In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NPhard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction.
In Hfree graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NPhard and APXhard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomialtime solvability results for small graphs H such as P5, P6, the claw, or the fork.
We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an Hfree graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1 – ε) of the optimum in time exponential in a polynomial of log  V(G)  and ε−1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APXhard in Hfree graphs, the problem admits a quasipolynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.
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On the Hardness and Inapproximability of Recognizing Wheeler Graphs
In recent years several compressed indexes based on variants of the BurrowsWheeler transformation have been introduced. Some of these are used to index structures far more complex than a single string, as was originally done with the FMindex [Ferragina and Manzini, J. ACM 2005]. As such, there has been an increasing effort to better understand under which conditions such an indexing scheme is possible. This has led to the introduction of Wheeler graphs [Gagie et al., Theor. Comput. Sci., 2017]. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs and that Wheeler graphs can be indexed in a way which is spaceefficient. Hence, being able to recognize whether a given graph is a Wheeler graph, or being able to approximate a given graph by a Wheeler graph, could have numerous applications in indexing. Here we resolve the open question of whether there exists an efficient algorithm for recognizing if a given graph is a Wheeler graph. We present  The problem of recognizing whether a given graph G=(V,E) is a Wheeler graph is NPcomplete for any edge label alphabet of size sigma >= 2, even when G is a DAG. This holds even on a restricted, subset of graphs called dNFA's for d >= 5. This is in contrast to recent results demonstrating the problem can be solved in polynomial time for dNFA's where d <= 2. We also show the recognition problem can be solved in linear time for sigma =1;  There exists an 2^{e log sigma + O(n + e)} time exact algorithm where n = V and e = E. This algorithm relies on graph isomorphism being computable in strictly subexponential time;  We define an optimization variant of the problem called Wheeler Graph Violation, abbreviated WGV, where the aim is to remove the minimum number of edges in order to obtain a Wheeler graph. We show WGV is APXhard, even when G is a DAG, implying there exists a constant C >= 1 for which there is no Capproximation algorithm (unless P = NP). Also, conditioned on the Unique Games Conjecture, for all C >= 1, it is NPhard to find a Capproximation;  We define the Wheeler Subgraph problem, abbreviated WS, where the aim is to find the largest subgraph which is a Wheeler Graph (the dual of the WGV). In contrast to WGV, we prove that the WS problem is in APX for sigma=O(1); The above findings suggest that most problems under this theme are computationally difficult. However, we identify a class of graphs for which the recognition problem is polynomialtime solvable, raising the open question of which parameters determine this problem's difficulty.
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 NSFPAR ID:
 10164524
 Date Published:
 Journal Name:
 27th Annual European Symposium on Algorithms (ESA) 2019
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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