1,472 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Scalable Distributed Algorithms for Size-Constrained Submodular Maximization in the MapReduce and Adaptive Complexity Models

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    Distributed maximization of a submodular function in the MapReduce model has received much attention, culminating in two frameworks that allow a centralized algorithm to be run in the MR setting without loss of approximation, as long as the centralized algorithm satisfies a certain consistency property - which had only been shown to be satisfied by the standard greedy and continous greedy algorithms. A separate line of work has studied parallelizability of submodular maximization in the adaptive complexity model, where each thread may have access to the entire ground set. For the size-constrained maximization of a monotone and submodular function, we show that several sublinearly adaptive algorithms satisfy the consistency property required to work in the MR setting, which yields highly practical parallelizable and distributed algorithms. Also, we develop the first linear-time distributed algorithm for this problem with constant MR rounds. Finally, we provide a method to increase the maximum cardinality constraint for MR algorithms at the cost of additional MR rounds.Comment: 45 pages, 6 figure

    Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

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    We address the computation of the degrees of minors of a noncommutative symbolic matrix of form A[c]:=k=1mAktckxk, A[c] := \sum_{k=1}^m A_k t^{c_k} x_k, where AkA_k are matrices over a field K\mathbb{K}, xix_i are noncommutative variables, ckc_k are integer weights, and tt is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of A[c]A[c] of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes

    Decision-making with gaussian processes: sampling strategies and monte carlo methods

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    We study Gaussian processes and their application to decision-making in the real world. We begin by reviewing the foundations of Bayesian decision theory and show how these ideas give rise to methods such as Bayesian optimization. We investigate practical techniques for carrying out these strategies, with an emphasis on estimating and maximizing acquisition functions. Finally, we introduce pathwise approaches to conditioning Gaussian processes and demonstrate key benefits for representing random variables in this manner.Open Acces

    Budget-Feasible Market Design for Biodiversity Conservation: Considering Incentives and Spatial Coordination

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    How to best incentivize farmers to conserve biodiversity on private land is an important policy question. Conservation auctions provide a mechanism to elicit farmers\u27 opportunity costs, but their design is challenging and often suffer from low participation due to strategic complexity. Conservation auctions should ideally be incentive-compatible, address spatial synergies that maximize biodiversity gains, and respect the predefined budget of the government. Recent advances in mechanism design suggest budget-feasible auctions, but little is known about average-case efficiency. Based on this line of research, we introduce an incentive-compatible conservation auction mechanism that considers the bid taker\u27s spatial synergies and respects budget. The results are compared against the celebrated Vickrey-Clarke-Groves mechanism. Our numerical results estimate the efficiency loss that can be expected for different assumptions on the synergistic values of the government. They provide evidence that budget-feasible mechanisms provide a new tool for policymakers in this domain

    Popularity Ratio Maximization: Surpassing Competitors through Influence Propagation

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    In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment (PA) model for popularity growth with the Independent Cascade (IC) model for influence propagation in social networks called PA-IC model. In PA-IC, a popular item and a novice item grab shares of popularity from the natural popularity growth via the PA model, while the novice item tries to gain extra popularity via influence cascade in a social network. The {\em popularity ratio} is defined as the ratio of the popularity measure between the novice item and the popular item. We formulate {\em Popularity Ratio Maximization (PRM)} as the problem of selecting seeds in multiple rounds to maximize the popularity ratio in the end. We analyze the popularity ratio and show that it is monotone but not submodular. To provide an effective solution, we devise a surrogate objective function and show that empirically it is very close to the original objective function while theoretically, it is monotone and submodular. We design two efficient algorithms, one for the overlapping influence and non-overlapping seeds (across rounds) setting and the other for the non-overlapping influence and overlapping seed setting, and further discuss how to deal with other models and problem variants. Our empirical evaluation further demonstrates that the proposed PRM-IMM method consistently achieves the best popularity promotion compared to other methods. Our theoretical and empirical analyses shed light on the interplay between influence maximization and preferential attachment in social networks.Comment: 22 pages, 8 figures, to be appear SIGMOD 202

    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

    When Bidders Are DAOs

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    In a typical decentralized autonomous organization (DAO), people organize themselves into a group that is programmatically managed. DAOs can act as bidders in auctions, with a DAO's bid treated by the auctioneer as if it had been submitted by an individual, without regard to the internal structure of the DAO. We study auctions in which the bidders are DAOs. More precisely, we consider the design of two-level auctions in which the "participants" are groups of bidders rather than individuals. Bidders form DAOs to pool resources, but must then also negotiate the terms by which the DAO's winnings are shared. We model the outcome of a DAO's negotiations by an aggregation function (which aggregates DAO members' bids into a single group bid), and a budget-balanced cost-sharing mechanism (that determines DAO members' access to the DAO's allocation and distributes the total payment demanded from the DAO to its members). We pursue two-level mechanisms that are incentive-compatible (with truthful bidding a dominant strategy for members of each DAO) and approximately welfare-optimal. We prove that, even in the case of a single-item auction, incentive-compatible welfare maximization is not possible: No matter what the outer mechanism and the cost-sharing mechanisms used by DAOs, the welfare of the resulting two-level mechanism can be a lnn\approx \ln n factor less than optimal. We complement this lower bound with a natural two-level mechanism that achieves a matching approximate welfare guarantee. Our upper bound also extends to multi-item auctions where individuals have additive valuations. Finally, we show that our positive results cannot be extended much further: Even in multi-item settings with unit-demand bidders, truthful two-level mechanisms form a highly restricted class and as a consequence cannot guarantee any non-trivial approximation of the maximum social welfare

    Independent Sets in Elimination Graphs with a Submodular Objective

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    Maximum weight independent set (MWIS) admits a 1k\frac1k-approximation in inductively kk-independent graphs and a 12k\frac{1}{2k}-approximation in kk-perfectly orientable graphs. These are a a parameterized class of graphs that generalize kk-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph G=(V,E)G=(V,E) and a non-negative submodular function f:2VR+f: 2^V \rightarrow \mathbb{R}_+, the goal is to approximately solve maxSIGf(S)\max_{S \in \mathcal{I}_G} f(S) where IG\mathcal{I}_G is the set of independent sets of GG. We obtain an Ω(1k)\Omega(\frac1k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1e(k+1)\frac{1}{e(k+1)}. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively kk-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.Comment: Extended abstract to appear in Proceedings of APPROX 2023. v2 corrects technical typos in few place
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