273 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Faster Discrete Convex Function Minimization with Predictions: The M-Convex Case

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    Recent years have seen a growing interest in accelerating optimization algorithms with machine-learned predictions. Sakaue and Oki (NeurIPS 2022) have developed a general framework that warm-starts the L-convex function minimization method with predictions, revealing the idea's usefulness for various discrete optimization problems. In this paper, we present a framework for using predictions to accelerate M-convex function minimization, thus complementing previous research and extending the range of discrete optimization algorithms that can benefit from predictions. Our framework is particularly effective for an important subclass called laminar convex minimization, which appears in many operations research applications. Our methods can improve time complexity bounds upon the best worst-case results by using predictions and even have potential to go beyond a lower-bound result

    On linear, fractional, and submodular optimization

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    In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree

    LP-based approximation algorithms for partial-ordered scheduling and matroid augmentation

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    In this thesis, we study two NP-hard problems from Combinatorial Optimization, from the perspective of approximation algorithms. The first problem we study is called Partial-Order Scheduling on Parallel Machines, which we abbreviate to PO Scheduling. Here, we are given a partially ordered set of jobs which we want to schedule to a set of machines. Each job has some weight and some processing time associated to it. On each machine, the order of the jobs scheduled to it must agree with the given partial order, i.e., a job can only be started once all its predecessors scheduled to the same machine have been completed. However, two jobs scheduled to different machines are not constrained in any way. Thus, PO Scheduling deviates from the well-studied problem of precedence-constrained scheduling in this regard. The goal of PO Scheduling is to find a feasible schedule which minimizes the sum of weighted completion times of the jobs. PO Scheduling generalizes an already NP-hard version of scheduling introduced by Bosman, Frascaria, Olver, Sitters and Stougie [3], where they ask the same question as in PO Scheduling for the case where the jobs are totally ordered. The authors above present a constant-factor approximation algorithm for their problem. We conjecture that there is a constant-factor approximation algorithm for PO Scheduling as well. While we do not solve the problem, we give approximation algorithms for the special case that the partial order consists of disjoint totally ordered chains of linearly bounded length. Additionally, we give a structural result for optimal schedules in the case that the partial order consists of disjoint, backwardly ordered (with regard to the Smith-ratio) chains. We point towards some potential research directions. For the Weighted Tree Augmentation Problem, we are given a graph with a distinguished spanning tree. Each non tree-edge has a cost associated to it. The goal is to find a cost-minimal set of edges such that when we add them to the tree-edges, the resulting graph is 2-edge-connected. Weighted tree augmentation is NP-hard. There has been recent progress in decreasing the best-known approximation factor for the problem by Traub and Zenklusen to (1.5 + ε) [51, 52]. We study a generalization of weighted tree augmentation, called the Weighted Matroid Augmentation Problem, which we abbreviate to WMAP. In WMAP, we consider a matroid with a distinguished basis and a cost function on the non-basis elements. The goal is to find a cost-minimal set such that the union of the fundamental circuits of the elements in the set with regard to the distinguished basis cover that basis. We conjecture that there is a 2-approximation algorithm for the problem in the case that the matroid is regular. While we do not solve the problem, we give an approximation algorithm for the special case of the cographic matroid and show that there is no constant-factor approximation algorithm for WMAP for representable matroids unless P = NP

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    The Predicted-Deletion Dynamic Model: Taking Advantage of ML Predictions, for Free

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    The main bottleneck in designing efficient dynamic algorithms is the unknown nature of the update sequence. In particular, there are some problems, like 3-vertex connectivity, planar digraph all pairs shortest paths, and others, where the separation in runtime between the best partially dynamic solutions and the best fully dynamic solutions is polynomial, sometimes even exponential. In this paper, we formulate the predicted-deletion dynamic model, motivated by a recent line of empirical work about predicting edge updates in dynamic graphs. In this model, edges are inserted and deleted online, and when an edge is inserted, it is accompanied by a "prediction" of its deletion time. This models real world settings where services may have access to historical data or other information about an input and can subsequently use such information make predictions about user behavior. The model is also of theoretical interest, as it interpolates between the partially dynamic and fully dynamic settings, and provides a natural extension of the algorithms with predictions paradigm to the dynamic setting. We give a novel framework for this model that "lifts" partially dynamic algorithms into the fully dynamic setting with little overhead. We use our framework to obtain improved efficiency bounds over the state-of-the-art dynamic algorithms for a variety of problems. In particular, we design algorithms that have amortized update time that scales with a partially dynamic algorithm, with high probability, when the predictions are of high quality. On the flip side, our algorithms do no worse than existing fully-dynamic algorithms when the predictions are of low quality. Furthermore, our algorithms exhibit a graceful trade-off between the two cases. Thus, we are able to take advantage of ML predictions asymptotically "for free.'

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    A Tight Competitive Ratio for Online Submodular Welfare Maximization

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    In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given nn bidders each equipped with a general (not necessarily monotone) submodular utility and mm items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of \nicefrac{1}{4}. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of 3−22≈0.1715733-2\sqrt{2}\approx 0.171573 to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of ≈0.27493\approx 0.27493, improving the previously known \nicefrac{1}{4} guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest

    A survey of parameterized algorithms and the complexity of edge modification

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    The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

    Coresets for Clustering with General Assignment Constraints

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    Designing small-sized \emph{coresets}, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We significantly extend and generalize the results of a very recent paper (Braverman et al., FOCS'22), by demonstrating that the idea of hierarchical uniform sampling (Chen, SICOMP'09; Braverman et al., FOCS'22) can be applied to efficiently construct coresets for a very general class of constrained clustering problems with general assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body B)\mathcal{B}). Our main theorem shows that a small-sized ϵ\epsilon-coreset exists as long as a complexity measure Lip(B)\mathsf{Lip}(\mathcal{B}) of the structure constraint, and the \emph{covering exponent} Λϵ(X)\Lambda_\epsilon(\mathcal{X}) for metric space (X,d)(\mathcal{X},d) are bounded. The complexity measure Lip(B)\mathsf{Lip}(\mathcal{B}) for convex body B\mathcal{B} is the Lipschitz constant of a certain transportation problem constrained in B\mathcal{B}, called \emph{optimal assignment transportation problem}. We prove nontrivial upper bounds of Lip(B)\mathsf{Lip}(\mathcal{B}) for various polytopes, including the general matroid basis polytopes, and laminar matroid polytopes (with better bound). As an application of our general theorem, we construct the first coreset for the fault-tolerant clustering problem (with or without capacity upper/lower bound) for the above metric spaces, in which the fault-tolerance requirement is captured by a uniform matroid basis polytope
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