Bisection of Bounded Treewidth Graphs by Convolutions

In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]). In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n)

Simultaneous Feedback Vertex Set: A Parameterized Perspective

Given a family of graphs $\mathcal{F}$, a graph $G$, and a positive integer $k$, the $\mathcal{F}$-Deletion problem asks whether we can delete at most $k$ vertices from $G$ to obtain a graph in $\mathcal{F}$. $\mathcal{F}$-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph $G = (V, \cup_{i=1}^{\alpha} E_{i})$, where the edge set of $G$ is partitioned into $\alpha$ color classes, is called an $\alpha$-edge-colored graph. A natural extension of the $\mathcal{F}$-Deletion problem to edge-colored graphs is the $\alpha$-Simultaneous $\mathcal{F}$-Deletion problem. In the latter problem, we are given an $\alpha$-edge-colored graph $G$ and the goal is to find a set $S$ of at most $k$ vertices such that each graph $G_i \setminus S$, where $G_i = (V, E_i)$ and $1 \leq i \leq \alpha$, is in $\mathcal{F}$. In this work, we study $\alpha$-Simultaneous $\mathcal{F}$-Deletion for $\mathcal{F}$ being the family of forests. In other words, we focus on the $\alpha$-Simultaneous Feedback Vertex Set ($\alpha$-SimFVS) problem. Algorithmically, we show that, like its classical counterpart, $\alpha$-SimFVS parameterized by $k$ is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant $\alpha$. In particular, we give an algorithm running in $2^{O(\alpha k)}n^{O(1)}$ time and a kernel with $O(\alpha k^{3(\alpha + 1)})$ vertices. The running time of our algorithm implies that $\alpha$-SimFVS is FPT even when $\alpha \in o(\log n)$. We complement this positive result by showing that for $\alpha \in O(\log n)$, where $n$ is the number of vertices in the input graph, $\alpha$-SimFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014)

The Effect of Interfacial Chemical Bonding in TiO2-SiO2 Composites on their Photocatalytic NOx Abatement Performance

The authors gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council (Grant Ref: EP/M003299/1) and the Natural Science Foundation of China (No. 51461135005) International Joint Research Project (EPSRC-NSFC).Peer reviewedPublisher PD

Photocatalyst efficiencies in concrete technology : the effect of photocatalyst placement

The authors gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council (Grant Ref: EP/M003299/1) and the Natural Science Foundation of China (No. 51461135005, No. 51478370) International Joint Research Project (EPSRC-NSFC).Peer reviewedPublisher PD

Self-Repairing Disk Arrays

As the prices of magnetic storage continue to decrease, the cost of replacing failed disks becomes increasingly dominated by the cost of the service call itself. We propose to eliminate these calls by building disk arrays that contain enough spare disks to operate without any human intervention during their whole lifetime. To evaluate the feasibility of this approach, we have simulated the behavior of two-dimensional disk arrays with n parity disks and n(n-1)/2 data disks under realistic failure and repair assumptions. Our conclusion is that having n(n+1)/2 spare disks is more than enough to achieve a 99.999 percent probability of not losing data over four years. We observe that the same objectives cannot be reached with RAID level 6 organizations and would require RAID stripes that could tolerate triple disk failures.Comment: Part of ADAPT Workshop proceedings, 2015 (arXiv:1412.2347

Implementation of horizontal well CBM/ECBM technology and the assessment of effective CO2 storage capacity in a Scottish coalfield

Acknowledgements The authors wish to thank Composite Energy Ltd., the BG Group, Scottish Power and the Royal Bank of Scotland for their funding and contributions towards the research reported in this paper.Non peer reviewedPublisher PD