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We show, assuming the Strong Exponential Time Hypothesis, that for every $varepsilon > 0$, approximating directed Diameter on $m$-arc graphs within ratio $7/4 - varepsilon$ requires $m^{4/3 - o(1)}$ time. Our construction uses nonnegative edge weights but even holds for sparse digraphs, i.e., for which the number of vertices $n$ and the number of arcs $m$ satisfy $m = n log^{O(1)} n$. This is the first result that conditionally rules out a near-linear time $5/3$-approximation for Diameter.
We show, assuming the Strong Exponential Time Hypothesis, that for every $varepsilon > 0$, approximating undirected unweighted Diameter on $n$-vertex $n^{1+o(1)}$-edge graphs within ratio $7/4 - varepsilon$ requires $m^{4/3 - o(1)}$ time. This is the
The problem of finding a common refinement of a set of rooted trees with common leaf set $L$ appears naturally in mathematical phylogenetics whenever poorly resolved information on the same taxa from different sources is to be reconciled. This consti
Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an
We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively