Motivo: Fast Motif Counting via Succinct Color Coding and Adaptive Sampling

The randomized technique of color coding is behind state-of-the-art algorithms for estimating graph motif counts. Those algorithms, however, are not yet capable of scaling well to very large graphs with billions of edges. In this paper we develop novel tools for the `motif counting via color coding' framework. As a result, our new algorithm, Motivo, is able to scale well to larger graphs while at the same time provide more accurate graphlet counts than ever before. This is achieved thanks to two types of improvements. First, we design new succinct data structures that support fast common color coding operations, and a biased coloring trick that trades accuracy versus running time and memory usage. These adaptations drastically reduce the time and memory requirements of color coding. Second, we develop an adaptive graphlet sampling strategy, based on a fractional set cover problem, that breaks the additive approximation barrier of standard sampling. This strategy gives multiplicative approximations for all graphlets at once, allowing us to count not only the most frequent graphlets but also extremely rare ones. To give an idea of the improvements, in $40$ minutes Motivo counts $7$-nodes motifs on a graph with $65$M nodes and $1.8$B edges; this is $30$ and $500$ times larger than the state of the art, respectively in terms of nodes and edges. On the accuracy side, in one hour Motivo produces accurate counts of $\approx \! 10.000$ distinct $8$-node motifs on graphs where state-of-the-art algorithms fail even to find the second most frequent motif. Our method requires just a high-end desktop machine. These results show how color coding can bring motif mining to the realm of truly massive graphs using only ordinary hardware.Comment: 13 page

Tracks from hell - when finding a proof may be easier than checking it

We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them

Beeping a Maximal Independent Set

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.Comment: arXiv admin note: substantial text overlap with arXiv:1108.192

Locality of not-so-weak coloring

Many graph problems are locally checkable: a solution is globally feasible if it looks valid in all constant-radius neighborhoods. This idea is formalized in the concept of locally checkable labelings (LCLs), introduced by Naor and Stockmeyer (1995). Recently, Chang et al. (2016) showed that in bounded-degree graphs, every LCL problem belongs to one of the following classes: - "Easy": solvable in $O(\log^* n)$ rounds with both deterministic and randomized distributed algorithms. - "Hard": requires at least $\Omega(\log n)$ rounds with deterministic and $\Omega(\log \log n)$ rounds with randomized distributed algorithms. Hence for any parameterized LCL problem, when we move from local problems towards global problems, there is some point at which complexity suddenly jumps from easy to hard. For example, for vertex coloring in $d$-regular graphs it is now known that this jump is at precisely $d$ colors: coloring with $d+1$ colors is easy, while coloring with $d$ colors is hard. However, it is currently poorly understood where this jump takes place when one looks at defective colorings. To study this question, we define $k$-partial $c$-coloring as follows: nodes are labeled with numbers between $1$ and $c$, and every node is incident to at least $k$ properly colored edges. It is known that $1$-partial $2$-coloring (a.k.a. weak $2$-coloring) is easy for any $d \ge 1$. As our main result, we show that $k$-partial $2$-coloring becomes hard as soon as $k \ge 2$, no matter how large a $d$ we have. We also show that this is fundamentally different from $k$-partial $3$-coloring: no matter which $k \ge 3$ we choose, the problem is always hard for $d = k$ but it becomes easy when $d \gg k$. The same was known previously for partial $c$-coloring with $c \ge 4$, but the case of $c < 4$ was open

Exact bounds for distributed graph colouring

We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with $n$ colours, by prior work it is known that we can find a proper 3-colouring in $\frac{1}{2} \log^*(n) \pm O(1)$ communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many $n$ the time complexity is precisely $\frac{1}{2} \log^* n$ communication rounds.Comment: 16 pages, 3 figure

Viability Assessment in Liver Transplantation—What Is the Impact of Dynamic Organ Preservation?

Based on the continuous increase of donor risk, with a majority of organs classified as marginal, quality assessment and prediction of liver function is of utmost importance. This is also caused by the notoriously lack of effective replacement of a failing liver by a device or intensive care treatment. While various parameters of liver function and injury are well-known from clinical practice, the majority of specific tests require prolonged diagnostic time and are more difficult to assess ex situ. In addition, viability assessment of procured organs needs time, because the development of the full picture of cellular injury and the initiation of repair processes depends on metabolic active tissue and reoxygenation with full blood over several hours or days. Measuring injury during cold storage preservation is therefore unlikely to predict the viability after transplantation. In contrast, dynamic organ preservation strategies offer a great opportunity to assess organs before implantation through analysis of recirculating perfusates, bile and perfused liver tissue. Accordingly, several parameters targeting hepatocyte or cholangiocyte function or metabolism have been recently suggested as potential viability tests before organ transplantation. We summarize here a current status of respective machine perfusion tests, and report their clinical relevance

Frameworks for logically classifying polynomial-time optimisation problems.

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems