Color-flow decomposition of QCD amplitudes

We introduce a new color decomposition for multi-parton amplitudes in QCD, free of fundamental-representation matrices and structure constants. This decomposition has a physical interpretation in terms of the flow of color, which makes it ideal for merging with shower Monte-Carlo programs. The color-flow decomposition allows for very efficient evaluation of amplitudes with many quarks and gluons, many times faster than the standard color decomposition based on fundamental-representation matrices. This will increase the speed of event generators for multi-jet processes, which are the principal backgrounds to signals of new physics at colliders.Comment: 23 pages, 11 figures, version to appear on Phys. Rev.

A practical fpt algorithm for Flow Decomposition and transcript assembly

The Flow Decomposition problem, which asks for the smallest set of weighted paths that "covers" a flow on a DAG, has recently been used as an important computational step in transcript assembly. We prove the problem is in FPT when parameterized by the number of paths by giving a practical linear fpt algorithm. Further, we implement and engineer a Flow Decomposition solver based on this algorithm, and evaluate its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic. Finally, we contextualize our design choices with two hardness results related to preprocessing and weight recovery. Specifically, $k$-Flow Decomposition does not admit polynomial kernels under standard complexity assumptions, and the related problem of assigning (known) weights to a given set of paths is NP-hard.Comment: Introduces software package Toboggan: Version 1.0. http://dx.doi.org/10.5281/zenodo.82163

Flow Decomposition for Multi-User Channels - Part I

A framework based on the idea of flow decomposition is proposed to characterize the decode-forward region for general multi-source, multi-relay, all-cast channels with independent input distributions. The region is difficult to characterize directly when deadlocks occur between two relay nodes, in which both relays benefit by decoding after each other. Rate-vectors in the decode-forward region depend ambiguously on the outcomes of all deadlocks in the channel. The region is characterized indirectly in two phases. The first phase assumes relays can operate non-causally. It is shown that every rate-vector in the decode-forward region corresponds to a set of flow decompositions, which describe the messages decoded at each node with respect to the messages forwarded by all the other nodes. The second phase imposes causal restrictions on the relays. Given an arbitrary set of (possibly non-causal) flow decompositions, necessary and sufficient conditions are derived for the existence of an equivalent set of causal flow decompositions that achieves the same rate-vector region

Flow Decomposition With Subpath Constraints

Flow network decomposition is a natural model for problems where we are given a flow network arising from superimposing a set of weighted paths and would like to recover the underlying data, i.e., decompose the flow into the original paths and their weights. Thus, variations on flow decomposition are often used as subroutines in multiassembly problems such as RNA transcript assembly. In practice, we frequently have access to information beyond flow values in the form of subpaths, and many tools incorporate these heuristically. But despite acknowledging their utility in practice, previous work has not formally addressed the effect of subpath constraints on the accuracy of flow network decomposition approaches. We formalize the flow decomposition with subpath constraints problem, give the first algorithms for it, and study its usefulness for recovering ground truth decompositions. For finding a minimum decomposition, we propose both a heuristic and an FPTalgorithm. Experiments on RNA transcript datasets show that for instances with larger solution path sets, the addition of subpath constraints finds 13% more ground truth solutions when minimal decompositions are found exactly, and 30% more ground truth solutions when minimal decompositions are found heuristically.Peer reviewe

Generating QCD amplitudes in the color-flow basis with MadGraph

We propose to make use of the off-shell recursive relations with the color-flow decomposition in the calculation of QCD amplitudes on MadGraph. We introduce colored quarks and their interactions with nine gluons in the color-flow basis plus an Abelian gluon on MadGraph, such that it generates helicity amplitudes in the color-flow basis with off-shell recursive formulae for multi-gluon sub-amplitudes. We demonstrate calculations of up to 5-jet processes such as $gg\rightarrow 5g$, $u\bar{u}\rightarrow 5g$ and $uu\rightarrow uuggg$. Although our demonstration is limited, it paves the way to evaluate amplitudes with more quark lines and gluons with Madgraph.Comment: 29 pages, 13 figure

Canonical information flow decomposition among neural structure subsets

Partial directed coherence (PDC) and directed coherence (DC) which describe complementary aspects of the directed information flow between pairs of univariate components that belong to a vector of simultaneously observed time series have recently been generalized as bPDC/bDC respectively to portray the relationship between subsets of component vectors (Takahashi, 2009; Faes and Nollo, 2013). This generalization is specially important for neuroscience applications as one often wishes to address the link between the set of time series from an observed ROI (region of interest) with respect to series from some other physiologically relevant ROI. bPDC/bDC are limited, however, in that several time series within a given subset may be irrelevant or may even interact opposingly with respect to one another leading to interpretation difficulties. To address this, we propose an alternative measure, termed cPDC/cDC, employing canonical decomposition to reveal the main frequency domain modes of interaction between the vector subsets. We also show bPDC/bDC and cPDC/cDC are related and possess mutual information rate interpretations. Numerical examples and a real data set illustrate the concepts. The present contribution provides what is seemingly the first canonical decomposition of information flow in the frequency domain

Efficient Minimum Flow Decomposition via Integer Linear Programming

Extended version of RECOMB 2022 paperMinimum flow decomposition (MFD) is an NP-hard problem asking to decompose a network flow into a minimum set of paths (together with associated weights). Variants of it are powerful models in multiassembly problems in Bioinformatics, such as RNA assembly. Owing to its hardness, practical multiassembly tools either use heuristics or solve simpler, polynomial time-solvable versions of the problem, which may yield solutions that are not minimal or do not perfectly decompose the flow. Here, we provide the first fast and exact solver for MFD on acyclic flow networks, based on Integer Linear Programming (ILP). Key to our approach is an encoding of all the exponentially many solution paths using only a quadratic number of variables. We also extend our ILP formulation to many practical variants, such as incorporating longer or paired-end reads, or minimizing flow errors. On both simulated and real-flow splicing graphs, our approach solves any instance inPeer reviewe

Resilient flow decomposition of unicast connections with network coding

In this paper we close the gap between end-to-end diversity coding and intra-session network coding for unicast connections resilient against single link failures. In particular, we show that coding operations are sufficient to perform at the source and receiver if the user data can be split into at most two parts over the filed GF(2). Our proof is purely combinatorial and based on standard graph and network flow techniques. It is a linear time construction that defines the route of subflows A, B and A+B between the source and destination nodes. The proposed resilient flow decomposition method generalizes the 1+1 protection and the end-to-end diversity coding approaches while keeping both of their benefits. It provides a simple yet resource efficient protection method feasible in 2-connected backbone topologies. Since the core switches do not need to be modified, this result can bring benefits to current transport networks.Comment: submitted to IEEE International Symposium on Information Theory (ISIT) 201