Afrouz Jabal Ameli

Afrouz Jabal Ameli, Tomohiro Koana, Jesper Nederlof et al.

The Strongly Connected Steiner Subgraph (SCSS) problem is a well-studied network design problem that asks for a minimum subgraph that strongly connects a given set of terminals. In this paper, we present several new algorithmic and complexity results for SCSS. As our main result, we show that SCSS can be solved in time $17^{\mathrm{tw}} n^{O(1)}$ on directed graphs with $n$ vertices when a tree decomposition of the underlying graph of width $\mathrm{tw}$ is provided. This improves over a natural $\mathrm{tw}^{O(\mathrm{tw})}n^{O(1)}$ time algorithm, and is the first algorithm with this kind of running time for a problem involving strong connectivity. Second, we give an exact exponential-time algorithm that solves SCSS in $2^n n^{O(1)}$ time, improving the known bounds for general directed graphs. Finally, we investigate kernelization with respect to vertex cover. We prove that SCSS does not admit a polynomial kernel when parameterized by the size of a vertex cover, unless the polynomial hierarchy collapses. In contrast, we show that the closely related Strongly Connected Spanning Subgraph problem does admit a polynomial kernel under the same parameterization.

Afrouz Jabal Ameli, Jesper Nederlof, Shengzhe Wang

We provide improved space-time tradeoffs for permutation problems over additively idempotent semi-rings. In particular, there is an algorithm for the Traveling Salesperson Problem that solves $N$-vertex instances using space $S$ and time $T$ where $S\cdot T \leq 3.7493^{N}$. This improves a previous work by Koivisto and Parviainen [SODA'10] where $S\cdot T \leq 3.9271^N$, and overcomes a barrier they identified, as their bound was shown to be optimal within their framework. To get our results, we introduce a new parameter of a set system that we call the chain efficiency. This relates the number of maximal chains contained in the set system with the cardinality of the system. We show that set systems of high efficiency imply efficient space-time tradeoffs for permutation problems, and give constructions of set systems with high chain efficiency, disproving a conjecture by Johnson, Leader and Russel [Comb. Probab. Comput.'15].