Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2

Fast Algorithms for Join Operations on Tree Decompositions

Treewidth is a measure of how tree-like a graph is. It has many important algorithmic applications because many NP-hard problems on general graphs become tractable when restricted to graphs of bounded treewidth. Algorithms for problems on graphs of bounded treewidth mostly are dynamic programming algorithms using the structure of a tree decomposition of the graph. The bottleneck in the worst-case run time of these algorithms often is the computations for the so called join nodes in the associated nice tree decomposition. In this paper, we review two different approaches that have appeared in the literature about computations for the join nodes: one using fast zeta and M\"obius transforms and one using fast Fourier transforms. We combine these approaches to obtain new, faster algorithms for a broad class of vertex subset problems known as the [\sigma,\rho]-domination problems. Our main result is that we show how to solve [\sigma,\rho]-domination problems in $O(s^{t+2} t n^2 (t\log(s)+\log(n)))$ arithmetic operations. Here, t is the treewidth, s is the (fixed) number of states required to represent partial solutions of the specific [\sigma,\rho]-domination problem, and n is the number of vertices in the graph. This reduces the polynomial factors involved compared to the previously best time bound (van Rooij, Bodlaender, Rossmanith, ESA 2009) of $O( s^{t+2} (st)^{2(s-2)} n^3 )$ arithmetic operations. In particular, this removes the dependence of the degree of the polynomial on the fixed number of states~$s$.Comment: An earlier version appeared in "Treewidth, Kernels, and Algorithms. Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday" LNCS 1216

Maximizing Happiness in Graphs of Bounded Clique-Width

Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Reddy '18 about parameterization by the distance to threshold graphs by showing that MHE is NP-complete on threshold graphs. Hence, it is not even in XP when parameterized by clique-width, since threshold graphs have clique-width at most two. As a complement for this result we provide a $n^{\mathcal{O}(\ell \cdot \operatorname{cw})}$ algorithm for MHE, where $\ell$ is the number of colors and $\operatorname{cw}$ is the clique-width of the input graph. We also construct an FPT algorithm for MHV with running time $\mathcal{O}^*((\ell+1)^{\mathcal{O}(\operatorname{cw})})$, where $\ell$ is the number of colors in the input. Additionally, we show $\mathcal{O}(\ell n^2)$ algorithm for MHV on interval graphs.Comment: Accepted to LATIN 202

Hamiltonicity below Dirac's condition

Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time $c^k \cdot n^{O(1)}$, for some fixed constant $c$, if at least $n-k$ vertices have degree at least $n/2$, or if all vertices have degree at least $n/2-k$. The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH). The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with $O(k)$ vertices can be found in polynomial time

Sp6 and Sp8 transcription factors control AER formation and dorsal-ventral patterning in limb development

The formation and maintenance of the apical ectodermal ridge (AER) is critical for the outgrowth and patterning of the vertebrate limb. The induction of the AER is a complex process that relies on integrated interactions among the Fgf, Wnt, and Bmp signaling pathways that operate within the ectoderm and between the ectoderm and the mesoderm of the early limb bud. The transcription factors Sp6 and Sp8 are expressed in the limb ectoderm and AER during limb development. Sp6 mutant mice display a mild syndactyly phenotype while Sp8 mutants exhibit severe limb truncations. Both mutants show defects in AER maturation and in dorsal-ventral patterning. To gain further insights into the role Sp6 and Sp8 play in limb development, we have produced mice lacking both Sp6 and Sp8 activity in the limb ectoderm. Remarkably, the elimination or significant reduction in Sp6;Sp8 gene dosage leads to tetra-amelia; initial budding occurs, but neither Fgf8 nor En1 are activated. Mutants bearing a single functional allele of Sp8 (Sp6-/-;Sp8+/-) exhibit a split-hand/foot malformation phenotype with double dorsal digit tips probably due to an irregular and immature AER that is not maintained in the center of the bud and on the abnormal expansion of Wnt7a expression to the ventral ectoderm. Our data are compatible with Sp6 and Sp8 working together and in a dose-dependent manner as indispensable mediators of Wnt/βcatenin and Bmp signaling in the limb ectoderm. We suggest that the function of these factors links proximal-distal and dorsal-ventral patterning

Genome-wide screens identify Toxoplasma gondii determinants of parasite fitness in IFNγ-activated murine macrophages

Macrophages play an essential role in the early immune response against Toxoplasma and are the cell type preferentially infected by the parasite in vivo. Interferon gamma (IFNγ) elicits a variety of anti-Toxoplasma activities in macrophages. Using a genome-wide CRISPR screen we identify 353 Toxoplasma genes that determine parasite fitness in naїve or IFNγ-activated murine macrophages, seven of which are further confirmed. We show that one of these genes encodes dense granule protein GRA45, which has a chaperone-like domain, is critical for correct localization of GRAs into the PVM and secretion of GRA effectors into the host cytoplasm. Parasites lacking GRA45 are more susceptible to IFNγ-mediated growth inhibition and have reduced virulence in mice. Together, we identify and characterize an important chaperone-like GRA in Toxoplasma and provide a resource for the community to further explore the function of Toxoplasma genes that determine fitness in IFNγ-activated macrophages

Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space TeX in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into TeX can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices

On the parameterized complexity of party nominations

by Neeldhara Misr