When an infectious disease spreads, how to quickly vaccinate with a limited budget per time step to reduce the impact of the virus is very important. Specifically, vaccination will be carried out in every time step, a...
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We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function f by a linear combination fn of n elements from a dictionary. When the target function is containe...
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Taking a good decision about the assignment of each job becomes crucial in the era of hybrid and heterogeneous computing systems, such as personal machines equipped with CPUs and GPUs, or HPC platforms composed of mul...
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ISBN:
(纸本)9783031695766;9783031695773
Taking a good decision about the assignment of each job becomes crucial in the era of hybrid and heterogeneous computing systems, such as personal machines equipped with CPUs and GPUs, or HPC platforms composed of multiple generations of processors. In this study, we focus on the fundamental makespan minimization problem of scheduling jobs subject to precedence constraints on a platform composed of q different families of machines. Each family is constituted of identical parallel machines. The processing time of a job depends on the family where it is allocated. We propose an algorithm that guarantees an approximation ratio of q + 1 + 2 root q - 1, which improves upon the existing upper bound of q(q+1). In particular, this algorithm achieves a ratio of 5 in the special case of a machine composed only of CPUs and GPUs. This specific scenario with q = 2 families of machines has been widely studied by the scientific community in recent years. The best known lower and upper bounds known so far were 3 and 5.83, respectively.
Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weig...
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ISBN:
(纸本)9781611977929
Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weighted, axis-aligned rectangles such that the total weight of the rectangles in the cover is minimized. Already the unit-weight case is known to be NP-hard and the general problem has, to the best of our knowledge, not been studied experimentally before. We show a new basic property of optimal solutions of the weighted problem. This allows us to speed up existing algorithms for the unit-weight case, obtain an improved ILP formulation for both the weighted and unweighted problem, and develop several approximation algorithms and heuristics for the weighted case. All our algorithms are evaluated in a large experimental study on 186 837 polygons combined with six cost functions, which provides evidence that our algorithms are both fast and yield close-to-optimal solutions in practice.
Expectation propagation (EP) achieves near-optimal performance for large-scale multiple-input multiple-output (L-MIMO) detection, however, at the expense of unaffordable matrix inversions. To tackle the issue, several...
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Expectation propagation (EP) achieves near-optimal performance for large-scale multiple-input multiple-output (L-MIMO) detection, however, at the expense of unaffordable matrix inversions. To tackle the issue, several low-complexity EP detectors have been proposed. However, they all fail to exploit the properties of channel matrices, thus resulting in unsatisfactory performance in non-ideal scenarios. To this end, in this paper, a block-diagonal Neumann-series-based expectation propagation approximation (BD-NS-EPA) algorithm is proposed, which is applicable for both ideal uncorrelated channels and the correlated channels with multiple-antenna user equipment system. First, a block-diagonal-based Neumann iteration is employed, which skillfully exerts the main information of the channels while reducing computational cost. An adjustable sorting message updating scheme then is introduced to reduce the update of redundant nodes during iterations. Numerical results show that, for 128 & times;32 MIMO with the non-ideal channel, the proposed algorithm exhibits 0.3 dB away from the original EP when bit error-rate (BER) = 10(& minus;3), at the cost of mere 3% normalized complexity. The implementation results on SMIC 65-nm CMOS technology suggest that the proposed detector can achieve 1.252 Gbps/W and 0.275 Mbps/kGE hardware efficiency, further demonstrating that the proposed detectors can achieve a good trade-off between error-rate performance and hardware efficiency.
The feedback arc set problem is one of the most fundamental and well-studied ranking problems where n objects are to be ordered based on their pairwise comparison. The problem enjoys several efficient approximation al...
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ISBN:
(纸本)1577358872
The feedback arc set problem is one of the most fundamental and well-studied ranking problems where n objects are to be ordered based on their pairwise comparison. The problem enjoys several efficient approximation algorithms in the offline setting. Unfortunately, online there are strong lower bounds on the competitive ratio establishing that no algorithm can perform well in the worst case. This paper introduces a new beyond-worst-case model for online feedback arc set. In the model, a sample of the input is given to the algorithm offline before the remaining instance is revealed online. This models the case in practice where yesterday's data is available and is similar to today's online instance. This sample is drawn from a known distribution which may not be uniform. We design an online algorithm with strong theoretical guarantees. The algorithm has a small constant competitive ratio when the sample is uniform-if not, we show we can recover the same result by adding a provably minimal sample. Empirical results validate the theory and show that such algorithms can be used on temporal data to obtain strong results.
Counterfactual explanations shed light on the decisions of black-box models by explaining how an input can be altered to obtain a favourable decision from the model (e.g., when a loan application has been rejected). H...
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ISBN:
(纸本)1577358872
Counterfactual explanations shed light on the decisions of black-box models by explaining how an input can be altered to obtain a favourable decision from the model (e.g., when a loan application has been rejected). However, as noted recently, counterfactual explainers may lack robustness in the sense that a minor change in the input can cause a major change in the explanation. This can cause confusion on the user side and open the door for adversarial attacks. In this paper, we study some sources of non-robustness. While there are fundamental reasons for why an explainer that returns a single counterfactual cannot be robust in all instances, we show that some interesting robustness guarantees can be given by reporting multiple rather than a single counterfactual. Unfortunately, the number of counterfactuals that need to be reported for the theoretical guarantees to hold can be prohibitively large. We therefore propose an approximation algorithm that uses a diversity criterion to select a feasible number of most relevant explanations and study its robustness empirically. Our experiments indicate that our method improves the state-of-the-art in generating robust explanations, while maintaining other desirable properties and providing competitive computational performance.
We provide a simple (1 - O(1/root k))-selectable Online Contention Resolution Scheme for k-uniform matroids against a fixed-order adversary. If A(i) and G(i) denote the set of selected elements and the set of realized...
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ISBN:
(纸本)9783959773096
We provide a simple (1 - O(1/root k))-selectable Online Contention Resolution Scheme for k-uniform matroids against a fixed-order adversary. If A(i) and G(i) denote the set of selected elements and the set of realized active elements among the first i (respectively), our algorithm selects with probability 1- 1/root k any active element i such that |A(i-1)|+ 1 <= (1-1/root k) center dot E[|G(i)|]+ root k. This implies a (1- O(1/root k)) prophet inequality against fixed-order adversaries for k-uniform matroids that is considerably simpler than previous algorithms [2, 4, 18]. We also prove that no OCRS can be (1 - Omega(root log k/k))-selectable for k-uniform matroids against an almighty adversary. This guarantee is matched by the (known) simple greedy algorithm that selects every active element with probability 1 - Theta(root log k/k) [17].
This letter presents PANTR, an efficient solver for nonconvex constrained optimization problems, that is well-suited as an inner solver for an augmented Lagrangian method. The proposed scheme combines forward-backward...
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This letter presents PANTR, an efficient solver for nonconvex constrained optimization problems, that is well-suited as an inner solver for an augmented Lagrangian method. The proposed scheme combines forward-backward iterations with solutions to trust-region subproblems: the former ensures global convergence, whereas the latter enables fast update directions. We discuss how the algorithm is able to exploit exact Hessian information of the smooth objective term through a linear Newton approximation, while benefiting from the structure of box-constraints or $\ell _{1}$ -regularization. An open-source C++ implementation of PANTR is made available as part of the NLP solver library ALPAQA. Finally, the effectiveness of the proposed method is demonstrated in nonlinear model predictive control applications.
In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla (FOCS 2002), the input is a complete graph where edges are labeled either + or -, and the goal is to find a partition of the vertice...
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ISBN:
(纸本)9798400703836
In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla (FOCS 2002), the input is a complete graph where edges are labeled either + or -, and the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the -edges within parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev (STOC 2015) gave a 2.06-approximation by providing a near-optimal rounding of the standard LP, and Cohen-Addad, Lee, Li, and Newman (FOCS 2022, 2023) finally bypassed the integrality gap of 2 for this LP giving a 1.73-approximation for the problem. While introducing new ideas for Correlation Clustering, their algorithm is more complicated than typical approximation algorithms in the following two aspects: (1) It is based on two different relaxations with separate rounding algorithms connected by the round-or-cut procedure. (2) Each of the rounding algorithms has to separately handle seemingly inevitable correlated rounding errors, coming from correlated rounding of Sherali-Adams and other strong LP relaxations. In order to create a simple and unified framework for Correlation Clustering similar to those for typical approximate optimization tasks, we propose the cluster LP as a strong linear program that might tightly capture the approximability of Correlation Clustering. It unifies all the previous relaxations for the problem. It is exponential-sized, but we show that it can be (1 + epsilon)-approximately solved in polynomial time for any epsilon > 0, providing the framework for designing rounding algorithms without worrying about correlated rounding errors;these errors are handled uniformly in solving the relaxation.
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