Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

Satgraphs and independent domination. Part 1

AbstractA graph G is called a satgraph if there exists a partition A∪B=V(G) such that•A induces a clique [possibly, A=∅],•B induces a matching [i.e., G(B) is a 1-regular subgraph, possibly, B=∅], and•there are no triangles (a,b,b′), where a∈A and b,b′∈B.We also introduce the hereditary closure of SAT, denoted by HSAT [hereditary satgraphs]. The class HSAT contains split graphs. In turn, HSAT is contained in the class of all (1,2)-split graphs [A. Gyárfás, Generalized split graphs and Ramsey numbers, J. Combin. Theory Ser. A 81 (2) (1998) 255–261], the latter being still not characterized. We characterize satgraphs in terms of forbidden induced subgraphs.There exist close connections between satgraphs and the satisfiability problem [SAT]. In fact, SAT is linear-time equivalent to finding the independent domination number in the corresponding satgraph. It follows that the independent domination problem is NP-complete for the hereditary satgraphs. In particular, it is NP-complete for perfect graphs

Contractible stability spaces and faithful braid group actions

We prove that any `finite-type' component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi--Yau-$N$ category $\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$ case. Other classes of triangulated categories with finite-type components in their stability spaces include locally-finite triangulated categories with finite rank Grothendieck group and discrete derived categories of finite global dimension.Comment: Final version, to appear in Geom. Topo

On the first-order transduction quasiorder of hereditary classes of graphs

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures, and several important class properties can be defined in terms of transductions. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. This normal form allows to prove, among other results, that the local variants of (monadic) stability and (monadic) dependence are equivalent to their non-local versions. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of additive classes is also a bounded distributive join-semilattice. We characterize transductions of paths, cubic graphs, and cubic trees in terms of bandwidth, bounded degree, and treewidth. We establish that the classes of all graphs with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder and leave open whether the same holds for the classes of all graphs with treewidth at most $k$. We identify the obstructions for a class to be a transduction of a class with bounded degree, leading to an interesting transduction duality formulation. Eventually, we discuss a notion of dense analogs of sparse transduction-preserved class properties, and propose several related conjectures

On 2-representation infinite algebras arising from dimer models

The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a $2$-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. In this paper, we show that internal perfect matchings, which correspond to toric exceptional divisors on a crepant resolution of a $3$-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Moreover, combining this result with the theorems due to Amiot-Iyama-Reiten, we show that the stable category of graded maximal Cohen-Macaulay modules admits a tilting object for any $3$-dimensional Gorenstein toric isolated singularity. We then show that all internal perfect matchings corresponding to the same toric exceptional divisor are transformed into each other using the mutations of perfect matchings, and this induces derived equivalences of $2$-representation infinite algebras.Comment: 28 pages, v2: improved some proof