Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of $k$ terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying graph metric, but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is, to what extent can prior knowledge of the underlying graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than $\Omega(\log k)$. Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey theory in high-dimensional hypercubes. Then we switch to the stochastic model, where the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples. Read More: https://epubs.siam.org/doi/10.1137/16M109669

SOLVING PROCESS PLANNING AND SCHEDULING PROBLEMS USING THE CONCEPT OF MAXIMUM WEIGHTED INDEPENDENT SET

Process planning and scheduling (PPS) is an essential and practical topic but a very intractable problem in manufacturing systems. Many research studies use iterative methods to solve such problems; however, they cannot achieve satisfactory results in both quality and computational speed. Other studies formulate scheduling problems as a graph coloring problem (GCP) or its extensions, but these formulations are limited to certain types of scheduling problems. In this dissertation, we propose a novel approach to formulate a general type of the PPS problem with resource allocation and process planning integrated towards a typical objective, minimizing the makespan. The PPS problem is formulated into an undirected weighted conflicting graph, where nodes represent operations and their resources; edges represent constraints, and weight factors are guidelines for the node selection at each time slot. Then, the Maximum Weighted Independent Set (MWIS) problem, which considers a graph with weights assigned to nodes and seeks to discover the “heaviest” independent set, that is, a set of nodes with maximum total weight so that no two nodes in the set are connected by an edge, can be solved to find the best set of operations with their desired resources for each discrete time slot. This proposed approach solves the PPS problem directly (a direct method in computational mathematics context). We establish that the proposed approach always returns a feasible optimum or near-optimum solution to the PPS problem. The performance of the proposed approach for the PPS problem depends on the accuracy and computational speed of solving the MWIS problem. We propose a divide-and-conquer algorithm structure with relatively low complexity for solving the MWIS problem. An exact MWIS algorithm and an All Maximal Independent Set Listing (AMISL) algorithm are developed based on this algorithm structure. The proposed algorithm structure can also be used to compose the exact MWIS algorithm with existing approximation MWIS algorithms. This is an effective way to improve the accuracy of existing approximation MWIS algorithms or improve the computational speed of the exact MWIS algorithm. All eight algorithms for the MWIS problem, the exact MWIS algorithm, the AMISL algorithm, two approximation algorithms from the literature, and four composed algorithms, are tested on the test instances based on the PPS application environment. The different configurations of the proposed approach for solving the PPS problem are tested on a real-world PPS example and further designated test instances to evaluate the scalability, accuracy, and robustness

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Central Florida Future, Vol. 21 No. 46, February 28, 1989

Alafaya widening delayed; CONDOM CATASTROPHE (photo); Nader to college students: help make a better world; Vandals destroy more light poles; SPORTS: The UCF women\u27s basketball team closed the regular season with a win over local rival Rollins. Pat Crocklin closes his UCF basketball career in record style.https://stars.library.ucf.edu/centralfloridafuture/1908/thumbnail.jp