scholarly journals Defending with Shared Resources on a Network

2020 ◽  
Vol 34 (02) ◽  
pp. 2111-2118
Author(s):  
Minming Li ◽  
Long Tran-Thanh ◽  
Xiaowei Wu
Keyword(s):  
Lower Bound ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
General Problem ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Np Hard ◽  
Shared Resources ◽  
Target Node ◽  
Special Cases

In this paper we consider a defending problem on a network. In the model, the defender holds a total defending resource of R, which can be distributed to the nodes of the network. The defending resource allocated to a node can be shared by its neighbors. There is a weight associated with every edge that represents the efficiency defending resources are shared between neighboring nodes. We consider the setting when each attack can affect not only the target node, but its neighbors as well. Assuming that nodes in the network have different treasures to defend and different defending requirements, the defender aims at allocating the defending resource to the nodes to minimize the loss due to attack. We give polynomial time exact algorithms for two important special cases of the network defending problem. For the case when an attack can only affect the target node, we present an LP-based exact algorithm. For the case when defending resources cannot be shared, we present a max-flow-based exact algorithm. We show that the general problem is NP-hard, and we give a 2-approximation algorithm based on LP-rounding. Moreover, by giving a matching lower bound of 2 on the integrality gap on the LP relaxation, we show that our rounding is tight.

New algorithms for pattern matching with wildcards and length constraints

2015 ◽  
Vol 07 (03) ◽  
pp. 1550032 ◽  
Author(s):  
Abdullah N. Arslan ◽  
Betsy George ◽  
Kirsten Stor
Keyword(s):  
Pattern Matching ◽  
Polynomial Time ◽  
Original Problem ◽  
Heuristic Algorithms ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Worst Case ◽  
Special Cases ◽  
New Algorithms

The pattern matching with wildcards and length constraints problem is an interesting problem in the literature whose computational complexity is still open. There are polynomial time exact algorithms for its special cases. There are heuristic algorithms, and online algorithms that do not guarantee an optimal solution to the original problem. We consider two special cases of the problem for which we develop offline solutions. We give an algorithm for one case with provably better worst case time complexity compared to existing algorithms. We present the first exact algorithm for the second case. This algorithm uses integer linear programming (ILP) and it takes polynomial time under certain conditions.

Optimal Order Picking in Carousels Storage System

2012 ◽  
Vol 601 ◽  
pp. 347-353
Author(s):  
Xiong Zhi Wang ◽  
Guo Qing Wang
Keyword(s):  
Approximation Algorithm ◽  
General Problem ◽  
Storage System ◽  
Order Picking ◽  
Total Order ◽  
Experimental Results ◽  
Optimal Order ◽  
Np Hard ◽  
Special Case

We study the order picking problem in carousels system with a single picker. The objective is to find a picking scheduling to minimizing the total order picking time. After showing the problem being strongly in NP-Hard and finding two characteristics, we construct an approximation algorithm for a special case (two carousels) and a heuristics for the general problem. Experimental results verify that the solutions are quickly and steadily achieved and show its better performance.

APPROXIMATION ALGORITHMS FOR SCHEDULING MALLEABLE TASKS UNDER PRECEDENCE CONSTRAINTS

2002 ◽  
Vol 13 (04) ◽  
pp. 613-627 ◽  
Author(s):  
RENAUD LEPÈRE ◽  
DENIS TRYSTRAM ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Precedence Constraints ◽  
Performance Guarantee ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Special Cases ◽  
Special Case ◽  
Polynomial Time Approximation

This work presents approximation algorithms for scheduling the tasks of a parallel application that are subject to precedence constraints. The considered tasks are malleable which means that they may be executed on a varying number of processors in parallel. The considered objective criterion is the makespan, i.e., the largest task completion time. We demonstrate a close relationship between this scheduling problem and one of its subproblems, the allotment problem. By exploiting this relationship, we design a polynomial time approximation algorithm with performance guarantee arbitrarily close to [Formula: see text] for the special case of series parallel precedence constraints and for the special case of precedence constraints of bounded width. These special cases cover the important situation of tree structured precedence constraints. For arbitrary precedence constraints, we give a polynomial time approximation algorithm with performance guarantee [Formula: see text].

scholarly journals UNIQUENESS, INTRACTABILITY AND EXACT ALGORITHMS: REFLECTIONS ON LEVEL-K PHYLOGENETIC NETWORKS

2009 ◽  
Vol 07 (04) ◽  
pp. 597-623 ◽  
Author(s):  
LEO VAN IERSEL ◽  
STEVEN KELK ◽  
MATTHIAS MNICH
Keyword(s):  
Gene Transfer ◽  
Horizontal Gene Transfer ◽  
Phylogenetic Trees ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Phylogenetic Networks ◽  
Np Hard ◽  
Level 1

Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone so-called reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how non-treelike the evolution can be, with level-0 networks being trees. We study the problem of constructing level-k phylogenetic networks from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for each k, a level-k network that is uniquely defined by its triplets. We demonstrate the applicability of this result by using it to prove that (1) for all k ≥ 1 it is NP-hard to construct a level-k network consistent with all input triplets, and (2) for all k ≥ 0 it is NP-hard to construct a level-k network consistent with a maximum number of input triplets, even when the input is dense. As a response to this intractability, we give an exact algorithm for constructing level-1 networks consistent with a maximum number of input triplets.

scholarly journals Parameterized Algorithms for Power-Efficiently Connecting Wireless Sensor Networks: Theory and Experiments

2021 ◽  
Author(s):  
Matthias Bentert ◽  
René van Bevern ◽  
André Nichterlein ◽  
Rolf Niedermeier ◽  
Pavel V. Smirnov
Keyword(s):  
Communication Network ◽  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Sensor Network ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Wireless Sensor ◽  
Connected Components ◽  
Np Hard ◽  
Transmission Power

We study a problem of energy-efficiently connecting a symmetric wireless communication network: given an n-vertex graph with edge weights, find a connected spanning subgraph of minimum cost, where the cost is determined by each vertex paying the heaviest edge incident to it in the subgraph. The problem is known to be NP-hard. Strengthening this hardness result, we show that even o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard. Moreover, we show that under the exponential time hypothesis, there are no exact algorithms running in 2o(n) time or in [Formula: see text] time for any computable function f. We also show that the special case of connecting c network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size cO(1) unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects O(log n)-connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components because of sensor faults. In experiments, the algorithm outperforms CPLEX with known integer linear programming (ILP) formulations when n is sufficiently large compared with c. Summary of Contribution: Wireless sensor networks are used to monitor air pollution, water pollution, and machine health; in forest fire and landslide detection; and in natural disaster prevention. Sensors in wireless sensor networks are often battery-powered and disposable, so one may be interested in lowering the energy consumption of the sensors in order to achieve a long lifetime of the network. We study the min-power symmetric connectivity problem, which models the task of assigning transmission powers to sensors so as to achieve a connected communication network with minimum total power consumption. The problem is NP-hard. We provide perhaps the first parameterized complexity study of optimal and approximate solutions for the problem. Our algorithms work in polynomial time in the scenario where one has to reconnect a sensor network with n sensors and O(log n)-connected components by means of a minimum transmission power increase or if one can find transmission power lower bounds that already yield a network with O(log n)-connected components. In experiments, we show that, in this scenario, our algorithms outperform previously known exact algorithms based on ILP formulations.

scholarly journals On the Max-Min Fair Stochastic Allocation of Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 2070-2078
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita
Keyword(s):  
Approximate Solution ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Indivisible Goods ◽  
Additive Utility ◽  
Ex Ante ◽  
Stochastic Setting

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.

ON PLANAR MEDIANOID COMPETITIVE LOCATION PROBLEMS WITH MANHATTAN DISTANCE

2013 ◽  
Vol 30 (02) ◽  
pp. 1250050 ◽  
Author(s):  
KE FU ◽  
ZHAOWEI MIAO ◽  
JIAYAN XU
Keyword(s):  
Polynomial Time ◽  
General Problem ◽  
Location Problem ◽  
Location Problems ◽  
Manhattan Distance ◽  
Competitive Location ◽  
Customer Preferences ◽  
Special Cases ◽  
Present Solution ◽  
Solution Methods

A medianoid problem is a competitive location problem that determines the locations of a number of new service facilities that are competing with existing facilities for service to customers. This paper studies the medianoid problem on the plane with Manhattan distance. For the medianoid problem with binary customer preferences, i.e., a case where customers choose the closest facility to satisfy their entire demand, we show that the general problem is NP-hard and present solution methods to solve various special cases in polynomial time. We also show that the problem with partially binary customer preferences can be solved with a similar approach we develop for the model with binary customer preferences.

APPROXIMATING 3D POINTS WITH CYLINDRICAL SEGMENTS

2004 ◽  
Vol 14 (03) ◽  
pp. 189-201 ◽  
Author(s):  
BINHAI ZHU
Keyword(s):  
Approximation Algorithm ◽  
Computational Biology ◽  
Polynomial Time ◽  
General Problem ◽  
Approximation Scheme ◽  
Polynomial Time Approximation Scheme ◽  
Geometric Problem ◽  
Time Approximation ◽  
Optimal Value ◽  
Neural Maps

In this paper, we study a 3D geometric problem originated from computing neural maps in the computational biology community: Given a set S of n points in 3D, compute k cylindrical segments (with different radii, orientations, lengths and no segment penetrates another) enclosing S such that the sum of their radii is minimized. There is no known result in this direction except when k=1. The general problem is strongly NP-hard and we obtain a polynomial time approximation scheme (PTAS) for any fixed k>1 in O(n3k-2/δ4k-3) time by returning k cylindrical segments with sum of radii at most (1+δ) of the corresponding optimal value, if exist. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k=1.

scholarly journals Weighted Maxmin Fair Share Allocation of Indivisible Chores

2019 ◽  
Author(s):  
Haris Aziz ◽  
Hau Chan ◽  
Bo Li
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Time Constant ◽  
Polynomial Time ◽  
Natural Generalization ◽  
Linear Program ◽  
Fair Share ◽  
Special Cases

We initiate the study of indivisible chore allocation for agents with asymmetric shares. The fairness concept we focus on is the weighted natural generalization of maxmin share: WMMS fairness and OWMMS fairness. We first highlight the fact that commonly-used algorithms that work well for allocation of goods to asymmetric agents, and even for chores to symmetric agents do not provide good approximations for allocation of chores to asymmetric agents under WMMS. As a consequence, we present a novel polynomial-time constant-approximation algorithm, via linear program, for OWMMS. For two special cases: the binary valuation case and the 2-agent case, we provide exact or better constant-approximation algorithms.

scholarly journals A Polynomial-Time Exact Algorithm for k -Depot Capacitated Vehicle Routing Problem on a Tree

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Liang Xu ◽  
Tong Zhang ◽  
Rantao Hu ◽  
Jialing Guo
Keyword(s):  
Approximation Algorithm ◽  
Vehicle Routing ◽  
Polynomial Time ◽  
Vehicle Routing Problem ◽  
Exact Algorithm ◽  
Fixed Number ◽  
Routing Problem ◽  
Capacitated Vehicle Routing Problem ◽  
Shaped Graph ◽  
Capacitated Vehicle

This paper proposes a polynomial-time exact algorithm for the k -depot capacitated vehicle routing problem on a tree for fixed k ( k -depot CVRPT for short), which involves dispatching a fixed number of k capacitated vehicles in depots on a tree-shaped graph to serve customers with the objective of minimizing total distance traveled. The polynomial-time exact algorithm improves the 2-approximation algorithm when k is a constant.

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