Neste trabalho abordamos o problema de coloraÃÃo de vÃrtices via programaÃÃo inteira. Uma versÃo expandida da formulaÃÃo por conjuntos independentes à utilizada para abrigar outras sub-estruturas do grafos alÃm dos vÃrtices. Cada uma dessas sub-estruturas define uma restriÃÃo que determina quantos conjuntos independentes sÃo necessarios para cobrir aquele subgrafo. Experimentos com um mÃtodo de geraÃÃo de cortes e colunas para o problema sÃo feitos para determinar um limite inferior para um conjunto de instÃncias classicas para esse problema a biblioteca DIMACS. / In this work the vertex coloring problem is approached via integer programming. A tighter version of the independent set formulation is used, where the vertex-related constraints are substituted by subgraph-related constraints. Each constraint establishes a lower bound on the number of independent sets intersecting a subgraph H. It is shown a sufficient condition for this inequality to define a facet of the associated polytope. Basically, H is required to be color critical, not included in another color critical subgraph, and to have a connected complement. Also, the column generation algorithm proposed by Mehotra and Trick (INFORMS Journal in Computing, 1996) is adapted to allow the addition of cutting planes and to provide lower bounds along the process, which may abbreviate its end. Some computational experiments are reported.

RODRIGUES, Carlos Diego. Limites inferiores para o problema de coloração de vértices via geração de cortes e colunas. 2008. 79 f. : Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Computação, Fortaleza-CE, 2005. / Submitted by guaracy araujo (guaraa3355@gmail.com) on 2016-06-08T19:45:39Z No. of bitstreams: 1 2008_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) / Approved for entry into archive by guaracy araujo (guaraa3355@gmail.com) on 2016-06-08T19:45:58Z (GMT) No. of bitstreams: 1 2008_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) / Made available in DSpace on 2016-06-08T19:45:58Z (GMT). No. of bitstreams: 1 2008_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) Previous issue date: 2008 / In this work the vertex coloring problem is approached via integer programming. A tighter version of the independent set formulation is used, where the vertex-related constraints are substituted by subgraph-related constraints. Each constraint establishes a lower bound on the number of independent sets intersecting a subgraph H. It is shown a sufficient condition for this inequality to define a facet of the associated polytope. Basically, H is required to be color critical, not included in another color critical subgraph, and to have a connected complement. Also, the column generation algorithm proposed by Mehotra and Trick (INFORMS Journal in Computing, 1996) is adapted to allow the addition of cutting planes and to provide lower bounds along the process, which may abbreviate its end. Some computational experiments are reported. / Neste trabalho abordamos o problema de coloração de vértices via programação inteira. Uma versão expandida da formulação por conjuntos independentes é utilizada para abrigar outras sub-estruturas do grafos além dos vértices. Cada uma dessas sub-estruturas define uma restrição que determina quantos conjuntos independentes são necessarios para cobrir aquele subgrafo. Experimentos com um método de geração de cortes e colunas para o problema são feitos para determinar um limite inferior para um conjunto de instâncias classicas para esse problema a biblioteca DIMACS.

RODRIGUES, Carlos Diego. Limites inferiores para o problema de coloração de vértices via geração de cortes e colunas. 2008. 79 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2008. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-11T11:50:40Z No. of bitstreams: 1 2005_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-14T15:28:23Z (GMT) No. of bitstreams: 1 2005_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) / Made available in DSpace on 2016-07-14T15:28:23Z (GMT). No. of bitstreams: 1 2005_dis_cdrodrigues.pdf: 545679 bytes, checksum: 7cceeca6a76bce10cbde4bfb4ef0ee02 (MD5) Previous issue date: 2008 / In this work the vertex coloring problem is approached via integer programming. A tighter version of the independent set formulation is used, where the vertex-related constraints are substituted by subgraph-related constraints. Each constraint establishes a lower bound on the number of independent sets intersecting a subgraph H. It is shown a sufficient condition for this inequality to define a facet of the associated polytope. Basically, H is required to be color critical, not included in another color critical subgraph, and to have a connected complement. Also, the column generation algorithm proposed by Mehotra and Trick (INFORMS Journal in Computing, 1996) is adapted to allow the addition of cutting planes and to provide lower bounds along the process, which may abbreviate its end. Some computational experiments are reported. / Neste trabalho abordamos o problema de coloração de vértices via programação inteira. Uma versão expandida da formulação por conjuntos independentes é utilizada para abrigar outras sub-estruturas do grafos além dos vértices. Cada uma dessas sub-estruturas define uma restrição que determina quantos conjuntos independentes são necessarios para cobrir aquele subgrafo. Experimentos com um método de geração de cortes e colunas para o problema são feitos para determinar um limite inferior para um conjunto de instâncias classicas para esse problema a biblioteca DIMACS.

This work introduces two new parameterizations of graph problems generalizing vertex cover which fill part of the space between vertex cover and clique width in the hierarchy of graf parameterizations. We also study parameterized complexity of Hamiltonian path and cycle, vertex coloring, precoloring extension and equitable coloring parameterized by these two parameterizations. With the exception of precoloring extension which is W[1]-hard in one case, all the other problems listed above are tractable for both parameterizations. The boundary between tractability and intractability of these problems can therefore be moved closer to parameterization by clique width.

CAMPOS, Victor Almeida. Um estudo do politopo e dos limites inferiores gerados pela formulação de coloração dos representantes. 2005. 108 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2005. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-12T18:37:10Z No. of bitstreams: 1 2005_dis_vacampos.pdf: 624425 bytes, checksum: 13eb092def3e5c973c883bf32b893ba8 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-22T12:41:39Z (GMT) No. of bitstreams: 1 2005_dis_vacampos.pdf: 624425 bytes, checksum: 13eb092def3e5c973c883bf32b893ba8 (MD5) / Made available in DSpace on 2016-07-22T12:41:39Z (GMT). No. of bitstreams: 1 2005_dis_vacampos.pdf: 624425 bytes, checksum: 13eb092def3e5c973c883bf32b893ba8 (MD5) Previous issue date: 2005 / The vertex coloring problem is one of the most studied problems in graph theory for its relevance in practical and theoretical fields. From a theoretical point of view, it is a NP-Hard problem. Moreover, it is classified among the most difficult problems of NP- Hard in the sense that finding an approximation to the chromatic number is also NP-Hard. The importance of the coloring problem motivates searching for methods to find lower bounds close to the chromatic number. Historically, the first lower bounds used were obtained from the size of maximal cliques. More recently, relaxed integer programming formulations gained more attention. A formulation which found good lower bounds was the coloring problem through stable sets whose relaxed lower bound equals the fractional chromatic number. In this work, we make a comparison between the known integer programming formulations to motivate our choice for the Representatives formulation. We revise this formulation to remove symmetry and present a partial study of the polytope associated with the convex hull of its integer solutions. We discuss how to se the Representatives formulation to get lower bounds for the fractional chromatic number and we show how to get such lower bounds that differ at most by one unit to its exact value. / O problema de coloração de vértices é considerado um dos modelos mais estudados em teoria dos grafos pela sua relevância em campos práticos e teóricos. Do ponto de vista teórico, o problema de coloração é NP - Difícil. Além disto, foi classificado entre os problemas mais difíceis de NP, no sentido de que achar uma aproximação para o número cromático também é NP - Difícil. A importância do problema de coloração tem incentivado a investigar métodos para encontrar limitantes inferiores próximos do número cromático. Historicamente, os primeiros limitantes inferiores utilizados para resolvê-lo lidavam com cliques maximais. Mais recentemente, popularizou-se a utilização de relaxações lineares de formulações de programação inteira. Uma formulação que mostrou bons limitantes inferiores foi a formulação por conjuntos independentes, cujo valor de relaxação equivale ao número cromático fracionário. No presente trabalho, fazemos uma comparação entre as formulações de programação inteira conhecidas para indicar a escolha pela formulação dos representantes. Revisamos a formulação para remover simetrias existentes e apresentamos um estudo parcial do politopo associado ao fecho convexo de suas soluções inteiras. Discutimos como é possível utilizar a formulação dos representantes para gerar limites inferiores para o número cromático fracionário. Realizamos a implementação de um método de planos de corte para aproximar o número cromático fracionário e mostramos que podemos gerar limitantes inferiores que normalmente não diferem em mais de uma unidade.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O problema de coloraÃÃo de vÃrtices à considerado um dos modelos mais estudados em teoria dos grafos pela sua relevÃncia em campos prÃticos e teÃricos. Do ponto de vista teÃrico, o problema de coloraÃÃo à NP - DifÃcil. AlÃm disto, foi classificado entre os problemas mais difÃceis de NP, no sentido de que achar uma aproximaÃÃo para o nÃmero cromÃtico tambÃm à NP - DifÃcil. A importÃncia do problema de coloraÃÃo tem incentivado a investigar mÃtodos para encontrar limitantes inferiores prÃximos do nÃmero cromÃtico. Historicamente, os primeiros limitantes inferiores utilizados para resolvÃ-lo lidavam com cliques maximais. Mais recentemente, popularizou-se a utilizaÃÃo de relaxaÃÃes lineares de formulaÃÃes de programaÃÃo inteira. Uma formulaÃÃo que mostrou bons limitantes inferiores foi a formulaÃÃo por conjuntos independentes, cujo valor de relaxaÃÃo equivale ao nÃmero cromÃtico fracionÃrio. No presente trabalho, fazemos uma comparaÃÃo entre as formulaÃÃes de programaÃÃo inteira conhecidas para indicar a escolha pela formulaÃÃo dos representantes. Revisamos a formulaÃÃo para remover simetrias existentes e apresentamos um estudo parcial do politopo associado ao fecho convexo de suas soluÃÃes inteiras. Discutimos como à possÃvel utilizar a formulaÃÃo dos representantes para gerar limites inferiores para o nÃmero cromÃtico fracionÃrio. Realizamos a implementaÃÃo de um mÃtodo de planos de corte para aproximar o nÃmero cromÃtico fracionÃrio e mostramos que podemos gerar limitantes inferiores que normalmente nÃo diferem em mais de uma unidade. / The vertex coloring problem is one of the most studied problems in graph theory for its relevance in practical and theoretical fields. From a theoretical point of view, it is a NP-Hard problem. Moreover, it is classified among the most difficult problems of NP- Hard in the sense that finding an approximation to the chromatic number is also NP-Hard. The importance of the coloring problem motivates searching for methods to find lower bounds close to the chromatic number. Historically, the first lower bounds used were obtained from the size of maximal cliques. More recently, relaxed integer programming formulations gained more attention. A formulation which found good lower bounds was the coloring problem through stable sets whose relaxed lower bound equals the fractional chromatic number. In this work, we make a comparison between the known integer programming formulations to motivate our choice for the Representatives formulation. We revise this formulation to remove symmetry and present a partial study of the polytope associated with the convex hull of its integer solutions. We discuss how to se the Representatives formulation to get lower bounds for the fractional chromatic number and we show how to get such lower bounds that differ at most by one unit to its exact value.

Στα πλαίσια της διατριβής αναπτύξαμε δύο από τους πρώτους αλγορίθμους υπολογισμού μιας ε-προσεγγιστικής ισορροπίας Nash για την περίπτωση όπου το ε είναι κάποια σταθερά. Οι προσεγγίσεις που επιτυγχάνουν οι αλγόριθμοί μας είναι ε=3/4 και ε=(2+λ)/4 αντίστοιχα, όπου λ είναι το ελάχιστο, μεταξύ όλων των ισορροπιών Nash, κέρδος για έναν παίκτη. Επιπλέον, μελετήσαμε μια ευρεία κλάση τυχαίων παιγνίων δύο παικτών, για την οποία υπολογίσαμε μια πολύ καλή ε-προσεγγιστική ισορροπία Nash, με το ε να τείνει στο 0 καθώς το πλήθος των διαθέσιμων στρατηγικών των παικτών τείνει στο άπειρο. Οι αρχές της θεωρίας παιγνίων είναι χρήσιμες στην ανάλυση της επίδρασης που έχει στην καθολική απόδοση ενός συστήματος διαμοιραζόμενων πόρων η εγωιστική και ανταγωνιστική συμπεριφορά των χρηστών του. Προς την κατεύθυνση αυτή, εστιάσαμε στο πρόβλημα της εξισορρόπησης φορτίου. Μελετήσαμε διάφορα μοντέλα πληροφόρησης (π.χ. όταν όλα τα φορτία είναι άγνωστα ή όταν κάθε παίκτης γνωρίζει το μέγεθος του δικού του φορτίου) και αναλύσαμε για το καθένα το σύνολο και τις ιδιότητες των ισορροπιών Nash. Yπολογίσαμε επίσης φράγματα στο λόγο απόκλισης, ο οποίος εκφράζει την επίδραση που έχει στην απόδοση του συστήματος η εγωιστική συμπεριφορά των χρηστών του. Εκτός από τα υπολογιστικά θέματα που σχετίζονται με τη θεωρία παιγνίων, έχει ενδιαφέρον να μελετηθεί κατά πόσο μπορεί η θεωρία παιγνίων να βοηθήσει στην ανάπτυξη και ανάλυση αλγορίθμων για υπολογιστικά δύσκολα προβλήματα συνδυαστικής βελτιστοποίησης. Προς αυτήν την κατεύθυνση, μελετήσαμε από παιγνιοθεωρητική σκοπιά το πρόβλημα χρωματισμού των κορυφών ενός γραφήματος. Ορίσαμε κατάλληλα το παίγνιο χρωματισμού γραφήματος και αποδείξαμε ότι κάθε παίγνιο χρωματισμού γραφήματος έχει πάντα μια αγνή ισορροπία Nash, και ότι κάθε αγνή ισορροπία Nash αντιστοιχεί σε ορθό χρωματισμό του γραφήματος. Δείξαμε επίσης ότι υπάρχει πάντα μια αγνή ισορροπία Nash που χρησιμοποιεί βέλτιστο αριθμό χρωμάτων, δηλαδή ίσο με το χρωματικό αριθμό του γραφήματος. Επιπλέον, περιγράψαμε και αναλύσαμε έναν πολυωνυμικό αλγόριθμο που υπολογίζει μια αγνή ισορροπία Nash για ένα οποιοδήποτε παίγνιο χρωματισμού γραφήματος και χρησιμοποιεί συνολικά ένα πλήθος χρωμάτων που ικανοποιεί ταυτόχρονα τα περισσότερα κλασικά γνωστά φράγματα στο χρωματικό αριθμό. / We developed two algorithms for computing an e-approximate Nash equilibrium for the case where e is an absolute constant. The approximations achieved by our algorithms are e=3/4 and e=(2+l)/4 respectively, where $\lambda$ is the minimum, among all Nash equilibria, payoff of either player. Furthermore, we studied a wide class of random two player games, for which we showed how to compute an e-approximate Nash equilibrium, where e tends to zero as the number of strategies of the players tends to infinity. Game theoretic concepts are useful in determining the impact that selfish behavior plays on the global performance of a system involving selfish entities. Towards this direction, we focused on the problem of load balancing. We studied the case where the agents are not necessarily fully informed about the exact values of their loads. We focused on several models of information (e.g. when all agents know nothing about the loads, or when each agents knows her own load) and, for each model, we characterized the set of Nash equilibria and analyzed their properties. Moreover, we bounded the coordination ratio, a measure which captures the impact that selfish behavior has to the global performance of the system, in contrast to the performance achieved by an optimum centralized algorithm. Besides the computational issues related to game theory, it is interesting to investigate whether game theory can help us in developing and analyzing algorithms for computationally difficult combinatorial optimization problems. Towards this direction, we studied from a game theoretic point of view the problem of vertex coloring. In particular, we properly defined the graph coloring game and we proved that every graph coloring game has a pure Nash equilibrium, and each pure Nash equilibrium corresponds to a proper coloring of the graph. We also showed that there exists a pure Nash equilibrium that uses an optimum number of colors, i.e. equal to the chromatic number. Furthermore, we developed and analyzed a polynomial time algorithm that computes a pure Nash equilibrium for any graph coloring game, using a number of colors satisfying most of the known classical bounds on the chromatic number.

Network coding has emerged as an attractive alternative to routing because of the through put improvement it provides by reducing the number of channel uses. In a wireless scenario, in addition, further improvement can be obtained through Physical layer Network Coding (PNC), a technique in which nodes are allowed to transmit simultaneously, instead of transmitting in orthogonal slots. In this thesis, the design and analysis of network coding schemes are considered, for wireless two-way relaying, multi-user Multiple Access Relay Channel (MARC) and wireline networks. In a wireless two-way relay channel with PNC, the simultaneous transmissions of user nodes result in Multiple Access Interference (MAI) at there lay node. The harmful effect of MAI is the presence of signal set dependent deep channel fade conditions, called singular fade states, under which the minimum distance of the effective constellation at the relay become zero. Adaptively changing the network coding map used at the relay according to channel conditions greatly reduces the impact of this MAI. In this work, we obtain these adaptive PNC maps, which are finite in number ,by completing partially filled Latin Squares and using graph vertex coloring. Having obtained the network coding maps, the set of all possible channel realizations is quantized into a finite number of regions, with a specific network coding map chosen in a particular region and such a quantization is obtained analytically for 2λ-PSK signal set. The performance of the adaptive PNC scheme for two-way relaying is analyzed and tight high SNR upper bounds are obtained for the average end-to-end symbol error probability, in terms of the average error probability of a point-to-point fading channel. The adaptive PNC scheme is generalized for two-way relaying with multiple antennas at the nodes. As an alternative to the adaptive PNC scheme for two-way relaying, a Distributed Space Time Coding (DSTC) scheme is proposed, which effectively re-moves the effect of singular fade states at the transmitting nodes itself without any Channel State Information at the Transmitter (CSIT), and without any need to change the PNC map as a function of channel fade conditions. It is shown that the singular fade states can be viewed equivalently as vector subspaces of C2, which are referred to as the singular fade subspaces. DSTC design criterion to minimize the number of singular fade subspaces and maximize the coding gain is formulated and explicit low decoding complexity DSTC designs are provided. For the K-user MARC, in which K source nodes want to transmit messages to a destination node D with the help of are lay node R, a new PNC scheme is proposed. Use of a many-to-one PNC map with conventional minimum squared Euclidean distance decoding at D, results in a loss of diversity order due to error propagation from the relay node. To counter this, we propose a novel low complexity decoder which offers the maximum diversity order of two. Next, we consider wire line networks and explore the connections between linear network coding, linear index coding and discrete polymatroids, which are the multi-set analogue of matroids. We define a discrete polymatroidal network and show that a fractional vector linear solution over a field Fq exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over Fq.An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over Fq for the discrete polymatroid, results in a fractional vector linear solution over Fq for the constructed network. It is shown that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. Multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.