Hardness and Approximation of Submodular Minimum Linear Ordering Problems

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $f(\cdot)$ due to an ordering $\sigma$ of the items (say $[n]$), i.e., $\min_{\sigma} \sum_{i\in [n]} f(E_{i,\sigma})$, where $E_{i,\sigma}$ is the set of items mapped by $\sigma$ to indices $[i]$. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012], using Lov\'asz extension of submodular functions. We show a $(2-\frac{1+\ell_{f}}{1+|E|})$-approximation for monotone submodular MLOP where $\ell_{f}=\frac{f(E)}{\max_{x\in E}f(\{x\})}$ satisfies $1 \leq \ell_f \leq |E|$. Our theory provides new approximation bounds for special cases of the problem, in particular a $(2-\frac{1+r(E)}{1+|E|})$-approximation for the matroid MLOP, where $f = r$ is the rank function of a matroid. We further show that minimum latency vertex cover (MLVC) is $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest

Efficient Data Collection in Multimedia Vehicular Sensing Platforms

Vehicles provide an ideal platform for urban sensing applications, as they can be equipped with all kinds of sensing devices that can continuously monitor the environment around the travelling vehicle. In this work we are particularly concerned with the use of vehicles as building blocks of a multimedia mobile sensor system able to capture camera snapshots of the streets to support traffic monitoring and urban surveillance tasks. However, cameras are high data-rate sensors while wireless infrastructures used for vehicular communications may face performance constraints. Thus, data redundancy mitigation is of paramount importance in such systems. To address this issue in this paper we exploit sub-modular optimisation techniques to design efficient and robust data collection schemes for multimedia vehicular sensor networks. We also explore an alternative approach for data collection that operates on longer time scales and relies only on localised decisions rather than centralised computations. We use network simulations with realistic vehicular mobility patterns to verify the performance gains of our proposed schemes compared to a baseline solution that ignores data redundancy. Simulation results show that our data collection techniques can ensure a more accurate coverage of the road network while significantly reducing the amount of transferred data