A flexible workforce or a versatile machine is employed to perform various types of operations. Often these resources are associated with setups. Whenever a worker or machine switches from processing one type of operation to another a setup time may be required although several operations of a same type can be processed in succession after a single setup. The presence of setups gives rise to the problem of choosing batch sizes that are neither too large nor too small. In the last one and a half decade, many researchers have addressed the problem of scheduling with batching. A majority of articles assumes that there is only one type of scarce resource, which is typically machine. Often there can be two scarce resources such as a worker and a machine or a machine and a tool. We propose a resource constrained scheduling model with a single operator and two or more machines. Whenever the operator changes machine, a setup time is required that may be sequence dependent or sequence independent. We consider the two cases of an open shop and a flow shop. In the open shop case, the order in which a job visits the machines is unrestricted. In the flow shop case, every job must visit the machines in the same order. We consider various scheduling objectives. For variable number of machines, many cases are intractable. We discuss some dominance properties that narrow down the search for an optimal schedule. We present a dynamic programming approach which solves a large number of cases. The running time of the dynamic program is polynomial for a fixed number of machines. For the case of two machines, we show that the dominance properties have a nice interpretation. We develop some algorithms and justify their use by establishing running times, comparing the running times with those of the existing algorithms, and testing the performance of the algorithms.

In this work we formalize a new natural objective (or cost) function for bi-clustering - Monochromatic bi-clustering. Our objective function is suitable for detecting meaningful homogenous clusters based on categorical valued input matrices. Such problems have arisen recently in systems biology where researchers have inferred functional classifications of biological agents based on their pairwise interactions. We analyze the computational complexity of the resulting optimization problems. We show that finding optimal solutions is NP-hard and complement this result by introducing a polynomial time approximation algorithm for this bi-clustering task. This is the first positive approximation guarantee for bi-clustering algorithms. We also show that bi-clustering with our objective function can be viewed as a generalization of correlation clustering.

According to the theory of NPcompleteness, many problems that have important realworld applications are NPhard. This excludes the possibility of solving them in polynomial time unless P=NP. A number of approaches have been proposed in dealing with NPhard problems, among them are approximation algorithms and parameterized algorithms. The study of approximation algorithms tries to find good enough solutions instead of optimal solutions in polynomial time, while parameterized algorithms try to give exact solutions when a natural parameter is small. In this thesis, we study the structural properties of parameterized computation and approximation algorithms for NP optimization problems. In particular, we investigate the relationship between parameterized complexity and polynomialtime approximation scheme (PTAS) for NP optimization problems. We give nice characterizations for two important subclasses in PTAS: Fully Polynomial Time Approximation Scheme (FPTAS) and Effcient Polynomial Time Approximation Scheme (EPTAS), using the theory of parameterized complexity. Our characterization of the class FPTAS has its advantages over the former characterizations, and our characterization of EPTAS is the first systematic investigation of this new but important approximation class. We develop new techniques to derive strong computational lower bounds for certain parameterized problems based on the theory of parameterized complexity. For example, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the clique problem could not be solved in time O(f (k)no(k)) for any function f . This lower bound matches the upper bound of the trivial algorithm that simply enumerates and checks all subsets of k vertices in the given graph of n vertices. We then extend our techniques to derive computational lower bounds for PTAS and EPTAS algorithms of NP optimization problems. We prove that certain NP optimization problems with known PTAS algorithms have no PTAS algorithms of running time O(f (1/Epsilon)no(1/Epsilon)) for any function f . Therefore, for these NP optimization problems, although theoretically they can be approximated in polynomial time to an arbitrarily small error bound Epsilon, they have no practically effective approximation algorithms for small error bound Epsilon. To our knowledge, this is the first time such lower bound results have been derived for PTAS algorithms. This seems to open a new direction for the study of computational lower bounds on the approximability of NP optimization problems.

This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.

<div>We consider the models of sequential games over supply chain networks and production chain networks. In the supply chain model, we show that in particular, for series-parallel networks, there is a unique equilibrium. </div><div>We provide a polynomial time algorithm to compute the equilibrium and study the impact of the network structure to the total trade flow at equilibrium. Our results shed light on the trade-off between competition, production cost, and double marginalization. </div><div><br></div><div>In the production chain model, we investigated sequential decisions and delegation options over three agents, chain, and tree networks. Our main contribution is showing the value of delegation and how to maximumly leverage the middleman's aligned interests with the principal. In particular, we provide a polynomial time algorithm to find the optimal delegation structure and the corresponding necessary contract payments for the principal. Furthermore, we analyzed the trade-off of the delegation and gave a deeper insight into the value of delegation in different conditions. Several questions are left for future research such as what's the optimal delegation structures in general tree and how to build the model that agents can try multiple times until the task is successful. </div>

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18701号 / 情博第551号 / 新制||情||97(附属図書館) / 31634 / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 阿久津 達也, 教授 山本 章博, 教授 岡部 寿男 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

Delay is known to be a critical performance metric for various real-world routing applications including multimedia communication and freight delivery. Provisioning delay-minimal (or at least delay-bounded) routing services for all traffic of an application is highly important. As a basic paradigm of networking, multi-path routing has been proven to be able to obtain lower delay performance than the single-path routing, since traffic congestions can be avoided. However, to our best knowledge, (i) many of existing delay-aware multi-path routing studies only consider the aggregate traffic delay. Considering that even the solution achieving the optimal aggregate traffic delay has a possibly unbounded delay performance for certain individual traffic unit, those studies may be insufficient in practice; besides, (ii) most existing studies which optimize or bound delays of all traffic are best-effort, where the achieved solutions have no theoretical performance guarantee. In this dissertation, we study four delay-aware multi-path routing problems, with the delay performances of all traffic taken into account. Three of them are in communication and one of them is in transportation. Note that our study differ from all related ones as we are the first to study the four fundamental problems to our best knowledge. Although we prove that our studied problems are all NP-hard, we design approximation algorithms with theoretical performance guarantee for solving each of them. To be specific, we claim the following contributions. Minimize maximum delay and average delay. First, we consider a single-unicast setting where in a multi-hop network a sender requires to use multiple paths to stream a flow at a fixed rate to a receiver. Two important delay metrics are the average sender-to-receiver delay and the maximum sender-to-receiver delay. Existing results say that the two delay metrics of a flow cannot be both within bounded-ratio gaps to the optimal. In comparison, we design three different flow solutions, each of which can minimize the two delay metrics simultaneously within a $(1/epsilon)$-ratio gap to the optimal, at a cost of only delivering $(1-epsilon)$-fraction of the flow, for any user-defined $epsilonin(0,1)$. The gap $(1/epsilon)$ is proven to be at least near-tight, and we further show that our solutions can be extended to the multiple-unicast setting. Minimize Age-of-Information (AoI). Second, we consider a single-unicast setting where in a multi-hop network a sender requires to use multiple paths to periodically send a batch of data to a receiver. We study a newly proposed delay-sensitive networking performance metric, AoI, defined as the elapsed time since the generation of the last received data. We consider the problem of minimizing AoI subject to throughput requirements, which we prove is NP-hard. We note that our AoI problem differs from existing ones in that we are the first to consider the batch generation of data and multi-path communication. We develop both an optimal algorithm with a pseudo-polynomial time complexity and an approximation framework with a polynomial time complexity. Our framework can build upon any polynomial-time $alpha$-approximation algorithm of the maximum delay minimization problem, to construct an $(alpha+c)$-approximate solution for minimizing AoI. Here $c$ is a constant dependent on throughput requirements. Maximize network utility. Third, we consider a multiple-unicast setting where in a multi-hop network there exist many network users. Each user requires a sender to use multiple paths to stream a flow to a receiver, incurring an utility that is a function of the experienced maximum delay or the achieved throughput. Our objective is to maximize the aggregate utility of all users under throughput requirements and maximum delay constraints. We observe that it is NP-complete either to construct an optimal solution under relaxed maximum delay constraints or relaxed throughput requirements, or to figure out a feasible solution with all constraints satisfied. Hence it is non-trivial even to obtain approximate solutions satisfying relaxed constraints in a polynomial time. We develop a polynomial-time approximation algorithm. Our algorithm obtains solutions with constant approximation ratios under realistic conditions, at the cost of violating constraints by up to constant-ratios. Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks. Finally, we consider a common truck operation scenario where a truck is driving in a national highway network to fulfill multiple transportation tasks in order. We study an NP-hard timely eco-routing problem of minimizing total fuel consumption under task pickup and delivery time window constraints. We note that optimizing task execution times is a new challenging design space for saving fuel in our multi-task setting, and it differentiates our study from existing ones under the single-task setting. We design a fast and efficient heuristic. We characterize conditions under which the solution of our heuristic must be optimal, and further prove its optimality gap in case the conditions are not met. We simulate a heavy-duty truck driving across the US national highway system, and empirically observe that the fuel consumption achieved by our heuristic can be $22%$ less than that achieved by the fastest-/shortest- path baselines. Furthermore, the fuel saving of our heuristic as compared to the baselines is robust to the number of tasks. / Doctor of Philosophy / We consider a network modeled as a directed graph, where it takes time for data to traverse each link in the network. It models many critical applications both in the communication area and in the transportation field. For example, both the European education network and the US national highway network can be modeled as directed graphs. We consider a scenario where a source node is required to send multiple (a set of) data packets to a destination node through the network as fast as possible, possibly using multiple source-to-destination paths. In this dissertation we study four problems all of which try to figure out routing solutions to send the set of data packets, with an objective of minimizing experienced travel time or subject to travel time constraints. Although all of our four problems are NP-hard, we design approximation algorithms to solve them and obtain solutions with theoretically bounded gaps as compared to the optimal. The first three problems are in the communication area, and the last problem is in the transportation field. We claim the following specific contributions. Minimize maximum delay and average delay. First, we consider the setting of simultaneously minimizing the average travel time and the worst (largest) travel time of sending the set of data packets from source to destination. Existing results say that the two metrics of travel time cannot be minimized to be both within bounded-ratio gaps to the optimal. As a comparison, we design three different routing solutions, each of which can minimize the two metrics of travel time simultaneously within a constant bounded ratio-gap to the optimal, but at a cost of only delivering a portion of the data. Minimize Age-of-Information (AoI). Second, we consider the problem of minimizing a newly proposed travel-time-sensitive performance metric, i.e., AoI, which is the elapsed time since the generation of the last received data. Our AoI study differs from existing ones in that we are the first to consider a set of data and multi-path routing. We develop both an optimal algorithm with a pseudo-polynomial time complexity and an approximation framework with a polynomial time complexity. Maximize network utility. Third, we consider a more general setting with multiple source destination pairs. Each source incurs a utility that is a function of the experienced travel time or the achieved throughput to send data to its destination. Our objective is to maximize the aggregate utility under throughput requirements and travel time constraints. We develop a polynomial-time approximation algorithm, at the cost of violating constraints by up to constant-ratios. It is non-trivial to design such algorithms, as we prove that it is NPcomplete either to construct an optimal solution under relaxed delay constraints or relaxed throughput requirements, or to figure out a feasible solution with all constraints satisfied. Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks. Finally, we consider a truck and multiple transportation tasks in order, where each task requires the truck to pick up cargoes at a source timely, and deliver them to a destination timely. The need of coordinating task execution times is a new challenging design space for saving fuel in our multi-task setting, and it differentiates our study from existing ones under the single-task setting. We design an efficient heuristic. We characterize conditions under which the solution of our heuristic must be optimal, and further prove its performance gap as compared to the optimal in case the conditions are not met.

La complexité implicite est la caractérisation de classes de complexité par des restrictions syntaxiques sur des modèles de calcul. Plusieurs sous-systèmes de la logique linéaire caractérisant le temps polynomial ont été définis: ces systèmes sont corrects (les termes normalisent en temps polynomial) et complets (il est possible de simuler une machine de Turing pendant un nombre polynomial d'étapes). Un des buts sur le long terme est de donner statiquement des bornes de complexité. C’est pourquoi nous cherchons les caractérisations du temps polynomial les plus expressives possible. Notre principal outil est la sémantique des contextes: des jetons voyagent à travers le réseau selon certaines règles. Les chemins définis par ces jetons représentent la réduction du réseau. Contrairement aux travaux précédents, nous ne définissons pas directement des sous-systèmes de la logique linéaire. Nous définissons d'abord des relations -> sur les sous-termes des réseaux de preuves tel que: B -> C ssi ”le nombre de copies de B dépend du nombre de copies de C”. L’acyclicité de -> borne le nombre de copies de chaque sous-terme, donc la complexité du terme. Ensuite nous définissons des sous-systèmes de la logique linéaire assurant l’acyclicité de ->. Nous étudions aussi des caractérisations du temps élémentaire et primitif récursif. Dans le but d’adapter nos sous-systèmes de la logique linéaire à des langages plus riches, nous adaptons la sémantique des contextes aux réseaux d’interaction, utilisés comme langage cible pour de petits langage de programmation. Nous utilisons cette sémantique des contexte pour définir une sémantique dénotationnelle sur les réseaux d’interactions. / Implicit computational complexity is the characterization of complexity classes by syntactic restrictions on computation models. Several subsystems of linear logic characterizing polynomial time have been defined : these systems are sound (terms normalize in polynomial time) and complete (it is possible to simulate a Turing machine during a polynomial number of steps). One of the long term goals is to statically prove complexity bounds. This is why we are looking for the most expressive characterizations possible. Our main tool is context semantics : tokens travel across proof-nets (programs of linear logic) according to some rules. The paths defined by these tokens represent the reduction of the proof-net.Contrary to previous works, we do not directly define subsystems of linear logic. We first define relations -> on subterms of proof-nets such that: B -> C means \the number of copies of B depends on the number of copies of C". The acyclicity of -> allows us to bound the number of copies of any subterm, this bounds the complexity of the term. Then, we define subsystems of linear logic guaranteeing the acyclicity of ->. We also study characterizations of elementary time and primitive recursive time. In orderto adapt our linear logic subsystems to richer languages, we adapt the context semantics to interaction nets, used as a target language for small programming languages. We use this context semantics to define a denotational semantics on interaction nets.

Nous nous intéressons dans cette thèse à la problématique de la décision collective. L’objectif est de déterminer une solution de compromis pour des problèmes soumis à de multiples points de vue. Les problèmes considérés sont de nature combinatoire. Plus précisément, il s’agit de la classe des systèmes d’ensembles qui ont une structure de matroïde. La théorie des matroïdes est centrale en optimisation combinatoire, elle a permis d’unifier des structures apparemment séparées comme les arbres et les couplages dans les graphes et elle a engendré des algorithmes efficaces pour résoudre des problèmes d’optimisation non triviaux en temps polynomial. Nous nous intéressons à fournir des algorithmes d’approximation polynomiaux centralisés et décentralisés avec garantie de performance pour déterminer une solution de compromis qui est une base du matroïde. La solution de compromis doit également être équitable pour tous les membres de la collectivité. Nous portons un intérêt particulier au problème de partage équitable de biens indivisibles qui est une thématique importante en choix social computationnel et dont le problème se modélise par les matroïdes. / In this thesis, we are interested in collective decision-making. The objective is to find a tradeoff solution for problems that are evaluated by multiple points of view. We consider problems having a matroid structure. Matroid theory is significant in combinatorial optimization, it helped to unify apparently separated structures like forests and matchings in graphs and it includes efficient algorithms for solving non-trivial optimization problems in polynomial time. We are interested to provide polynomial time centralized and decentralized approximation algorithms for finding a tradeoff solution which is a base of the matroid. The tradeoff solution must also be fair for all the members of the community. We are particularly interested in the issue of the fair division of indivisible goods which is central in computational social choice and that can be modeled by matroids.