Large Cuts with Local Algorithms on Triangle-Free Graphs

We study the problem of finding large cuts in $d$-regular triangle-free graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a cut of expected size $(1/2 + 0.177/\sqrt{d})m$, where $m$ is the number of edges. We give a simpler algorithm that does much better: it finds a cut of expected size $(1/2 + 0.28125/\sqrt{d})m$. As a corollary, this shows that in any $d$-regular triangle-free graph there exists a cut of at least this size. Our algorithm can be interpreted as a very efficient randomised distributed algorithm: each node needs to produce only one random bit, and the algorithm runs in one synchronous communication round. This work is also a case study of applying computational techniques in the design of distributed algorithms: our algorithm was designed by a computer program that searched for optimal algorithms for small values of $d$.Comment: 1+17 pages, 8 figure

Large Cuts with Local Algorithms on Triangle-Free Graphs

Let G be a d-regular triangle-free graph with in edges. We present an algorithm which finds a cut in G with at least (1/2 + 0.28125/root d)rn edges in expectation, improving upon Shearer's classic result. In particular, this implies that any d-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of G. Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed d, there exists a weighted neighbourhood graph N-d such that there is a one-to-one correspondence between heavy cuts of N-d and randomised local algorithms that find large cuts in any d-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in d-regular graphs: we can compute the optimal cut of N-d to attain a lower bound on the maximum cut size of any d-regular triangle-free graph.Peer reviewe

A tail-like assembly at the portal vertex in intact herpes simplex type-1 virions

Herpes viruses are prevalent and well characterized human pathogens. Despite extensive study, much remains to be learned about the structure of the genome packaging and release machinery in the capsids of these large and complex double-stranded DNA viruses. However, such machinery is well characterized in tailed bacteriophage, which share a common evolutionary origin with herpesvirus. In tailed bacteriophage, the genome exits from the virus particle through a portal and is transferred into the host cell by a complex apparatus (i.e. the tail) located at the portal vertex. Here we use electron cryo-tomography of human herpes simplex type-1 (HSV-1) virions to reveal a previously unsuspected feature at the portal vertex, which extends across the HSV-1 tegument layer to form a connection between the capsid and the viral membrane. The location of this assembly suggests that it plays a role in genome release into the nucleus and is also important for virion architecture

Closure properties of pattern languages

Pattern languages are a well-established class of languages, but very little is known about their closure properties. In the present paper we establish a large number of closure properties of the terminal-free pattern languages, and we characterise when the union of two terminal-free pattern languages is again a terminal-free pattern language. We demonstrate that the equivalent question for general pattern languages is characterised differently, and that it is linked to some of the most prominent open problems for pattern languages. We also provide fundamental insights into a well-known construction of E-pattern languages as unions of NE-pattern languages, and vice versa

Closure properties of pattern languages

Pattern languages are a well-established class of languages that is particularly popular in algorithmic learning theory, but very little is known about their closure properties. In the present paper we establish a large number of closure properties of the terminal-free pattern languages, and we characterise when the union of two terminal-free pattern languages is again a terminal-free pattern language. We demonstrate that the equivalent question for general pattern languages is characterised differently, and that it is linked to some of the most prominent open problems for pattern languages. We also provide fundamental insights into a well-known construction of E-pattern languages as unions of NE-pattern languages, and vice versa. © 2014 Springer International Publishing Switzerland

Brief Announcement: Temporal Locality in Online Algorithms

Online algorithms make decisions based on past inputs, with the goal of being competitive against an algorithm that sees also future inputs. In this work, we introduce time-local online algorithms; these are online algorithms in which the output at any given time is a function of only T latest inputs. Our main observation is that time-local online algorithms are closely connected to local distributed graph algorithms: distributed algorithms make decisions based on the local information in the spatial dimension, while time-local online algorithms make decisions based on the local information in the temporal dimension. We formalize this connection, and show how we can directly use the tools developed to study distributed approximability of graph optimization problems to prove upper and lower bounds on the competitive ratio achieved with time-local online algorithms. Moreover, we show how to use computational techniques to synthesize optimal time-local algorithms

Input-Dynamic Distributed Algorithms for Communication Networks

Consider a distributed task where the communication network is fixed but the local inputs given to the nodes of the distributed system may change over time. In this work, we explore the following question: if some of the local inputs change, can an existing solution be updated efficiently, in a dynamic and distributed manner? To address this question, we define the batch dynamic CONGEST model in which we are given a bandwidth-limited communication network and a dynamic edge labelling defines the problem input. The task is to maintain a solution to a graph problem on the labeled graph under batch changes. We investigate, when a batch of $\alpha$ edge label changes arrive, -- how much time as a function of $\alpha$ we need to update an existing solution, and -- how much information the nodes have to keep in local memory between batches in order to update the solution quickly. Our work lays the foundations for the theory of input-dynamic distributed network algorithms. We give a general picture of the complexity landscape in this model, design both universal algorithms and algorithms for concrete problems, and present a general framework for lower bounds. In particular, we derive non-trivial upper bounds for two selected, contrasting problems: maintaining a minimum spanning tree and detecting cliques