Increasing global competition, rapidly changing markets, and greater consumer awareness have altered the way in which corporations do business, To become more efficient, many industries have sought to model some opera...
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Increasing global competition, rapidly changing markets, and greater consumer awareness have altered the way in which corporations do business, To become more efficient, many industries have sought to model some operational aspects by gigantic optimization problems, It is not atypical to encounter models that capture 10(6) separate ''yes'' or ''no'' decisions to be made, Although one could, in principle, try all 2(106) possible solutions to find the optimal one, such a method would be impractically slow. Unfortunately, for most of these models, no algorithms are known that find optimal solutions with reasonable computation times, Typically, industry must rely on solutions of unguaranteed quality that are constructed in an ad hoc manner, Fortunately, for some of these models there are good approximation algorithms: algorithms that produce solutions quickly that are provably close to optimal, Over the past 6 years, there has been a sequence of major breakthroughs in our understanding of the design of approximation algorithms and of limits to obtaining such performance guarantees;this area has been one of the most flourishing areas of discrete mathematics and theoretical computer science.
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homol...
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Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in nearlinear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we present approximation algorithms for theAirport and Railwayproblem (AR) on several classes of graphs. The AR problem, introduced as reported byAdamaszek et al. (in: Ollinger, Vollmer (eds) 33rd sympo...
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In this paper, we present approximation algorithms for theAirport and Railwayproblem (AR) on several classes of graphs. The AR problem, introduced as reported byAdamaszek et al. (in: Ollinger, Vollmer (eds) 33rd symposium on theoretical aspectsof computer science (STACS 2016). Leibniz international proceedings in informatics(LIPIcs), Dagstuhl, 2016), is a combination of theCapacitated Facility Loca-tionproblem (CFL) and theNetwork Design Problem(NDP). An AR instanceconsists of a set of points (cities)Vin a metricd(., .), each of which is associatedwith a non-negative costfvand a numberk, which represent respectively the cost ofestablishing an airport (facility) in the corresponding point, and the universal airportcapacity. A feasible solution is a network of airports and railways providing servicesto all cities without violating any capacity, where railways are edges connecting pairsof points, with their costs equivalent to the distance between the respective *** objective is to find such a network with the least cost. In other words, find aforest, each component having at mostkpoints and one open facility, minimizing thetotal cost of edges and airport opening costs. As reported by Adamaszek et al. (in:Ollinger, Vollmer (eds) 33rd symposium on theoretical aspects of computer science(STACS 2016). Leibniz international proceedings in informatics (LIPIcs), Dagstuhl,2016) presented a PTAS for AR in the two-dimensional Euclidean metricR2witha uniform opening cost. In subsequent work (as reported by Adamaszek et al. (in:Niedermeier, Vall & eacute;e (eds) 35th symposium on theoretical aspects of computer science(STACS 2018). Leibniz international proceedings in informatics (LIPIcs), Dagstuhl,2018).) presented a bicriteria43(2+1 alpha)-approximation algorithm for AR with non-uniform opening costs but violating the airport capacity by a factor of 1 + alpha, i.e.(1+alpha)k capacity where 0 < alpha <= 1, a(2 + k/k-1 + epsilon)-approximation algorithm and a bicrite-ria Q
Given an edge-weighted (metric/general) complete graph with n vertices, where n mod k = 0, maximum weight (metric/general) k-cycle/path partition problem is to find a set of nk vertex disjoint k-cycles/paths such that...
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Given an edge-weighted (metric/general) complete graph with n vertices, where n mod k = 0, maximum weight (metric/general) k-cycle/path partition problem is to find a set of nk vertex disjoint k-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric k-cycle partition, we improve the previous approximation ratio from 35 to 710 for k = 5, and from 78 (1- 1 /k ) (2) for k >5 to ( 7 /8 - 1/8k )(1 -1/k) for constant odd k> 5and to 78 (1- 1/k + k(k-1) )for even k> 5. For metric k-path partition, we improve approximation ratio from 7 8 (1 - 1k )to 27k(2)-48k+16 /32k(2)-36k-24 for k is an element of {6, 8,10}. For the case of k = 4, we improve the approximation ratio from 43 to 56 for metric 4-cycle partition, from 2/3 to 3/4 for general 4-cycle partition, and from 3 /4 to 14 /17 for metric 4-path partition.
In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph G = (V, E;w, f;r) and k construction teams, ...
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In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph G = (V, E;w, f;r) and k construction teams, having nonuniform construction speeds lambda(1), lambda(2), ... , lambda(k), where r is an element of V is a fixed common root, w : E -> R+ is an edge-weight function, satisfying the triangle inequality, and f: V -> R-0(+) (i.e., R+ boolean OR approximation) is a vertex-weight function with f (r) = 0, we are asked to find k trees for these k construction teams, each tree having the same root r, and collectively covering all vertices in V, the objective is to minimize the maximum completion time of k construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting k paths for k trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant delta > 0, we first design a 58.3286(1 + delta)-approximation algorithm to solve the HRTC problem, and this algorithm runs in time O(n(2)(n + log n/delta) +log(w (E) + f (V))). Meanwhile, we present a simple 116.6572(1 + delta)-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a max{2 rho, 2 + rho - 2/k}-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time O(n(2)), where rho is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding max{2 rho, 2 + rho - 2/k}-approximation algorithm also resolves the HRPC problem.
In this paper, we have a set of weighted linear equalities and inequalities of the form A(1)x equivalent to 0(mod p), A2x not equivalent to 0(mod p) where all entries of A(1) and A(2) are in {- 1, 0, 1}. The objective...
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ISBN:
(纸本)9789819614899;9789819614905
In this paper, we have a set of weighted linear equalities and inequalities of the form A(1)x equivalent to 0(mod p), A2x not equivalent to 0(mod p) where all entries of A(1) and A(2) are in {- 1, 0, 1}. The objective is to assign each x(i) to Z(p) = {0,..., p-1} to maximize the total weight of the satisfied equalities and inequalities. This problem is a generalization of k-Correlation Clustering problem. We design an approximation algorithm with the approximation ratio max{a, (2-p)a+p-1/p}, where a is the weighted proportion of equalities in all equalities and inequalities. As a varies from 0 to 1, the approximation ratio varies from p-1/p to 1 and the minimum value is 1/2 when a is 1/2.
A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708–717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio 2, by a primal-d...
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ISBN:
(纸本)9783031813955
A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708–717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio 2, by a primal-dual algorithm with a reverse delete phase. Recently, Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation ratio 16 for a larger class of so called γ-pliable set families, that have much weaker uncrossing properties. In this paper we will improve the approximation ratio to 10. Using this result and other techniques, we also improve approximation ratios for the following two problems related to the Capacitatedk-Edge Connected Spanning Subgraph (Cap-k-ECSS) problem. Near Min-Cuts Cover: Given a graph G0=(V,E0) and an edge set E on V with costs, find a min-cost edge set J⊆E that covers all cuts with at most k-1 edges of the graph G0. We improve the approximation ratio from 16 to 10. We also obtain approximation ratio k-λ0+1+ϵ, where λ0 is the edge connectivity of G0, which is better than ratio 10 when k-λ0≤8.(k, q)-Flexible Graph Connectivity ((k, q)-FGC): Given a graph G=(V,E) with edge costs, a set U⊆E of "unsafe" edges, and integers k, q, find a min-cost subgraph H of G such that every cut of H has at least k safe edges or at least k+q edges. We will show that (k, 1)-FGC admits approximation ratio 3.5+ϵ if k is odd (improving previous ratio 4), that (k, 2)-FGC admits approximation ratio 7+ϵ (improving previous ratio 20), and that (k, 3)-FGC admits approximation ratio 16 for k even (improving previous ratio 22). We also show that for unit costs, (k, q)-FGC admits approximation ratio α+2qk, where α≈1+O(1/k) is an approximation ratio for the Min-Sizek-Edge-Connected Spanning Subgraph problem. Near Min-Cuts Cover: Given a graph G0=(V,E0) and an edge set E on V with costs, find a min-cost edge set J⊆E that covers all cuts with at most k-1 edges of the graph G0. We improve the approximation ratio from 16
The Knapsack Median problem was known to be W[2]-hard if parameterized by the maximal number of opened facilities in feasible solutions (denoted by k), implying that exactly solving this problem in FPT(k) time is unli...
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Identifying positive influence dominating set (PIDS) with the smallest cardinality can produce positive effect with the minimal cost on a social network. The purpose of this article is to propose new approximation alg...
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Identifying positive influence dominating set (PIDS) with the smallest cardinality can produce positive effect with the minimal cost on a social network. The purpose of this article is to propose new approximation algorithms for the minimum PIDS problem and its variants such as the minimum connected PIDS and the minimum PIDS of multiplex networks, with the aim of finding target sets with smaller cardinality. Through the design of novel submodular potential function, we theoretically prove that new approximation algorithms yield approximation ratios with same order compared with existing algorithms. We further demonstrate the performance of our algorithm by showcasing its efficacy on several real-world and publicly available instances of social networks, thereby providing additional evidence that our proposed algorithm can identify PIDS with smaller cardinality.
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