In a graph G, a vertex is said to dominate itself and all of its neighbors. Adominating set of G = (V, E) is a subset D of V such that every vertex in V is dominated by atleast one vertex in D. Domination and its vari...
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In a graph G, a vertex is said to dominate itself and all of its neighbors. Adominating set of G = (V, E) is a subset D of V such that every vertex in V is dominated by atleast one vertex in D. Domination and its variations have many applications, and have beenextensively studied in the literature, see [4,8,9]. Among the variations of domination, the k-tupledomination problem was introduced in [7,8]. For a fixed positive integer k, a k-tuple dominating setof G = (V, E) is a subset D_k of V such that every vertex in V is dominated by at least k verticesof D. The special case when k = 1 is the usual domination. The case when k = 2 is called doubledomination in [7] where exact values of the double domination number of some special graphs areobtained. The same paper also gives various bounds of double and k-tuple domination in terms ofother parameters.
In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our a...
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In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1-E)-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. More precisely, we show that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight, assuming that all given weights are integral. Furthermore, we can find a (1-E)-approximate solution via solving O(E-1logr) instances of the unweighted matroid intersection problem, where r is the smaller rank of the two given matroids. Our algorithms make use of the weight-splitting approach of Frank (J algorithms 2(4):328-336, 1981) and the geometric scaling scheme of Duan and Pettie (J ACM 61(1):1, 2014). Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In addition, we give a further application of our decomposition technique: we solve efficiently the rank-maximal matroid intersection problem, a problem motivated by matching problems under preferences.
Concave mixed- integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed- integer points in a polyhedral region. In this work we describe an algorithm that finds an - app...
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Concave mixed- integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed- integer points in a polyhedral region. In this work we describe an algorithm that finds an - approximate solution to a concave mixed- integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in 1/ , provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless P = NP.
We propose an anytime algorithm to compute successively better approximations of the optimum of Minimum Vertex Guard. Though the presentation is focused on polygons, the work may be directly extended to terrains along...
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We study an optimization problem which can be interesting in several network applications, especially in telecommunication. Given a capacitated directed network G in which a flow, related to a certain commodity, has t...
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We study an optimization problem which can be interesting in several network applications, especially in telecommunication. Given a capacitated directed network G in which a flow, related to a certain commodity, has to be sent from a given set of origins to a given set of destinations, we want to compute a maximum congested cut, i.e., a cut (S, T) of G which maximizes the ratio between the net demand of T and the capacity of the cut. We will show that a maximum congested cut can be computed in polynomial time by formulating the problem as a special maximum mean-weight cut problem, and solving it by Newton's method for linear fractional combinatorial optimization (Radzik, Complexity in Numerical Optimization, World Scientific, Singapore, 1993). The introduced formulation will be extended to the case in which the flow demands vary in a polyhedron D, and a cut has to be determined which maximizes the congestion with respect to the flow demands in D, addressing a robust version of the problem. We will prove that the robust maximum congested cut problem is coNP-Hard. Special cases solvable in polynomial time as well as approximation algorithms for some relevant hard cases will be proposed. The specialization of the proposed approaches to the well-studied class of Hose models (Duffield et al., Proc., Conf Appl Technol Architect Protocol Comp Comm 29 (1999), 95-108) will be discussed. Then, in the second part of the paper, we will investigate the case in which k commodities are given, and a maximum congested cut has to be computed in such a multicommodity flow context. This problem is NP-Hard. Some algorithmic considerations will be formulated in order to approximate the maximum congested cut in multicommodity networks, also addressing the robust version of the problem. A consequence of the stated results is a polynomial time algorithm for the sparsest cut problem in undirected graphs, for the special case in which k is O(log n), and an O(k/log n)-approximation algorithm f
We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algori...
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We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a 1 - epsilon fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor epsilon approaches 0 when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the tradeoff we obtain here that determines how fast epsilon approaches 0, is near optimal: We give a nearly matching lower bound showing that the tradeoff cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a 1 - epsilon fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. Further, if the algorithm is given one number per distribution, namely the optimal profit possible for each of the adversary's distribution, then we design an algorithm that achieves a 1 - epsilon fraction of the weighted average of the optimal profit of each distribution the adversary picks. In the offline setting we give a fast algorithm to solve very large linear programs (LPs) with both packing and covering constraints. We give algorithms to approximately solve (within a factor of 1 + epsilon) the mixed packing-covering problem with O(gamma m log(n/delta)/epsilon(2)) oracle calls where the constraint matrix of this LP has dimen- sion n x m, the success probability of the algorithm is 1 - delta, and gamma quantifies how significant a single request is when compared to the sum tot
We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p >= 2 is a fixed integer). We obtain an 2O(pk)nO(1)-time...
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We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p >= 2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP subset of coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. Finally, we describe a 2(p + 1)-approximation algorithm for the p-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p- vertex/edge-connected subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) [30] and Nagamochi and Ibaraki (1992) [27].(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).
We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the probl...
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We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities;we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and to the covering problem that are within a factor of 2 of each other. Each of these results implies an approximative min-max result. For the general case of arbitrary weights and capacities we describe an LP-based (2 + epsilon)-approximation algorithm for the covering problem. Finally, we show that, unless P = NP, the covering problem cannot be approximated in polynomial time within arbitrarily good precision.
We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Williamson et ...
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We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Williamson et al. in Combinatorica 15(3):435-454, 1995. https://***/10.1007/BF01299747). Williamson et al. prove an approximation ratio of two for connectivity augmentation problems where the connectivity requirements can be specified by uncrossable functions. They state: "Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems." Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result to problems in the area of network design. (1) A 16-approximation algorithm for augmenting a family of small cuts of a graph G. The previous best approximation ratio was O(log |V(G)|). (2) A 16 center dot (sic)k/u(min)(sic)-approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph G = (V, E) with edge costs c is an element of Q(>= 0)(E) and edge capacities u is an element of Z(>= 0)(E), find a minimum-cost subset of the edges F subset of E such that the capacity of any cut in (V, F) is at least k;u(min) (respectively, u(max)) denotes the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. u(max) <= k. The previous best approximation ratio was min(O(log |V|), k, 2u(max)). (3) A 20-approximation algorithm for the model o
We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-loca...
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We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-located at the endpoints of an edge for it to be processed and, therefore, this problem contains both edge covering and scheduling decisions. These new problems have motivating applications in traffic systems and military intelligence operations. We provide complexity results for the dynamic edge covering and scheduling problem over different types of networks. We then show that existing approximation algorithms for parallel machine scheduling problems can be leveraged to provide approximation algorithms for this new class of problems over certain types of networks.
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