Network-Based Vertex Dissolution

We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.Comment: Version accepted at SIAM Journal on Discrete Mathematic

Partitioning Perfect Graphs into Stars

The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor

Combinatorial auctions : theory, experiments, and practice

This doctoral dissertation contributes to theory, experiments, and practice in combinatorial auctions. Combinatorial auctions are multi-object auctions, that enable bidders to bid on packages of items. In Chapter 2, we theoretically investigate the classical winner determination problem in geometrical settings. Specifically, we consider combinatorial auctions of items that can be arranged in rows, and the objective is, given bids on subsets of items, to find a subset of bids that maximizes auction revenue. Possible practical applications include allocating pieces of land for real estate development, or seats in a theater or stadium. We investigate several geometrical structures and bid shapes, and provide either a polynomial dynamic programming algorithm or an NP-hardness proof, filling in several gaps in academic literature. In Chapter 3, we combine theory with experiments, investigating coordination and threshold problems in combinatorial auctions. Bidders on small packages of items are unable to outbid provisionally winning bids on large packages alone; despite free-rider incentives, both coordination and cooperation are required. Coordination because smaller bidders need to bid on disjoint packages, and cooperation because more than one bidder is required to outbid a larger package bid. We propose indices quantifying both the coordination and the threshold problem, that can be used in providing feedback or generating valuations for laboratory experiments. Additionally, we develop coalitional feedback that is specifically aimed at helping bidders to overcome coordination and threshold problems. We test this in an experimental setting using human bidders, varying feedback from provisionally winning bids and prices, to winning and deadness levels, and coalitional feedback. We find that in situations where threshold problems are severe, coalitional feedback increases economic efficiency, but in easy or insurmountable threshold problems that is not always the case. Finally, in Chapter 4, we combine theory with practice. Scheduling a conference, based on preferences expressed by conference participants, can be seen as a combinatorial auction with public goods. In a situation with public goods, the utility of the final selected goods (presentations scheduled in parallel) are "consumed" by all bidders (conference participants). Constructing a good conference schedule is important: they are an essential part of academic research and require significant investments (e.g.\ time and money) from their participants. We provide computational complexity results, along with a combined approach of assigning talks to rooms and time slots, grouping talks into sessions, and deciding on an optimal itinerary for each participant. Our goal is to maximize attendance, considering the common practice of session hopping. On a secondary level, we accommodate presenters’ availabilities. This personalized conference scheduling approach has been applied to construct the schedule of the MathSport (2013), MAPSP (2015 and 2017) and ORBEL (2017) conferences

Números de dominación y empaquetamiento de ciertas gráficas de fichas

Dada una gráfica G de orden n ≥ 3, se define a F2(G) como la gráfica cuyo conjunto de vértices consiste de todos los 2-subconjuntos de V (G), y dos vértices X, Y de F2(G) serán adyacentes si y solo si la diferencia simétrica de X y Y consta de dos vértices que son adyacentes en G. Hasta donde sabemos, la gráfica F2(G) fue introducida por Johns en 1988 [43]. Posteriormente, en 1991 Alavi et al. [2], la definieron con el nombre de gráfica de doble vértice. En 2002, T. Rudolph [62] redefinió la misma gráfica bajo el nombre de el cuadrado simétrico de G. Similarmente, si en lugar de los 2-subconjuntos de V (G), se consideran los k-subconjuntos de V (G), donde k є {2 ,..., n - 1}, entonces obtenemos la gráfica Fk(G) que se llama la k-potencia simétrica de G [9]. En [27] Fabila-Monroy et. al., a Fk(G) la llamaron gráfica de k-fichas de G. Como se puede verificar en [3, 4, 6, 8, 9, 17, 18, 27, 39, 48, 52, 37, 53, 62, 66, 67] y las referencias contenidas en esos artículos, el interés en las gráficas Fk(G) ha generado una gran cantidad de investigaciones en muchas ramas de las matemáticas discretas. Una de las líneas de investigación con mayor actividad consiste en estimar o determinar el valor exacto de diversos parámetros combinatorios de Fk(G). En este trabajo se presentan algunos resultados originales sobre la estimación de dos de estos parámetros: el número de dominación y el número de empaquetamiento de F2(Pn) y F3(Pn), en donde Pn denota a la gráfica camino de orden n. El resultado principal de esta tesis consiste en la determinación del valor exacto del número de empaquetamiento de F2(Pn). Este resultado tiene como consecuencia la confirmación de una conjetura de Rob Pratt sobre el tamaño máximo que puede tener un código binario de longitud n y peso constante 2 que es 1-corrector para una transposición adyacente, y ha sido publicado en [30]. En particular, esta conjetura proponía la función generatriz ordinaria correspondiente a la secuencia A085680 en la OEIS (The On-Line Encyclopedia of Integer Sequences) [59]

Combinatorial approaches for the trunk packing problem

In this thesis we consider a three-dimensional packing problem arising in industry. The task is to pack a maximum number of rigid boxes with side length ratios of 4 : 2 : 1 into an irregularly shaped container. Motivated by the structure of manually constructed packings so far, we pursue a discrete approach. We discretize the shape of the container as well as the set of possible box placements. This discrete packing problem can be reduced to a maximum stable set problem. First we formulate the problem as an integer linear program, which admittedly can only be solved to optimality within reasonable runtime for very small instances. Therefore, we present several heuristics based, for example, on the linear programming relaxation or on local search. Other heuristics generate tight packings for the core of the container, thereby reducing the problem to a set of smaller subproblems. We compare all presented algorithms on real data sets. We achieve very good results for the majority of instances and for some instances we even surpass the manually constructed solutions.In dieser Arbeit behandeln wir ein dreidimensionales Packungsproblem aus der Industrie. Die Aufgabe besteht darin, möglichst viele starre Quader mit einem Seitenverhältnis von 4 : 2 : 1 in einen unregelmäßig geformten Container zu packen. Motiviert durch die Struktur der bisher manuell erstellten Packungen verfolgen wir einen diskreten Lösungsansatz. Dazu diskretisieren wir sowohl die Form des Containers als auch die Platzierungsmöglichkeiten der Quader. Dieses diskrete Packungsproblem lässt sich auf die Berechnung einer größtmöglichen unabhängigen Knotenmenge reduzieren. Wir formulieren das Problem zunächst als ganzzahliges lineares Programm, das allerdings nur für sehr kleine Instanzen mit angemessenem Rechenaufwand beweisbar optimal gelöst werden kann. Daher stellen wir verschiedene Heuristiken vor, die zum Beispiel auf einer Relaxierung des ganzzahligen linearen Programms oder lokaler Suche basieren. Andere Heuristiken generieren zunächst dichte Packungen für den Kern des Containers und reduzieren so das Problem auf eine Reihe kleinerer Teilprobleme. Wir vergleichen alle vorgestellten Algorithmen an Hand realer Datensätze. In der Mehrzahl der Fälle erreichen wir sehr gute Resultate, bei einigen Instanzen übertreffen wir sogar die manuell erstellten Packungen

Generalized Set and Graph Packing Problems

Many complex systems that exist in nature and society can be expressed in terms of networks (e.g., social networks, communication networks, biological networks, Web graph, among others). Usually a node represents an entity while an edge represents an interaction between two entities. A community arises in a network when two or more entities have common interests, e.g., related proteins, industrial sectors, groups of people, documents of a collection. There exist applications that model a community as a fixed graph H [98, 10, 119, 2, 142, 136]. Additionally, it is not expected that an entity of the network belongs to only one community; that is, communities tend to share their members. The community discovering or community detection problem consists on finding all communities in a given network. This problem has been extensively studied from a practical perspective [61, 137, 122, 116]. However, we believe that this problem also brings many interesting theoretical questions. Thus in this thesis, we will address this problem using a more rigorous approach. To that end, we first introduce graph problems that we consider capture well the community discovering problem. These graph problems generalize the classical H-Packing problem [88] in two different ways. In the H-Packing with t-Overlap problem, the goal is to find in a given graph G (the network) at least k subgraphs (the communities) isomorphic to a member of a family of graphs H (the community models) such that each pair of subgraphs overlaps in at most t vertices (the shared members). On the other hand, in the H-Packing with t-Membership problem instead of limiting the pairwise overlap, each vertex of G is contained in at most t subgraphs of the solution. For both problems each member of H has at most r vertices and m edges. An instance of the H-Packing with t-Overlap and t-Membership problems corresponds to an instance of the H-Packing problem for t = 0 and t = 1, respectively. We also restrict the overlap between the edges of the subgraphs in the solution instead of the vertices (called H-Packing with t-Edge Overlap and t-Edge Membership problems). Given the closeness of the r-Set Packing problem [87] to the H-Packing problem, we also consider overlap in the problem of packing disjoint sets of size at most r. As usual for set packing problems, given a collection S drawn from a universe U, we seek a sub-collection S'⊆S consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U belongs to at most t sets of S' while in the r-Set Packing with t-Overlap each pair of sets in S' overlaps in at most t elements. For both problems, each set of S has at most r elements. We refer to all the problems introduced in this thesis simply as packing problems with overlap. Also, we group as the family of t-Overlap problems: H-Packing with t-Overlap, H-Packing with t-Edge Overlap, and r-Set Packing with t-Overlap. While we call the family of t-Membership problems: H-Packing with t-Membership, H-Packing with t-Edge Membership, and r-Set Packing with t-Membership. The classical H-Packing and r-Set Packing problems are NP-complete [87, 88]. We will show in this thesis that allowing overlap in a packing does not make the problems "easier". More precisely, we show that the H-Packing with t-Membership and the r-Set Packing with t-Membership are NP-complete when H = {H'} and H' is an arbitrary connected graph with at least three vertices and r≥3, respectively. Parameterized complexity, introduced by Downey and Fellows [44], is an exciting and interesting approach to deal with NP-complete problems. The underlying idea of this approach is to isolate some aspects or parts of the input (known as the parameters) to investigate whether these parameters make the problem tractable or intractable. The main goal of this thesis is to study the parameterized complexity of our packing problems with overlap. We set up as a parameter k the size of the solution (number of communities), and we consider as fixed-constants r, m and t. We show that our problems admit polynomial kernels via two types of techniques: polynomial parametric transformations (PPTs) [16] and classical reduction algorithms [43]. PPTs are mainly used to show lower bounds and as far as we know they have not been used as extensively to obtain kernel results as classical kernelization techniques [96, 42]. Thus, we believe that employing PPTs is a promising approach to obtain kernel reductions for other problems as well. On the other hand, with non-trivial generalizations of kernelization algorithms for the classical H-Packing problem [114], we are able to improve our kernel sizes obtained via PPTs. These improved kernel sizes are equivalent to the kernel sizes for the disjoint version when t = 0 and t = 1 for the t-Overlap and t-Membership problems, respectively. We also obtain fixed-parameter algorithms for our packing problems with overlap (other than running brute force on the kernel). Our algorithms combine a search tree and a greedy localization technique and generalize a fixed-parameter algorithm for the problem of packing disjoint triangles [54]. In addition, we obtain faster FPT-algorithms by transforming our overlapping problems into an instance of the disjoint version of our problems. Finally, we introduce the Π-Packing with α()-Overlap problem to allow for more complex overlap constraints than the ones considered by the t-Overlap and t-Membership problems and also to include more general communities definitions. This problem seeks at least k induced subgraphs in a graph G subject to: each subgraph has at most r vertices and obeys a property Π (a community definition) and for any pair of subgraphs Hi,Hj, with i≠j, we have that α(Hi,Hj) = 0 holds (an overlap constraint). We show that the Π-Packing with α()-Overlap problem is fixed-parameter tractable provided that Π is computable in polynomial time in n and α() obeys some natural conditions. Motivated by practical applications we give several examples of α() functions which meet those conditions

Generalized Planar Matching

In this paper, we prove that maximum planar H-matching (the problem of determining the maximum number of node-disjointed copies of the fixed graph H contained in a variable planar graph G) is NP-complete for any connected planar graph H with three or more nodes. We also show that perfect planar H-matching is NP-complete for any connected outerplanar graph H with three or more nodes, and is, somewhat surprisingly, solvable in linear time for triangulated H with four or more nodes. The results generalize and unify several special-case results proved in the literature. The techniques can also be applied to solve a variety of problems, including the optimal tile salvage problem from wafer-scale integration. Although we prove that the optimal tile salvage problem and other like it are NP-complete, we also describe provably good approximation algorithms that are suitable for practical applications