arXiv subjects

Shamisa Nematollahi

Publications and source records attributed to Shamisa Nematollahi.

3 recordsLinked to original sources

arXiv2026

Buy-at-Bulk Facility Location on Trees

We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the $k$-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a $(1+ε)$-approximation for any $ε> 0$). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than $3/2$ on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of $1+ε$. As a consequence, we show that the problem has a $2$-approximation algorithm on trees.

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arXiv2026

Stochastic Multi-Robot Monitoring on Graphs under Markovian Mobility

We study a stochastic multi-robot monitoring problem on a connected graph $G=(V,E)$, where each robot moves according to a Markov chain on $G$ and monitors the closed neighborhood of its current vertex. The performance of $r$ robots is evaluated in steady state via two objectives: average-case coverage (the expected number of covered vertices) and worst-case coverage (the minimum coverage probability over all vertices). We consider three models: independent homogeneous strategies, where all robots share the same stationary distribution; independent heterogeneous strategies, where robots use different stationary distributions; and centralized strategies, allowing arbitrary correlations between robot locations. For the heterogeneous model, we prove that maximizing average coverage is NP-hard even for two robots, and that replicating an easy-to-compute optimal homogeneous strategy yields a $\left(1-\left(1-\frac{1}{r}\right)^r\right)$-approximation for both objective functions in the heterogeneous setting; moreover, no polynomial-time algorithm can achieve a ratio better than $1-\nicefrac{1}{e}$ unless $\text{P}=\text{NP}$. Centralized strategies can exploit correlations to reduce redundancy. We develop a hierarchy of approximation factors: for any positive integer $r'\le r$, writing $r=hr'+b$ with $0\le b<r'$, block coordination yields a $1-\left(1-\frac{r'}{r}\right)^h\left(1-\frac{b}{r}\right)$ approximation for both objectives. We also establish NP-hardness and a tight $1-\nicefrac{1}{e}$ inapproximability bound. Moreover, we prove diminishing-returns properties with respect to the number of robots: a non-increasing-ratio property holds for the average-case objective in all settings, but not for the heterogeneous worst-case objective.

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arXiv2025

Fixed-Parameter Tractable Submodular Maximization over a Matroid

In this paper, we design fixed-parameter tractable (FPT) algorithms for (non-monotone) submodular maximization subject to a matroid constraint, where the matroid rank $r$ is treated as a fixed parameter that is independent of the total number of elements $n$. We provide two FPT algorithms: one for the offline setting and another for the random-order streaming setting. Our streaming algorithm achieves a $\frac{1}{2}-\varepsilon$ approximation using $\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)$ memory, while our offline algorithm obtains a $1-\frac{1}{e}-\varepsilon$ approximation with $n\cdot 2^{\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)}$ runtime and $\widetilde{O}\left(\frac{r}{\textrm{poly}(\varepsilon)}\right)$ memory. Both approximation factors are near-optimal in their respective settings, given existing hardness results. In particular, our offline algorithm demonstrates that--unlike in the polynomial-time regime--there is essentially no separation between monotone and non-monotone submodular maximization under a matroid constraint in the FPT framework.

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