Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

We provide CONGEST model algorithms for approximating minimum weighted vertex cover and the maximum weighted matching. For bipartite graphs, we show that a $(1+\varepsilon)$-approximate weighted vertex cover can be computed deterministically in polylogarithmic time. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least a $\lambda$-fraction of the total weight, one can compute a $(2-2\lambda +\varepsilon)$-approximate weighted vertex cover in polylogarithmic time in the CONGEST model. Our result in particular implies that in graphs of arboricity $a$, one can compute a $(2-1/a+\varepsilon)$-approximate weighted vertex cover. For maximum weighted matchings, we show that a $(1-\varepsilon)$-approximate solution can be computed deterministically in polylogarithmic CONGEST rounds (for constant $\varepsilon$). We also provide a more efficient randomized algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the unweighted case. Finally, we show that even in the LOCAL model and in bipartite graphs of degree $\leq 3$, if $\varepsilon<\varepsilon_0$ for some constant $\varepsilon_0>0$, then computing a $(1+\varepsilon)$-approximation for the unweighted minimum vertex cover problem requires $\Omega\big(\frac{\log n}{\varepsilon}\big)$ rounds. This generalizes aresult of [G\"o\"os, Suomela; DISC '12], who showed that computing a $(1+\varepsilon_0)$-approximation in such graphs requires $\Omega(\log n)$ rounds

Applications of matching theory in constraint programming

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Efficient Propagators for Global Constraints

We study in this thesis three well known global constraints. The All-Different constraint restricts a set of variables to be assigned to distinct values. The global cardinality constraint (GCC) ensures that a value v is assigned to at least lv variables and to at most uv variables among a set of given variables where lv and uv are non-negative integers such that lv &le; uv. The Inter-Distance constraint ensures that all variables, among a set of variables x1, . . . , xn, are pairwise distant from p, i. e. |xi - xj| &ge; p for all i &ne; j. The All-Different constraint, the GCC, and the Inter-Distance constraint are largely used in scheduling problems. For instance, in scheduling problems where tasks with unit processing time compete for a single resource, we have an All-Different constraint on the starting time variables. When there are k resources, we have a GCC with lv = 0 and uv = k over all starting time variables. Finally, if tasks have processing time t and compete for a single resource, we have an Inter-Distance constraint with p = t over all starting time variables. We present new propagators for the All-Different constraint, the GCC, and the Inter-Distance constraint i. e. , new filtering algorithms that reduce the search space according to these constraints. For a given consistency, our propagators outperform previous propagators both in practice and in theory. The gains in performance are achieved through judicious use of advanced data structures combined with novel results on the structural properties of the constraints

Hamiltonian cycles on Ammann-Beenker tilings

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for th

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

Design principles for linear systems: Stability and optimality

This thesis consists of two parts. Each of them deals with problems in the design of linear time-invariant systems with certain prescribed properties, such as stability and cost optimality. The first part addresses theoretical questions arising in the design of autonomous decentralized systems. The network topology of such a system describes which agents are able to interact with each other. We study the following problem: For a specified network topology, can one find a set of interaction laws that yield stable dynamics for the ensemble of agents? We restrict our analysis to systems with strictly linear dynamics. This problem can also be referred to as the structural stability problem, seen as the counterpart to the structural controllability problem. In mathematical terms, we consider vector spaces of real square matrices for which every entry is either fixed at zero, or an arbitrary real number. We call them sparse matrix spaces, abbreviated SMS, and examine under what conditions they contain matrices for which all eigenvalues have strictly negative real parts. We call an SMS with this property stable. We estimate the proportion of stable SMS when their size approaches infinity and when the locations of the free variables are chosen independently at random. Using graph theory techniques, we also develop polynomial-time algorithms for extension of a given stable SMS to a stable SMS with up to two additional nodes. In the second part, we consider linear time-invariant systems with control. The well-known linear quadratic regulator (LQR) provides feedback controller that stabilizes the system while minimizing a quadratic cost function in the state of the system and the magnitude of the control. The optimal actuator design problem then consists of choosing an actuator that minimizes the cost incurred by an LQR. While this procedure guarantees a low overall cost incurred, it only takes into account the magnitude of the control signals the regulator sends to the actuator. Physical actuators are, however, also limited in their ability to follow rapid change in control signals. We show in this thesis how to design actuators so that the high-frequency content of the control signals is limited, while insuring stability and optimality of the resulting closed-loop system. We also address optimal actuator design for linear systems with process noise. It is well-known that the control that minimizes a quadratic cost in the state and control for a system with linear dynamics corrupted by additive Gaussian noise is of feedback type and its design depends on the solution of an associated Riccati equation. We consider here the case where the noise is multiplicative, by which we mean that its intensity is dependent on the state. We show how to derive the actuator that minimizes a linear quadratic cost