scholarly journals A Fast and Exact Greedy Algorithm for the Core–Periphery Problem

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 94 ◽  
Author(s):  
Dario Fasino ◽  
Franca Rinaldi
Keyword(s):  
Structural Analysis ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Optimization Problem ◽  
Connected Subgraph ◽  
Optimal Solutions ◽  
The Core ◽  
Core Set ◽  
Key Concepts

The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce a well-connected subgraph and share connections with peripheral nodes, while the peripheral nodes are loosely connected to the core nodes and other peripheral nodes. We propose a polynomial-time algorithm to detect core–periphery structures in networks having a symmetric adjacency matrix. The core set is defined as the solution of a combinatorial optimization problem, which has a pleasant symmetry with respect to graph complementation. We provide a complete description of the optimal solutions to that problem and an exact and efficient algorithm to compute them. The proposed approach is extended to networks with loops and oriented edges. Numerical simulations are carried out on both synthetic and real-world networks to demonstrate the effectiveness and practicability of the proposed algorithm.

BATCHING MACHINE SCHEDULING WITH BICRITERIA: MAXIMUM COST AND MAKESPAN

2014 ◽  
Vol 31 (04) ◽  
pp. 1450025 ◽  
Author(s):  
CHENG HE ◽  
HAO LIN ◽  
JINJIANG YUAN ◽  
YUNDONG MU
Keyword(s):  
Processing Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Maximum Cost ◽  
Parallel Batching ◽  
Multicriteria Problem ◽  
Batching Machine

In this paper, the problem of minimizing maximum cost and makespan simultaneously on an unbounded parallel-batching machine is considered. An unbounded parallel-batching machine is a machine that can handle any number of jobs in a batch and the processing time of a batch is the largest processing time of jobs in the batch. The main goal of a multicriteria problem is to find Pareto optimal solutions. We present a polynomial-time algorithm to produce all Pareto optimal solutions of this bicriteria scheduling problem.

Optimal Screening of Populations with Heterogeneous Risk Profiles Under the Availability of Multiple Tests

2021 ◽  
Author(s):  
Hrayer Aprahamian ◽  
Hadi El-Amine
Keyword(s):  
Objective Function ◽  
Structural Properties ◽  
Polynomial Time ◽  
Network Flow ◽  
Optimization Problem ◽  
Flow Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Network Flow Problem ◽  
Multiple Tests

We study the design of large-scale group testing schemes under a heterogeneous population (i.e., subjects with potentially different risk) and with the availability of multiple tests. The objective is to classify the population as positive or negative for a given binary characteristic (e.g., the presence of an infectious disease) as efficiently and accurately as possible. Our approach examines components often neglected in the literature, such as the dependence of testing cost on the group size and the possibility of no testing, which are especially relevant within a heterogeneous setting. By developing key structural properties of the resulting optimization problem, we are able to reduce it to a network flow problem under a specific, yet not too restrictive, objective function. We then provide results that facilitate the construction of the resulting graph and finally provide a polynomial time algorithm. Our case study, on the screening of HIV in the United States, demonstrates the substantial benefits of the proposed approach over conventional screening methods. Summary of Contribution: This paper studies the problem of testing heterogeneous populations in groups in order to reduce costs and hence allow for the use of more efficient tests for high-risk groups. The resulting problem is a difficult combinatorial optimization problem that is NP-complete under a general objective. Using structural properties specific to our objective function, we show that the problem can be cast as a network flow problem and provide a polynomial time algorithm.

ON TWO APPROXIMATION ALGORITHMS FOR THE CLIQUE PROBLEM

1993 ◽  
Vol 04 (02) ◽  
pp. 117-133
Author(s):  
IAIN A. STEWART
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Deterministic Algorithm ◽  
Time Approximation ◽  
Nondeterministic Algorithm ◽  
Polynomial Time Approximation ◽  
Clique Problem

We look at well-known polynomial-time approximation algorithms for the optimization problem MAX-CLIQUE (“find the size of the largest clique in a graph”) with regard to how easy it is to compute the actual cliques yielded by these approximation algorithms. We show that even for two “pretty useless” deterministic polynomial-time approximation algorithms, it is unlikely that the resulting clique can be computed efficiently in parallel. We also show that for each non-deterministic algorithm, it is unlikely that there is some deterministic polynomial-time algorithm that decides whether any given vertex appears in some clique yielded by that nondeterministic algorithm.

scholarly journals ADVICE FOR SEMIFEASIBLE SETS AND THE COMPLEXITY-THEORETIC COST(LESSNESS) OF ALGEBRAIC PROPERTIES

2005 ◽  
Vol 16 (05) ◽  
pp. 913-928 ◽  
Author(s):  
PIOTR FALISZEWSKI ◽  
LANE A. HEMASPAANDRA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Worst Case ◽  
Advice Complexity ◽  
The Core ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Algebraic Properties

Informally put, the semifeasible sets are the class of sets having a polynomial-time algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. We provide a tutorial overview of the advice complexity of the semifeasible sets. No previous familiarity with either the semifeasible sets or advice complexity will be assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations. Karp and Lipton introduced advice complexity about a quarter of a century ago.18 Advice complexity asks, for a given power of interpreter, how many bits of "help" suffice to accept a given set. Thus, this is a notion that contains aspects both of descriptional/informational complexity and of computational complexity. We will see that for some powers of interpreter the (worst-case) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic. While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—so-called selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.

scholarly journals Convexity of b-matching Games

2020 ◽  
Author(s):  
Soh Kumabe ◽  
Takanori Maehara
Keyword(s):  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kidney Exchange ◽  
The Core ◽  
Emptiness Problem ◽  
Matching Games ◽  
Matching Game ◽  
Necessary And Sufficient

The b-matching game is a cooperative game defined on a graph. The game generalizes the matching game to allow each individual to have more than one partner. The game has several applications, such as the roommate assignment, the multi-item version of the seller-buyer assignment, and the international kidney exchange. Compared with the standard matching game, the b-matching game is computationally hard. In particular, the core non-emptiness problem and the core membership problem are co-NP-hard. Therefore, we focus on the convexity of the game, which is a sufficient condition of the core non-emptiness and often more tractable concept than the core non-emptiness. It also has several additional benefits. In this study, we give a necessary and sufficient condition of the convexity of the b-matching game. This condition also gives an O(n log n + m α(n)) time algorithm to determine whether a given game is convex or not, where n and m are the number of vertices and edges of a given graph, respectively, and α(・) is the inverse-Ackermann function. Using our characterization, we also give a polynomial-time algorithm to compute the Shapley value of a convex b-matching game.

Optimizing Gate Assignments at Airport

2014 ◽  
Vol 556-562 ◽  
pp. 4178-4184
Author(s):  
Pan Zheng ◽  
Jing Li ◽  
Ying Hui Liang
Keyword(s):  
Simulated Annealing ◽  
Combinatorial Optimization ◽  
Objective Function ◽  
Assignment Problem ◽  
Optimization Problem ◽  
High Efficiency ◽  
Heuristic Method ◽  
Combinatorial Optimization Problem ◽  
The Core ◽  
Gate Assignment

Airport gate assignment is to appoint a gate for the arrival or leave flight and to ensure that the flight is on schedule. Assigning the airport gate with high efficiency is a key task among the airport ground busywork. As the core of airport operation, aircraft gate assignment is known as a kind of complicated combinatorial optimization problem. In this paper, we consider the over-constrained Airport Gate Assignment Problem where the number of flights exceeds the number of gates available, and where the objective is to minimize the overall variance of slack time (OVST). According to the intrinsic characteristics of the objective function itself, we design a meta-heuristic method and simulated annealing to solve the problem. Finally, the illustrative examples show the validity of the proposed approach.

APPROXIMATE BLOCK SORTING

2006 ◽  
Vol 17 (02) ◽  
pp. 337-355 ◽  
Author(s):  
MEENA MAHAJAN ◽  
RAGHAVAN RAMA ◽  
VENKATESH RAMAN ◽  
S. VIJAYKUMAR
Keyword(s):  
Approximation Algorithm ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fixed Parameter Tractable ◽  
Identity Permutation ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Increasing Sequences ◽  
Better Than

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

scholarly journals Next generation cluster editing

2015 ◽  
Author(s):  
Thomas Bellitto ◽  
Tobias Marschall ◽  
Alexander Schönhuth ◽  
Gunnar W Klau
Keyword(s):  
Structural Variation ◽  
Optimization Problem ◽  
Recall Performance ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Structural Variations ◽  
One Dimensional ◽  
Cluster Editing ◽  
A Genome ◽  
Time Linear

Genomic structural variations play key roles in genetic diversity and disease. Despite recent advances in structural variation discovery, many variants are yet to be discovered. Midsize insertions and deletions pose particularly involved algorithmic challenges. The recent CLEVER algorithm addressed these challenges with a statistical model on cliques in a graph whose nodes are read alignments and whose edges arise from a statistical test on length and overlap of read alignments. However, the resulting read alignment clusters tend to be too small and are heavily overlapping, which leads to losses in recall performance rates. Here we present a model based on weighted cluster editing, which alleviates these issues: clusters are provably non-overlapping and tend to be larger. In order to render the inherent optimization problem tractable on all read alignments of a genome, we present a novel, principled heuristic, which runs in time linear in the length of the genome. The heuristic is based on an exact polynomial-time algorithm for weighted cluster editing in one-dimensional point graphs. We demonstrate that the new model improves recall rates achieved by CLEVER.

scholarly journals Connected Subgraph Defense Games

Algorithmica ◽  
2021 ◽  
Author(s):  
Eleni C. Akrida ◽  
Argyrios Deligkas ◽  
Themistoklis Melissourgos ◽  
Paul G. Spirakis
Keyword(s):  
Polynomial Time ◽  
Nash Equilibria ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Connected Subgraph ◽  
Expected Number ◽  
Worst Case ◽  
Induced Subgraph ◽  
Other Hand

AbstractWe study a security game over a network played between a defender and kattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $$\lambda $$ λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $$\lambda $$ λ constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is $${\mathtt {NP}}$$ NP -hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any $$\lambda $$ λ ; this is the worst equilibrium defense ratio over all graphs.

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