Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph $G=(U,V,E)$ is convex if the vertices in $V$ can be linearly ordered such that for each vertex $u\in U$, the neighbors of $u$ are consecutive in the ordering of $V$. An induced matching $H$ of $G$ is a matching such that no edge of $E$ connects endpoints of two different edges of $H$. We show that in a convex bipartite graph with $n$ vertices and $m$ weighted edges, an induced matching of maximum total weight can be computed in $O(n+m)$ time. An unweighted convex bipartite graph has a representation of size $O(n)$ that records for each vertex $u\in U$ the first and last neighbor in the ordering of $V$. Given such a compact representation, we compute an induced matching of maximum cardinality in $O(n)$ time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in $O(n)$ time. If no compact representation is given, the cover can be computed in $O(n+m)$ time. All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of $O(n^2)$

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before

Robust Assignments via Ear Decompositions and Randomized Rounding

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape

Connected matchings in special families of graphs.

A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, Thomassé first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ¼, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D