A Faster Parameterized Algorithm for Treedepth

The width measure \emph{treedepth}, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which---given as input an $n$-vertex graph, a tree decomposition of the graph of width $w$, and an integer $t$---decides Treedepth, i.e. whether the treedepth of the graph is at most $t$, in time $2^{O(wt)} \cdot n$. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time $2^{2^{O(t)}} \cdot n$ and $2^{O(t^2)} \cdot n$, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nesetril as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed $t$) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of $2^{O(t \log t)}\cdot n$ for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track

About Treedepth and Related Notions

In this thesis we present several results relating to treedepth. First, we provide the fastest linear-time fpt algorithm to compute the treedepth of a graph. It decides if a graph has treedepth d in time 2^O(d^2) n. In the process we answer an open question by Nešetřil and Ossona de Mendez, which asked for a simple linear-time fpt algorithm.We then proceed to compare treewidth to treedepth. We give lower bounds for the running time and space consumption of any dynamic programming algorithm (for a reasonable definition of dynamic programming on tree/path/treedepth decompositions which we introduce) for the problems Vertex Cover, 3-Coloring and Dominating Set on either a tree, a path or a treedepth decomposition. These bounds match the best known running times for these problems to date. It is not difficult to see that there are linear-time fpt algorithms for Vertex Cover and 3-Coloring parameterized by a given treedepth decomposition of depth d with a space consumption bounded by poly(d) log n. We show the same is possible for Dominating Set. We analyze the random intersection graph model, which attempts to model real-world networks where the connections between actors represent underlying shared attributes. We show that this model, when configured such that it generates degenerate graphs, produces with high probability graphs which belong to a class of bounded expansion, otherwise the graphs are asymptotically almost surely somewhere dense. We then present an algorithm for motif/subgraph counting on bounded expansion graphs which exploits a characterization of bounded expansion graph classes via decompositions into parts of bounded treedepth. Finally, we present a heuristic to compute tree decompositions which starts by computing a treedepth decomposition and show that it is competitive against other known heuristics

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded tree- or pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2−ϵ)d·logO(1)n for Vertex Cover, (3−ϵ)d·logO(1)n for 3-Coloring and (3−ϵ)d·logO(1)n for Dominating Set for any ϵ>0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for Dominating Set on graphs of treedepth d: A pure branching algorithm that runs in time dO(d2)·n and uses space O(d3logd+dlogn) and a hybrid of branching and dynamic programming that achieves a running time of O(3dlogd·n) while using O(2ddlogd+dlogn) space

Toboggan

<p>Initial release of Toboggan: a practical fpt algorithm for Flow Decomposition and transcript assembly.<br> Toboggan is a Python software tool for efficiently computing optimal flow decompositions, intended for use in high-throughput settings. The scalability of the underlying algorithm comes from parameterizing the Flow Decomposition problem by the number of paths in the decomposition, which is quite small in many real-world instances.</p