Eight-Fifth Approximation for TSP Paths

We prove the approximation ratio 8/5 for the metric $\{s,t\}$-path-TSP problem, and more generally for shortest connected $T$-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides $\{s,t\}$-tour out of those constructed from a family \Fscr_{>0} of trees having a convex combination dominated by an optimal solution $x^*$ of the fractional relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for such an $\{s,t\}$-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected $T$-joins, for $|T|\ge 4$. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing $x^*/2$ in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in \Fscr_{>0} into an $\{s,t\}$-path (or more generally, into a $T$-join) and its complement, which induces a decomposition of $x^*$. This decomposition can be refined and then efficiently used to complete $x^*/2$ without using linear programming or particular properties of $T$, but by adding to each cut deficient for $x^*/2$ an individually tailored explicitly given vector, inherent in $x^*$. A simple example shows that the Best of Many Christofides algorithm may not find a shorter $\{s,t\}$-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change

The Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

Introducing and enhancing competition to improve solid waste management in Barcelona [WP]

Over the last two decades, Barcelona has implemented a far-reaching reform of the city’s solid waste management. In 2000, the city was divided in four zones, with four separate solid waste collection contracts being awarded to private firms, with none being allowed to obtain more than two zones, a rule that was revised in 2009 to just one contract per firm. This division of the market via exclusive territories sought to enhance competition in the expectation of the convergence of relative costs, efficiency and service quality throughout the city. Based on monthly observations of costs and outputs between 2015 and 2018, this paper analyzes and evaluates the creation of lots as a tool of competition. We find that firms producing in larger zones report higher costs, that increased competition was not sufficient to lead to converging costs, and that none of the firms operate under increasing returns to scale. As such, we recommend creating an additional zone. We further suggest that if one of the zones were to be subject to public production, and adopted a mixed delivery provision strategy, the ability of the regulator to deal with asymmetric information would improve and a more reliable system could be created

The influence of independent local parties on spending: Evidence from Dutch municipalities

Do independent local parties make different decisions on municipal finances compared to their national counterparts? In this paper, we empirically analyze whether independent local parties affect public finances in Dutch municipalities. Using a matching strategy, we compare municipalities that are similar in their observable characteristics except for the presence of an independent local party majority in the municipal council. We provide evidence that shows that municipalities with independent local majorities indeed differ in terms of local spending, specifically they spend more on categories of Local Public Administration, Public Health and Environment and Culture and Recreation which are arguably more local-oriented. We extend our analysis by looking at the local effects of local independent majorities. Using a regression kink design, we find consistent results if we look at the changes that take place once the majority share of the seats in the municipal council has been reached by independent local parties

Does inter-municipal cooperation really reduce delivery costs? An empirical evaluation of the role of scale economies, transaction costs, and governance arragements [WP]

Inter-municipal cooperation in public service delivery has attracted the interest of local authorities seeking to reform public service provision in recent years. Cost saving has been among the most important drivers of such cooperation. However, the empirical results from the literature on inter- municipal cooperation and its associated costs offer contradictory outcomes in this regard. The boom in empirical studies addressing this question over the last decade offers insights into the factors that might explain the discrepancy in reported outcomes. With this objective in mind, we conduct a meta-regression analysis that considers all existing multivariate empirical studies of this matter. We formulate several hypotheses regarding scale economies, transaction costs, and governance of cooperation, based on the prior theoretical literature. While we find no clear indications of the role played by transaction costs in the relationship between cooperation and service delivery costs, we find strong evidence that population size and governance are significant in explaining the relationship. Specifically, small populations and delegation to a higher tier of government seem to offer cost advantages to municipalities when opting to cooperate. Furthermore, we build an extension of our model by disentangling service-related transaction costs based on asset specificity and ease of measurability of the service

Does inter-municipal cooperation really reduce delivery costs? an empirical evaluation of the role of scale economies, transaction costs, and governance arrangements

Inter-municipal cooperation in public service delivery has attracted the interest of local authorities seeking to reform public service provision. Cost saving, together with better quality and coordination, has been among the most important drivers of such cooperation. However, the empirical results on inter-municipal cooperation and its associated costs offer divergent outcomes. By conducting a meta-regression analysis, we seek to explain this discrepancy. We formulate several hypotheses regarding scale economies, transaction costs, and governance of cooperation. While we find no clear indications of the role played by transaction costs in the relationship between cooperation and service delivery costs, we find strong evidence that population size and governance are significant in explaining the relationship. Specifically, small populations and delegation to a higher tier of government seem to offer cost advantages to cooperating municipalities. As an extension of our model, we seek to disentangle service-related transaction costs based on asset specificity and ease of measurability of the service

Secretagogues Modulate the Calcium Concentration in the Endoplasmic Reticulum of Insulin-secreting Cells STUDIES IN AEQUORIN-EXPRESSING INTACT AND PERMEABILIZED INS-1 CELLS

The precise regulation of the Ca2+ concentration in the endoplasmic reticulum ([Ca2+]er) is important for protein processing and signal transduction. In the pancreatic beta-cell, dysregulation of [Ca2+]er may cause impaired insulin secretion. The Ca2+-sensitive photoprotein aequorin mutated to lower its Ca2+ affinity was stably expressed in the endoplasmic reticulum (ER) of rat insulinoma INS-1 cells. The steady state [Ca2+]er was 267 +/- 9 microM. Both the Ca2+-ATPase inhibitor cyclopiazonic acid and 4-chloro-m-cresol, an activator of ryanodine receptors, caused an almost complete emptying of ER Ca2+. The inositol 1,4,5-trisphosphate generating agonists, carbachol, and ATP, reduced [Ca2+]er by 20-25%. Insulin secretagogues that raise cytosolic [Ca2+] by membrane depolarization increased [Ca2+]er in the potency order K+ >> glucose > leucine, paralleling their actions in the cytosolic compartment. Glucose, which augmented [Ca2+]er by about 25%, potentiated the Ca2+-mobilizing effect of carbachol, explaining the corresponding observation in cytosolic [Ca2+]. The filling of ER Ca2+ by glucose is not directly mediated by ATP production as shown by the continuous monitoring of cytosolic ATP in luciferase expressing cells. Both glucose and K+ increase [Ca2+]er, but only the former generated whereas the latter consumed ATP. Nonetheless, drastic lowering of cellular ATP with a mitochondrial uncoupler resulted in a marked decrease in [Ca2+]er, emphasizing the requirement for mitochondrially derived ATP above a critical threshold concentration. Using alpha-toxin permeabilized cells in the presence of ATP, glucose 6-phosphate did not change [Ca2+]er, invalidating the hypothesis that glucose acts through this metabolite. Therefore, insulin secretagogues that primarily stimulate Ca2+ influx, elevate [Ca2+]er to ensure beta-cell homeostasis

A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure

Optimal General Matchings

Given a graph $G=(V,E)$ and for each vertex $v \in V$ a subset $B(v)$ of the set $\{0,1,\ldots, d_G(v)\}$, where $d_G(v)$ denotes the degree of vertex $v$ in the graph $G$, a $B$-factor of $G$ is any set $F \subseteq E$ such that $d_F(v) \in B(v)$ for each vertex $v$, where $d_F(v)$ denotes the number of edges of $F$ incident to $v$. The general factor problem asks the existence of a $B$-factor in a given graph. A set $B(v)$ is said to have a {\em gap of length} $p$ if there exists a natural number $k \in B(v)$ such that $k+1, \ldots, k+p \notin B(v)$ and $k+p+1 \in B(v)$. Without any restrictions the general factor problem is NP-complete. However, if no set $B(v)$ contains a gap of length greater than $1$, then the problem can be solved in polynomial time and Cornuejols \cite{Cor} presented an algorithm for finding a $B$-factor, if it exists. In this paper we consider a weighted version of the general factor problem, in which each edge has a nonnegative weight and we are interested in finding a $B$-factor of maximum (or minimum) weight. In particular, this version comprises the minimum/maximum cardinality variant of the general factor problem, where we want to find a $B$-factor having a minimum/maximum number of edges. We present an algorithm for the maximum/minimum weight $B$-factor for the case when no set $B(v)$ contains a gap of length greater than $1$. This also yields the first polynomial time algorithm for the maximum/minimum cardinality $B$-factor for this case

Approximation hardness of Travelling Salesman via weighted amplifiers

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from 123/122 ([9]) to 117/116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems