The first moment and second central moments of the portfolio return, a.k.a. mean and variance, have been widely employed to assess the expected profit and risk of the portfolio. Investors pursue higher mean and lower ...
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The first moment and second central moments of the portfolio return, a.k.a. mean and variance, have been widely employed to assess the expected profit and risk of the portfolio. Investors pursue higher mean and lower variance when designing the portfolios. The two moments can well describe the distribution of the portfolio return when it follows the Gaussian distribution. However, the real world distribution of assets return is usually asymmetric and heavy-tailed, which is far from being a Gaussian distribution. The asymmetry and the heavy-tailedness are characterized by the third and fourth central moments, i.e., skewness and kurtosis, respectively. Higher skewness and lower kurtosis are preferred to reduce the probability of extreme losses. However, incorporating high-order moments in the portfolio design is very difficult due to their non-convexity and rapidly increasing computational cost with the dimension. In this paper, we propose a very efficient and convergence-provable algorithm framework based on the successive convex approximation (SCA) algorithm to solve high-order portfolios. The efficiency of the proposed algorithm framework is demonstrated by the numerical experiments.
We study a revenue maximization problem in the context of social networks. Namely, we generalize a model introduced by Alon, Mansour, and Tennenholtz [2] that captures inequity aversion, i.e., it captures the fact tha...
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We study a revenue maximization problem in the context of social networks. Namely, we generalize a model introduced by Alon, Mansour, and Tennenholtz [2] that captures inequity aversion, i.e., it captures the fact that prices offered to neighboring nodes should not differ significantly. We first provide approximation algorithms for a natural class of instances, where the total revenue is the sum of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for instance, to settings where the seller will only consider a fixed number of discount types or special offers. To complement our positive results, we resolve one of the open questions posed in [2] by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be large, or even when the number of allowed distinct prices is as small as three. Finally, we study extensions of the model regarding the demand type of the clients. (C) 2021 Elsevier B.V. All rights reserved.
Given a social network G, a cost associated with each user, and an influence threshold η, the minimum cost seed selection problem (MCSS) aims to find a set of seeds that minimizes the total cost to reach η users. Ex...
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Given a social network G, a cost associated with each user, and an influence threshold η, the minimum cost seed selection problem (MCSS) aims to find a set of seeds that minimizes the total cost to reach η users. Existing works are mainly devoted to providing an expected coverage guarantee on reaching η, classified as MCSS-ECG, where their solutions either rely on an impractical influence oracle or cannot attain the expected influence threshold. More importantly, due to the expected coverage guarantee, the actual influence in a campaign may drift from the threshold evidently. Thus, the advertisers would like to request for a probability guarantee of reaching η. This motivates us to further solve the MCSS problem with a probabilistic coverage guarantee, termed *** this paper, we first propose our algorithm CLEAR to solve MCSS-ECG, which reaches the expected influence threshold without any influence oracle or influence shortfall but a practical approximation ratio. However, the ratio involves an unknown term (i.e., the optimal cost). Thus, we further devise the STAR method to derive a lower bound of the optimal cost and then obtain the first explicit approximation ratio for MCSS-ECG. In MCSS-PCG, it is necessary to estimate the probability that the current seeds reach η, to decide when to stop seed selection. To achieve this, we design a new technique named MRR, which provides efficient probability estimation with a theoretical guarantee. With MRR in hand, we propose our algorithm SCORE for MCSS-PCG, whose performance guarantee is derived by measuring the gap between MCSS-ECG and MCSS-PCG, and applying the theoretical results in MCSS-ECG. Finally, extensive experiments demonstrate that our algorithms achieve up to two orders of magnitude speed-up compared to alternatives while meeting the requirement of MCSS-PCG with the smallest cost.
Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data. An important tool for data analysis is clustering. When t...
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Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data. An important tool for data analysis is clustering. When the data set is uncertain, we can model them as a set of probabilistic points each formalized as a probability distribution function which describes the possible locations of the points. In this paper, we study k-center problem for probabilistic points in a general metric space. First, we present a fast greedy approximation algorithm that builds k centers using a farthest-first traversal in k iterations. This algorithm improves the previous approximation factor of the unrestricted assigned k-center problem from 10 (see [1]) to 6. Next, we restrict the centers to be selected from all the probabilistic locations of the given points and we show that an optimal solution for this restricted setting is a 2-approximation factor solution for an optimal solution of the assigned k-center problem with expected distance assignment. Using this idea, we improve the approximation factor of the unrestricted assigned k-center problem to 4 by increasing the running time. The algorithm also runs in polynomial time when k is a constant. Additionally, we implement our algorithms on three real data sets. The experimental results show that in practice the approximation factors of our algorithms are better than in theory for these data sets. Also we compare the results of our algorithm with the previous works and discuss about the achieved results. At the end, we present our theoretical results for probabilistic k-median clustering.
In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k +...
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In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial (3/2 + epsilon) - approximation for CycAP, for any constant > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.
A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of findin...
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A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the Top-k-Overlapping Densest Subgraphs problem. Given an integer k >= 1 and a parameter lambda > 0, the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter lambda and the distance between each pair of subgraphs in the solution. The Top-k-Overlapping Densest Subgraphs problem has been shown to admit a 1/10-factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to 1/2, when k is smaller than the number of vertices in the graph, and to 2/3, when k is a constant. For the computational complexity, we show that the problem is NP-hard even when k = 3.
We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the 'winding'...
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We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the 'winding' technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. algorithms (SODA16), 514-527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.
We study the approximability of two related problems on graphs with n nodes and m edges: n-Pairs Shortest Paths (n-PSP), where the goal is to find a shortest path between O(n) prespecified pairs, and All Node Shortest...
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We study the approximability of two related problems on graphs with n nodes and m edges: n-Pairs Shortest Paths (n-PSP), where the goal is to find a shortest path between O(n) prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate n-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate n-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of n-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer κ ≥ 1, under the combinatorial 4κ-clique hypothesis, there is no combinatorial algorithm for unweighted undirected n-PSP with approximation ratio better than 1 + 1/κ that runs in O(m2−2/(κ+1)n1/(κ+1)−Ε) time. This nearly matches an upper bound implied by the result of Agarwal (2014). Our algorithms use a surprisingly wide range of techniques, including techniques from the girth problem, distance oracles, approximate APSP, spanners, fault-tolerant spanners, and link-cut trees. A highlight of our algorithmic results is that one can solve both n-PSP and ANSC in Õ(m + n3/2+Ε) time1 with approximation factor 2 + Ε (and additive error that is function of Ε), for any constant Ε > 0. For n-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for n-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms. Additionally, for ANSC, for all integers κ ≥ 1, we extend the very recent almost k-approximation algorithm for the girth problem
Let P be a set of points in Rd (or some other metric space), where each point p ∈ P has an associated transmission range, denoted ρ(p). The range assignment ρ induces a directed communication graph Gρ(P) on P, whi...
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ISBN:
(纸本)9783959772365
Let P be a set of points in Rd (or some other metric space), where each point p ∈ P has an associated transmission range, denoted ρ(p). The range assignment ρ induces a directed communication graph Gρ(P) on P, which contains an edge (p, q) iff |pq| ≤ ρ(p). In the broadcast range-assignment problem, the goal is to assign the ranges such that Gρ(P) contains an arborescence rooted at a designated root node and the cost Σ p∈P ρ(p)2 of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution - the number of ranges that are modified when a point is inserted into or deleted from P - and its approximation ratio. To this end we introduce the concept of k-stable algorithms, which are algorithms that modify the range of at most k points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm alg that, for any given fixed parameter ϵ > 0, is k(ϵ)-stable and that maintains a solution with approximation ratio 1+ϵ, where the stability parameter k(ϵ) only depends on ϵ and not on the size of P. We study such trade-offs in three settings. For the problem in R1, we present a SAS with k(ϵ) = O(1/ϵ). Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have k(ϵ) = ω(1/ϵ). We also present algorithms with very small stability parameters: a 1-stable (6 + 2√5)-approximation algorithm - this algorithm can only handle insertions - a (trivial) 2-stable 2-approximation algorithm, and a 3-stable 1.97-approximation algorithm. For the problem in S1 (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in S1, we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in R1. For the problem in R2, we also prove that no SAS exists, an
The Dyck language, which consists of well-balanced sequences of parentheses, is one of the most fundamental context-free languages. The Dyck edit distance quantifies the number of edits (character insertions, deletion...
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ISBN:
(纸本)9783959772358
The Dyck language, which consists of well-balanced sequences of parentheses, is one of the most fundamental context-free languages. The Dyck edit distance quantifies the number of edits (character insertions, deletions, and substitutions) required to make a given length-n parenthesis sequence well-balanced. RNA Folding involves a similar problem, where a closing parenthesis can match an opening parenthesis of the same type irrespective of their ordering. For example, in RNA Folding, both () and)(are valid matches, whereas the Dyck language only allows () as a match. Both of these problems have been studied extensively in the literature. Using fast matrix multiplication, it is possible to compute their exact solutions in time O(n2.687) (Chi, Duan, Xie, Zhang, STOC'22), and a (1 + ϵ)-multiplicative approximation is known with a running time of Ω(n2.372). The impracticality of fast matrix multiplication often makes combinatorial algorithms much more desirable. Unfortunately, it is known that the problems of (exactly) computing the Dyck edit distance and the folding distance are at least as hard as Boolean matrix multiplication. Thereby, they are unlikely to admit truly subcubic-time combinatorial algorithms. In terms of fast approximation algorithms that are combinatorial in nature, the state of the art for Dyck edit distance is an O(log n)-factor approximation algorithm that runs in near-linear time (Saha, FOCS'14), whereas for RNA Folding only an ϵn-additive approximation in (Equation presented) time (Saha, FOCS'17) is known. In this paper, we make substantial improvements to the state of the art for Dyck edit distance (with any number of parenthesis types). We design a constant-factor approximation algorithm that runs in Õ(n1.971) time (the first constant-factor approximation in subquadratic time). Moreover, we develop a (1 + ϵ)-factor approximation algorithm running in (Equation presented) time, which improves upon the earlier additive approximation. Finally, we de
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