démontrons toutes les relations de dominance entre eux. L'aboutissement de nos recherches est le modèle L8. Deuxièmement, nous résolvons ce modèle L8 à l'aide d'un algorithme de séparation et évaluation progressive (Branch and Cut - algorithme BC1). Nous développons plusieurs outils dont nous présentons ici trois branchements, sept pré-traitements, six familles de coupes dont trois que nous généralisons. Ces outils nous permettent déjà de battre le solveur CPLEX par un facteur de 3 à 58 sur nos graphes de référence. Troisièmement, nous développons une meilleure variante du modèle L8 : le modèle L8+ et utilisons une approche avec génération de colonnes (Branch, Price and Cut – algorithme BPC1). Dans la foulée, nous développons cinq familles de coupes et nous généralisons quatre d'entre-elles. Cette nouvelle approche, plus rapide que la première d'un facteur de 2 à 4, nous permet d'être de 2 à 133 fois plus rapide que le solveur CPLEX en utilisant le modèle L8+ sur nos graphes de référence. Quatrièmement, nous améliorons notre approche de génération de colonnes (Branch, Price and Cut – algorithme BPC2) avec une évaluation implicite du dual. Les plus grandes instances du PPCC que nous arrivons à résoudre dans un délai maximal d'une heure comprennent des graphes de 11 sommets et/ou de 55 arêtes, ce qui correspond approximativement à des instances du problème du voyageur de commerce cumulatif à 110 sommets. Mots clefs Tournées sur les arcs, fonction cumulative, problème du postier chinois cumulatif.----------Abstract The subject of this Ph.D. thesis is the Cumulative Chinese Postman Problem (CCPP). We focus on the delay of the service of each arc. This introduces a cumulative and dynamic aspect in the objective function therefore changing the structure of the Chinese Postman Problem. We prove that this problem is strongly NP-hard and reducible to a version of the Cumulative Travelling Salesman Problem. This problem is, to our knowledge, entirely new. The study of this problem was initiated in our master thesis. Our main goal in this thesis is to solve this problem exactly with the help of the tools of linear integer programming. Our contribution is manifold. First, we develop twenty different models. However, in this thesis, we only discuss and compare theoretically and experimentally the best eight models. We prove all dominance relations among them. Model L8 stands out as the best model. Secondly, we solve this model L8 with a Branch and Cut (algorithm BC1). Throughout our study, we develop several tools among which three branching rules, seven presolving algorithms, six families of cuts (three of them generalized). These tools alone allow us to solve the problem faster than CPLEX by a factor of 3 to 58 on our test graphs. Thirdly, we develop an improved model L8+ and use a column generation approach - a Branch, Price and Cut (algorithm BPC1). We also develop five new families of cuts (four of them generalized). This new approach is faster than the previous one by a factor of 2 to 4 and is faster than CPLEX with the new model L8+ by a factor of 2 to 133 on our test graphs. Fourthly, we improve our Branch, Price and Cut algorithm (algorithm BPC2) by using an implicit evaluation of the dual. The largest instances for which we are able to solve the CCPP in less than one hour include graphs with 11 nodes and/or 55 edges which correspond approximately to instances of the Cumulative Travelling Salesman Problem with 110 nodes

    Competent genetic-evolutionary optimization of water distribution systems

    Get PDF
    A genetic algorithm has been applied to the optimal design and rehabilitation of a water distribution system. Many of the previous applications have been limited to small water distribution systems, where the computer time used for solving the problem has been relatively small. In order to apply genetic and evolutionary optimization technique to a large-scale water distribution system, this paper employs one of competent genetic-evolutionary algorithms - a messy genetic algorithm to enhance the efficiency of an optimization procedure. A maximum flexibility is ensured by the formulation of a string and solution representation scheme, a fitness definition, and the integration of a well-developed hydraulic network solver that facilitate the application of a genetic algorithm to the optimization of a water distribution system. Two benchmark problems of water pipeline design and a real water distribution system are presented to demonstrate the application of the improved technique. The results obtained show that the number of the design trials required by the messy genetic algorithm is consistently fewer than the other genetic algorithms

    Directed Multicut with linearly ordered terminals

    Full text link
    Motivated by an application in network security, we investigate the following "linear" case of Directed Mutlicut. Let GG be a directed graph which includes some distinguished vertices t1,,tkt_1, \ldots, t_k. What is the size of the smallest edge cut which eliminates all paths from tit_i to tjt_j for all i<ji < j? We show that this problem is fixed-parameter tractable when parametrized in the cutset size pp via an algorithm running in O(4ppn4)O(4^p p n^4) time.Comment: 12 pages, 1 figur
    corecore