88 research outputs found
Compression with wildcards: All exact, or all minimal hitting sets
Our main objective is the COMPRESSED enumeration (based on wildcards) of all
minimal hitting sets of general hypergraphs. To the author's best knowledge the
only previous attempt towards compression, due to Toda [T], is based on BDD's
and much different from our techniques. Numerical experiments show that
traditional one-by-one enumeration schemes cannot compete against compressed
enumeration when the degree of compression is high. Our method works
particularly well in these two cases: Either compressing all exact hitting
sets, or all minimum-cardinality hitting sets. It also supports parallelization
and cut-off (i.e. restriction to all minimal hitting sets of cardinality at
most m).Comment: 30 pages, many Table
Application of hypergraphs in decomposition of discrete systems
seria: Lecture Notes in Control and Computer Science ; vol. 23
Exact Algorithms via Multivariate Subroutines
We consider the family of Phi-Subset problems, where the input consists of an instance I of size N over a universe U_I of size n and the task is to check whether the universe contains a subset with property Phi (e.g., Phi could be the property of being a feedback vertex set for the input graph of size at most k). Our main tool is a simple randomized algorithm which solves Phi-Subset in time (1+b-(1/c))^n N^(O(1)), provided that there is an algorithm for the Phi-Extension problem with running time b^{n-|X|} c^k N^{O(1)}. Here, the input for Phi-Extension is an instance I of size N over a universe U_I of size n, a subset X subseteq U_I, and an integer k, and the task is to check whether there is a set Y with X subseteq Y subseteq U_I and |Y X| <= k with property Phi.
We derandomize this algorithm at the cost of increasing the running time by a subexponential factor in n, and we adapt it to the enumeration setting where we need to enumerate all subsets of the universe with property Phi. This generalizes the results of Fomin et al. [STOC 2016] who proved the case where b=1.
As case studies, we use these results to design faster deterministic algorithms for:
- checking whether a graph has a feedback vertex set of size at most k
- enumerating all minimal feedback vertex sets
- enumerating all minimal vertex covers of size at most k, and
- enumerating all minimal 3-hitting sets.
We obtain these results by deriving new b^{n-|X|} c^k N^{O(1)}-time algorithms for the corresponding Phi-Extension problems (or enumeration variant). In some cases, this is done by adapting the analysis of an existing algorithm, or in other cases by designing a new algorithm. Our analyses are based on Measure and Conquer, but the value to minimize, 1+b-(1/c), is unconventional and requires non-convex optimization
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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