Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments

We consider the minimum spanning tree (MST) problem in an uncertainty model where uncertain edge weights can be explored at extra cost. The task is to find an MST by querying a minimum number of edges for their exact weight. This problem has received quite some attention from the algorithms theory community. In this paper, we conduct the first practical experiments for MST under uncertainty, theoretically compare three known algorithms, and compare theoretical with practical behavior of the algorithms. Among others, we observe that the average performance and the absolute number of queries are both far from the theoretical worst-case bounds. Furthermore, we investigate a known general preprocessing procedure and develop an implementation thereof that maximally reduces the data uncertainty. We also characterize a class of instances that is solved completely by our preprocessing. Our experiments are based on practical data from an application in telecommunications and uncertainty instances generated from the standard TSPLib graph library

The Complexity of Approximately Counting Retractions

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems

Counting Homomorphisms to K4K_4-minor-free Graphs, modulo 2

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$

Counting Small Induced Subgraphs with Hereditary Properties

We study the computational complexity of the problem #IndSub(\Phi) of counting k-vertex induced subgraphs of a graph G that satisfy a graph property \Phi. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): - If a hereditary property \Phi is true for all graphs, or if it is true only for finitely many graphs, then #IndSub(\Phi) is solvable in polynomial time. - Otherwise, #IndSub(\Phi) is #W[1]-complete when parameterised by k, and, assuming ETH, it cannot be solved in time f(k)*|G|^{o(k)} for any function f. This classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [TCS 02]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem. As additional result, we also present an exhaustive and explicit parameterised complexity classification for all properties that are invariant under homomorphic equivalence. By covering one of the most natural and general notions of closure, namely, closure under vertex-deletion (hereditary), we generalise some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of #IndSub(\Phi) for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [FOCS 20]

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time $c_{\sigma,\rho}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets

Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations

We study the complexity of approximating the number of answers to a small query \varphi in a large database D. We establish an exhaustive classification into tractable and intractable cases if \varphi is a conjunctive query possibly including disequalities and negations: - If there is a constant bound on the arity of \varphi, and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of \varphi is bounded. - If the arity is unbounded and \varphi does not have negations, then the problem has an FPTRAS if and only if the adaptive width of \varphi (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless NP =RP, there is no FPRAS even if the treewidth (and the adaptive width) is 1. However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021)

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as \textsc{Independent Set}, \textsc{Dominating Set}, \textsc{Independent Dominating Set}, and many others. For almost all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)} (if a tree decomposition of width \tw is given in the input). Let \sigMax denote the largest element of $\sigma$ if $\sigma$ is finite, or the largest missing integer $+1$ if $\sigma$ is cofinite; \rhoMax is defined analogously for $\rho$. Surprisingly, $c_{\sigma,\rho}$ is often significantly smaller than the natural bound \sigMax+\rhoMax+2 achieved by existing algorithms [van Rooij, 2020]. Toward defining $c_{\sigma,\rho}$, we say that $(\sigma, \rho)$ is \mname-structured if there is a pair $(\alpha,\beta)$ such that every integer in $\sigma$ equals $\alpha$ mod \mname, and every integer in $\rho$ equals $\beta$ mod \mname. Then, setting \begin{itemize} \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+1 if $(\sigma,\rho)$ is \mname-structured for some \mname \ge 3, or 2-structured with \sigMax\neq \rhoMax, or 2-structured with \sigMax=\rhoMax being odd, \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+2 if $(\sigma,\rho)$ is 2-structured, but not \mname-structured for any \mname \ge 3, and \sigMax=\rhoMax is even, and \item c_{\sigma,\rho}=\sigMax+\rhoMax+2 if $(\sigma,\rho)$ is not \mname-structured for any \mname\ge 2, \end{itemize} we provide algorithms counting $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)}. For example, for the \textsc{Exact Independent Dominating Set} problem (also known as \textsc{Perfect Code}) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, this improves the 3^\tw\cdot n^{\O(1)} algorithm of van Rooij to 2^\tw\cdot n^{\O(1)}. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial (except those having cofinite $\sigma$ with $\rho=\mathbb Z_{\ge0}$), and any $\varepsilon>0$, a (c_{\sigma,\rho}-\varepsilon)^\tw\cdot n^{\O(1)}-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (\#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets

Vacuum/Compression Valving (VCV) Using Parrafin-Wax on a Centrifugal Microfluidic CD Platform

This paper introduces novel vacuum/compression valves (VCVs) utilizing paraffin wax. A VCV is implemented by sealing the venting channel/hole with wax plugs (for normally-closed valve), or to be sealed by wax (for normally-open valve), and is activated by localized heating on the CD surface. We demonstrate that the VCV provides the advantages of avoiding unnecessary heating of the sample/reagents in the diagnostic process, allowing for vacuum sealing of the CD, and clear separation of the paraffin wax from the sample/reagents in the microfluidic process. As a proof of concept, the microfluidic processes of liquid flow switching and liquid metering is demonstrated with the VCV. Results show that the VCV lowers the required spinning frequency to perform the microfluidic processes with high accuracy and ease of control.open5

Multi-messenger observations of a binary neutron star merger

On 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of ~1.7 s with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg2 at a luminosity distance of 40+8-8 Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 Mo. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at ~40 Mpc) less than 11 hours after the merger by the One- Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ~10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position ~9 and ~16 days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta