3 research outputs found
Improved Inapproximability of VC Dimension and Littlestone's Dimension via (Unbalanced) Biclique
We study the complexity of computing (and approximating) VC Dimension and
Littlestone's Dimension when we are given the concept class explicitly. We give
a simple reduction from Maximum (Unbalanced) Biclique problem to approximating
VC Dimension and Littlestone's Dimension. With this connection, we derive a
range of hardness of approximation results and running time lower bounds. For
example, under the (randomized) Gap-Exponential Time Hypothesis or the
Strongish Planted Clique Hypothesis, we show a tight inapproximability result:
both dimensions are hard to approximate to within a factor of in
polynomial-time. These improve upon constant-factor inapproximability results
from [Manurangsi and Rubinstein, COLT 2017].Comment: To appear in ITCS 202
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Approximation and Hardness: Beyond P and NP
The theory of NP-hardness of approximation has led to numerous tight characterizations of approximability of hard combinatorial optimization problems. Nonetheless, there are many fundamental problems which are out of reach for these techniques, such as problems that can be solved (or approximated) in quasi-polynomial time, parameterized problems and problems in P.This dissertation continues the line of work that develops techniques to show inapproximability results for these problems. In the process, we provide hardness of approximation results for the following problems.- Problems Between P and NP: Dense Constraint Satisfaction Problems (CSPs), Densest k-Subgraph with Perfect Completeness, VC Dimension, and Littlestone's Dimension.- Parameterized Problems: k-Dominating Set, k-Clique, k-Biclique, Densest k-Subgraph, Parameterized 2-CSPs, Directed Steiner Network, k-Even Set, and k-Shortest Vector.- Problems in P: Closest Pair, and Maximum Inner Product.Some of our results, such as those for Densest k-Subgraph, Directed Steiner Network and Parameterized 2-CSP, also present the best known inapproximability factors for the problems, even in the (believed) NP-hard regime. Furthermore, our results for k-Dominating Set and k-Even Set resolve two long-standing open questions in the field of parameterized complexity
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Approximation and Hardness: Beyond P and NP
The theory of NP-hardness of approximation has led to numerous tight characterizations of approximability of hard combinatorial optimization problems. Nonetheless, there are many fundamental problems which are out of reach for these techniques, such as problems that can be solved (or approximated) in quasi-polynomial time, parameterized problems and problems in P.This dissertation continues the line of work that develops techniques to show inapproximability results for these problems. In the process, we provide hardness of approximation results for the following problems.- Problems Between P and NP: Dense Constraint Satisfaction Problems (CSPs), Densest k-Subgraph with Perfect Completeness, VC Dimension, and Littlestone's Dimension.- Parameterized Problems: k-Dominating Set, k-Clique, k-Biclique, Densest k-Subgraph, Parameterized 2-CSPs, Directed Steiner Network, k-Even Set, and k-Shortest Vector.- Problems in P: Closest Pair, and Maximum Inner Product.Some of our results, such as those for Densest k-Subgraph, Directed Steiner Network and Parameterized 2-CSP, also present the best known inapproximability factors for the problems, even in the (believed) NP-hard regime. Furthermore, our results for k-Dominating Set and k-Even Set resolve two long-standing open questions in the field of parameterized complexity