The independent set sequence of regular bipartite graphs

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite $G$. We consider the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges). Using bounds on the independent set polynomial $P(G,\lambda):=\sum_{t \geq 0} i_t(G)\lambda^t$ for these graphs, we obtain partial unimodality results in these cases. We then focus on the discrete hypercube $Q_d$, the graph on vertex set $\{0,1\}^d$ with two strings adjacent if they differ on exactly one coordinate. We obtain asymptotically tight estimates for $i_{t(d)}(Q_d)$ in the range $t(d)/2^{d-1} > 1-1/\sqrt{2}$, and nearly matching upper and lower bounds otherwise. We use these estimates to obtain a stronger partial unimodality result for the independent set sequence of $Q_d$.Comment: 18 pages, some typos from earlier versions corrected, this version to appear in Discrete Mathematic

A bipartite graph with non-unimodal independent set sequence

We show that the independent set sequence of a bipartite graph need not be unimodal

Two problems on independent sets in graphs

Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph $G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is unimodal. We provide evidence for this conjecture by showing that is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph $G(n,n,p)$, and show that for any fixed $p\in(0,1]$ its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for $p=\tilde{\Omega}(n^{-1/2})$. We also consider the problem of estimating $i(G)=\sum_{t \geq 0} i_t(G)$ for $G$ in various families. We give a sharp upper bound on the number of independent sets in an $n$-vertex graph with minimum degree $\delta$, for all fixed $\delta$ and sufficiently large $n$. Specifically, we show that the maximum is achieved uniquely by $K_{\delta, n-\delta}$, the complete bipartite graph with $\delta$ vertices in one partition class and $n-\delta$ in the other. We also present a weighted generalization: for all fixed $x>0$ and $\delta >0$, as long as $n=n(x,\delta)$ is large enough, if $G$ is a graph on $n$ vertices with minimum degree $\delta$ then $\sum_{t \geq 0} i_t(G)x^t \leq \sum_{t \geq 0} i_t(K_{\delta, n-\delta})x^t$ with equality if and only if $G=K_{\delta, n-\delta}$.Comment: 15 pages. Appeared in Discrete Mathematics in 201

GPU Accelerated Simulation of Transport Systems

Computer modelling and simulation of road networks are a vital tool used to evaluate, design and manage road network infrastructure. Road network simulations are however computationally expensive, with simulation runtime imposing limits on the scale and quantity of simulations performed within a reasonable time frame. This thesis examines the appropriateness of many-core processing architectures (such as GPUs) for the acceleration of microscopic and macroscopic road network simulation, and the potential impact on the choice of modelling approach. Fine-grained agent-based microscopic simulations of individual vehicles are parallelised using GPUs, achieving high performance through a novel graph-based communication strategy for data-parallel simulations. A minimal benchmark model and scalable road network are defined and used experimentally to evaluate performance compared to Aimsun, a commercial simulation tool for multi-core processors. Performance improvements of up to 67x are demonstrated for large scale simulations. High-level macroscopic simulations model network flow rather than individual vehicles. Although less computationally demanding than microscopic models, simulation runtimes can still be significant, often due to the calculation of many shortest paths. A novel Many-Source Shortest Path (MSSP) algorithm is proposed to concurrently find multiple shortest paths through sparse transport networks using GPUs. This is embedded within a commercial multi-core CPU macroscopic simulation tool, SATURN, and the performance evaluated on large-scale real-world road networks, demonstrating assignment performance improvements of up to 8.6x when comparing multi-processor GPU and CPU implementations. Finally, the impact of the performance improvements to both modelling techniques are evaluated using a common benchmark model and the relative improvements demonstrated by the benchmarking of each approach using different transport networks. These results suggest that GPUs will allow modellers to shift towards using finer-grained simulations for a broader range of modelling tasks