75 research outputs found
Algorithmic and enumerative aspects of the Moser-Tardos distribution
Moser & Tardos have developed a powerful algorithmic approach (henceforth
"MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its
variants is a search for "bad" events in a current configuration. In the
initial stage of MT, the variables are set independently. We examine the
distributions on these variables which arise during intermediate stages of MT.
We show that these configurations have a more or less "random" form, building
further on the "MT-distribution" concept of Haeupler et al. in understanding
the (intermediate and) output distribution of MT. This has a variety of
algorithmic applications; the most important is that bad events can be found
relatively quickly, improving upon MT across the complexity spectrum: it makes
some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which
are of basic combinatorial interest), gives lower-degree polynomial run-times
in some settings, transforms certain super-polynomial-time algorithms into
polynomial-time ones, and leads to Las Vegas algorithms for some coloring
problems for which only Monte Carlo algorithms were known.
We show that in certain conditions when the LLL condition is violated, a
variant of the MT algorithm can still produce a distribution which avoids most
of the bad events. We show in some cases this MT variant can run faster than
the original MT algorithm itself, and develop the first-known criterion for the
case of the asymmetric LLL. This can be used to find partial Latin transversals
-- improving upon earlier bounds of Stein (1975) -- among other applications.
We furthermore give applications in enumeration, showing that most applications
(where we aim for all or most of the bad events to be avoided) have many more
solutions than known before by proving that the MT-distribution has "large"
min-entropy and hence that its support-size is large
Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling
We devise an enumeration method for inclusion-wise minimal hitting sets in hypergraphs. It has delay O(mk* +1 · n2) and uses linear space. Hereby, n is the number of vertices, m the number of hyperedges, and k* the rank of the transversal hypergraph. In particular, on classes of hypergraphs for which the cardinality k* of the largest minimal hitting set is bounded, the delay is polynomial. The algorithm solves the extension problem for minimal hitting sets as a subroutine. We show that the extension problem is W[3]-complete when parameterised by the cardinality of the set which is to be extended. For the subroutine, we give an algorithm that is optimal under the exponential time hypothesis. Despite these lower bounds, we provide empirical evidence showing that the enumeration outperforms the theoretical worst-case guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations
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
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
Self-duality of bounded monotone boolean functions and related problems
AbstractIn this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2k2k2n) time. This implies an O(n2logn) algorithm for determining the self-duality of logn-DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n4(c+1)) algorithm for determining self-duality
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