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Independent sets of maximum weight beyond claw-free graphs and related problems

THEORETICAL COMPUTER SCIENCE 2025年 1032卷

作者： Brandstaedt, Andreas Lozin, Vadim Mosca, Raffaele Univ Rostock Inst Informat D-18051 Rostock Germany Univ Warwick Math Inst Coventry CV4 7AL England Univ G DAnnunzio Dipartimento Econ I-65121 Pescara Italy

The maximum weight independent set problem (WIS), which is known to be generally NP-hard, admits polynomial-time solutions when restricted to graphs in some special classes. In particular, due to the celebrated Edmonds' matching algorithm, WIS is solvable in polynomial time in the class of line graphs. This solution was extended to claw-free graphs and then further to fork- free graphs and to tclaw-free graphs, where tclaw is the graph consisting of t disjoint copies of the claw. The solution for tclaw-free graphs was obtained by generalizing Farber's approach to solve the problem for tK2-free graphs. In the present paper, we elaborate this approach further to develop a polynomial-time algorithm to solve the problem in the class of fork+tclaw-free graphs, generalizing both fork-free graphs and tclaw-free graphs, and in the class of P5 +tclaw-free graphs. We then apply the latter result to solve the more general problem of finding a d-regular induced subgraph of maximum weight in the class of P5 + tP3-free graphs in polynomial time for any natural d and t, extending some of the previously known solutions.

关键词： Maximum weight independent set problem Claw-free graphs Fork-free graphs Maximum regular induced subgraph problem polynomial-time algorithms P-5-freegraphs

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On the geometric red-blue set cover problem

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THEORETICAL COMPUTER SCIENCE 2025年 1033卷

作者： Madireddy, Raghunath Reddy Nandy, Subhas C. Pandit, Supantha Birla Inst Technol & Sci Pilani Hyderabad Campus Pilani India Indian Stat Inst Kolkata India Dhirubhai Ambani Inst Informat & Commun Technol Gandhinagar Gujarat India

Using various geometric objects, we study variations of the geometric Red-Blue Set Cover (RBSC) problem in & Ropf;2. Here, given two sets R and B of points, called red and blue points, respectively, and a set O of objects, the objective is to compute a subset O ' C O of objects such that O ' covers all the blue points in B and covers the minimum number of red points in R. We show that the RBSC problem with intervals on the real line is polynomial-time solvable. In & Ropf;2, the problem admits a polynomial-time algorithm for a particular case of axis-parallel lines when no two lines intersect at a red point. However, if the objects are horizontal lines and vertical segments, the problem becomes NP-hard. It remains NP-hard for axis-parallel unit length segments distributed arbitrarily in the plane. The problem is NP-hard for the axis-parallel rectangles even when (i) each rectangle in O is anchored at one of the given two parallel lines, and (ii) a horizontal line intersects all the rectangles in O. We show a variation of RBSC, called Special Red Blue Set Cover (SPECIAL-RBSC), to be APXhard. Next, we use this result to show APX-hardness of the following geometric variations of RBSC problem in & Ropf;2 where the objects are (i) axis-parallel rectangles containing the origin, (ii) axis-parallel strips, (iii) axis-parallel rectangles that are intersecting exactly zero or four times, (iv) axis-parallel line segments, and (v) downward shadows of line segments, by providing the encoding of these problems as the Special Red Blue Set Cover problem. These APX-hardness results are in the same line of work by Chan and Grant (2014), who provided the APX-hardness results for the geometric set cover problem for the above classes of objects.

关键词： Red-blue set cover Special red-blue set cover Anchored rectangles Segments polynomial-time algorithms APX-hard NP-hard

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Increasing the Maximum Capacity Path in a Network and Its Application for Improving the Connection Between Two Routers

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Tsinghua Science and Technology 2024年第3期29卷 753-765页

作者： Adrian M.Deaconu Javad Tayyebi Department of Mathematics and Computer Science Transilvania University of BrasovBrasov 500036Romania Department of Industrial Engineering Birjand University of TechnologyBirjand 000097Iran

Abstract:This paper addresses the problem of improving the optimal value of the Maximum Capacity Path(MCP)through expansion in a flexible network,and minimizing the involved *** only condition applied to the cost functions is to be non-decreasing *** is a non-restrictive condition,reflecting the reality in practice,and is considered for the first time in the ***,the total cost of expansion is a combination of max-type cost(e.g.,for supervision)and sum-type cost(*** building infrastructures,price of materials,price of labor,etc.).For this purpose,two types of strategies are combined:(l)increasing the capacity of the existing arcs,and(l)adding potential new *** different problems are introduced and *** the problems have immediate applications in Internet routing *** first one is to extend the network,so that the capacity of an McP in the modified network becomes equal to a prescribed value,therefore the cost of modifications is minimized.A strongly polynomial-time algorithm is deduced to solve this *** second problem is a network expansion under a budget constraint,so that the capacity of an McP is maximized.A weakly polynomial-time algorithm is presented to deal with *** the special case when all the costs are linear,a Meggido's parametric search technique is used to develop an algorithm for solving the problem in strongly polynomial *** new approach has a time complexity of O(n^(4)),which is better than the time complexity of O(n4 log(n)of the previously known method from literature.

关键词： Maximum Capacity Path(MCP) network expansion Internet routing polynomial-time algorithms

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STABLE MARRIAGE AND INDIFFERENCE

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DISCRETE APPLIED MATHEMATICS 1994年第3期48卷 261-272页

作者： IRVING, RW Computing Science Department University of Glasgow Glasgow G12 8QQ UK

It is well known that every instance of the classical stable marriage problem admits at least one stable matching, and that such a matching can be found in O(n2) time by application of the Gale/Shapley algorithm. In the classical version of the problem, each person must rank the members of the opposite sex in strict order of preference. In practical applications, a person may not wish (or be able) to choose between alternatives, thus allowing ties in the preference lists (or, more generally, allowing each preference list to be a partial order). With the introduction of such indifference, the notion of stability may be generalised in three obvious ways. For the weakest extension of stability, the same existence result holds, and essentially the same algorithm may be applied. In the other two cases, however, there is no guarantee that stable matchings exist. Nonetheless, in this paper, we describe polynomial-time algorithms that will establish, in either of these two cases, whether a matching of the appropriate kind exists, and if so will find such a matching.

关键词： STABLE MARRIAGE MATCHING polynomial-time algorithms

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Two-echelon lot-sizing with asymmetric information and continuous type space

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OMEGA-INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 2019年 87卷 158-176页

作者： Kerkkamp, R. B. O. van den Heuvel, W. Wagelmans, A. P. M. Erasmus Univ Econometr Inst Postbus 1738 NL-3000 DR Rotterdam Netherlands

We analyse a two-echelon discrete lot-sizing problem with a supplier and a retailer under information asymmetry. We assume that all cost parameters are time independent and that the retailer has single dimensional continuous private information, namely either his setup cost or his holding cost. The supplier uses mechanism design to determine a menu of contracts that minimises his expected costs, where each contract specifies the retailer's procurement plan and a side payment to the retailer. There is no restriction on the number of contracts in the menu. To optimally solve this principal-agent contracting problem we present a two-stage approach, based on a theoretical analysis. The first stage generates a list of procurement plans that is sufficient to solve the contracting problem to optimality. The second stage optimally assigns these plans to the retailer types and determines all side payments. The result is an optimal menu with finitely many contracts that pools retailer types. We identify cases for which the contracting problem can be solved in polynomial time and provide the corresponding algorithms. Furthermore, our analysis reveals that information asymmetry leads to atypical structures in the plans of the optimal menu, e.g., plans violating the zero-inventory property. Our solution approach and several results are directly applicable to more general problems as well. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Discrete lot-sizing Mechanism design Asymmetric information Combinatorial optimisation polynomial-time algorithms

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Non-isotonic routing metrics solvable to optimality via shortest path

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COMPUTER NETWORKS 2018年 145卷 89-95页

作者： Saad, Mohamed Univ Sharjah Dept Elect & Comp Engn Sharjah U Arab Emirates

It is a well-established fact that routing metrics cannot be optimized via shortest path algorithms unless they are both monotone and isotonic. In particular, monotonicity guarantees the convergence of the shortest path procedure, while isotonicity guarantees convergence to the optimal path. This paper presents a class of routing metrics that are not isotonic, yet can be solved to exact optimality via shortest path (or iterative shortest path) procedures. In particular, we consider routing metrics of the form "distance + 1/width" and "distance 'width", respectively, where the former is the default form of the composite metric of the interior gateway routing protocol (IGRP). To the best of our knowledge, for the first time we provide shortest-path-based algorithms that are guaranteed to minimize these metrics in spite of their non-isotonicity. This result implies that, contrary to common belief, the composite metric of the IGRP is in fact optimizable. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Routing Shortest path Isotonicity polynomial-time algorithms

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Green scheduling, flows and matchings

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THEORETICAL COMPUTER SCIENCE 2015年 579卷 126-136页

作者： Bampis, Evripidis Letsios, Dimitrios Lucarelli, Giorgio Univ Paris 06 Sorbonne Univ UMR 7606 LIP6 F-75005 Paris France

Recently, optimal combinatorial algorithms have been presented for the energy minimization multiprocessor speed-scaling problem with migrations [5,7]. These algorithms use repeated maximum-flow computations that allow the partition of the set of jobs into subsets in which all the jobs are executed at the same speed. The optimality of these algorithms is based on a series of technical lemmas showing that this partition and the corresponding speeds lead to the minimization of the energy consumption. In this paper, we show that both the algorithms and their analysis can be greatly simplified. In order to do this, we formulate the problem as a convex cost flow problem in an appropriate flow network. Furthermore, we show that our approach is useful to solve other problems in the dynamic speed-scaling setting. As an example, we consider the preemptive open-shop speed-scaling problem and we propose a polynomial-time algorithm for finding an optimal solution based on the computation of convex cost flows. We also propose a polynomial-time algorithm for minimizing a linear combination of the sum of the completion times of the jobs and the total energy consumption, for the non-preemptive multiprocessor speed-scaling problem. Instead of using convex cost flows, our algorithm is based on the computation of a minimum weighted maximum matching in an appropriate bipartite graph. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Speed-scaling polynomial-time algorithms Convex cost flows Weighted matchings

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Joint Optimal Routing and Power Allocation for Spectral Efficiency in Multihop Wireless Networks

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2014年第5期13卷 2530-2539页

作者： Saad, Mohamed Univ Sharjah Dept Elect & Comp Engn Sharjah U Arab Emirates

Given a multihop wireless network and a source-destination pair of nodes, this paper addresses the problem of jointly selecting a communication route and allocating transmit power levels, so that the end-to-end spectral efficiency of the route exceeds a desired threshold. Spectral-efficient routing has been subject to interest in the recent literature. The transmit power level, however, has been assumed to be known, and route selection was considered in isolation. This paper presents the first rigourously proven optimal, polynomial-time algorithms for two versions of the joint spectral-efficient routing and power allocation problem: sum-power minimization and maximum power minimization. The proposed algorithms rely on the divide-and-conquer principle and the Bellman-Ford algorithm for shortest (or widest) path computation. Our computational results further illustrate the efficiency of the proposed approach.

关键词： Multihop wireless networks spectrum-efficient routing power allocation polynomial-time algorithms divide-and-conquer Bellman-Ford

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Stability of the classification of returns to scale in FDH models

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第3期196卷 1223-1228页

作者： Soleimani-damaneh, M. Mostafaee, A. Azad Univ N Tehran Branch Dept Math Tehran Iran Univ Tehran Univ Coll Sci Sch Math Stat & Comp Sci Tehran Iran

This paper deals with the estimation of returns to scale (RTS) in free disposal hull (FDH) models and provides some stability intervals for preserving the RTS classification. It has been shown that the proposed stability intervals can be obtained via a polynomial-time algorithm based on the calculation of certain ratios of inputs and outputs, without solving any mathematical programming problem. The results of the study have been proved via some lemmas and theorems and have been illustrated by numerical examples and a real application. (c) 2008 Elsevier B.V. All rights reserved.

关键词： DEA FDH Returns-to-scale polynomial-time algorithms Stability

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Stable Matching with Uncertain Linear Preferences

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ALGORITHMICA 2020年第5期82卷 1410-1433页

作者： Aziz, Haris Biro, Peter Gaspers, Serge de Haan, Ronald Mattei, Nicholas Rastegari, Baharak UNSW Sydney Sydney NSW Australia Data61 Sydney NSW Australia Hungarian Acad Sci Budapest Hungary Corvinus Univ Budapest Budapest Hungary Univ Amsterdam ILLC Amsterdam Netherlands Tulane Univ New Orleans LA 70118 USA Univ Southampton Southampton Hants England

We consider the two-sided stable matching setting in which there may be uncertainty about the agents' preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model-for each agent, there is a probability distribution over linear preferences, (2) compact indifference model-for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model-there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

关键词： Stable matchings Stable marriage problem Uncertain preferences NP-hard problems polynomial-time algorithms

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