On the Approximability of Partial VC Dimension

  title={On the Approximability of Partial VC Dimension},
  author={Cristina Bazgan and Florent Foucaud and Florian Sikora},
We introduce the problem Partial VC Dimension that asks, given a hypergraph \(H=(X,E)\) and integers k and \(\ell \), whether one can select a set \(C\subseteq X\) of k vertices of H such that the set \(\{e\cap C, e\in E\}\) of distinct hyperedge-intersections with C has size at least \(\ell \). The sets \(e\cap C\) define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case \(\ell =2^k\), and of Distinguishing Transversal, which… 

Approximation and Hardness: Beyond P and NP

Author(s): Manurangsi, Pasin | Advisor(s): Trevisan, Luca; Raghavendra, Prasad | Abstract: The theory of NP-hardness of approximation has led to numerous tight characterizations of approximability of

Parameterized and approximation complexity of Partial VC Dimension

Hardness of Approximation Between P and NP

  • A. Rubinstein
  • Economics
    Hardness of Approximation Between P and NP
  • 2017
This book provides strong evidence that even finding an approximate Nash equilibrium is intractable, and proves several intractability theorems for different settings (two-player games and many- player games) and models (computational complexity, query complexity, and communication complexity).

Inapproximability of VC Dimension and Littlestone's Dimension

This work studies the complexity of computing the VC Dimension and Littlestone's Dimension and proves nearly matching lower bounds on the running time, that hold even for approximation algorithms.



Subexponential algorithms for partial cover problems

VC-dimension and Erdős-Pósa property

The VC-dimension of Set Systems Defined by Graphs

Parameterizations of Test Cover with Bounded Test Sizes

The parameterized complexity of Test- r-Cover, the restriction of Test Cover in which each edge contains at most r≥2 vertices, is studied and it is shown that it is para-NP-complete, as it is NP-hard to decide whether an instance of Tests-r-Cover has a test cover of size exactly.

Discriminating codes in bipartite graphs: bounds, extremal cardinalities, complexity

Borders on the minimum size of these codes are given, graphs where minimal discriminating codes have size close to the upper bound, or give the exact minimum size in particular graphs are investigated; an NP-completeness result is given.

Hitting Set in hypergraphs of low VC-dimension

A sharp threshold is identified where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard, and for set systems whose (primal or dual) VC-dimension is 1, it is shown that Hittingset is solvable in polynometric time.

Partial vs. Complete Domination: t-Dominating Set

A quintic problem kernel and a randomized $O((4+\varepsilon)^t\textit{poly}(n)$ algorithm is obtained, based on the divide-and-color method, which is rather intuitive and can be derandomized using a standard framework.

Hardness of Set Cover with Intersection 1

It is shown that the Set Covering problem with intersection 1 cannot be approximated within a o(log n) factor in random polynomial time unless N P ⊆ ZTIME(nO(log log n)), and it is observed that this problem is Max-SNP-Hard.

Parameterized Study of the Test Cover Problem

This paper carries out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity, and obtains four parameterizations for Test Cover that are fixed-parameter tractable (FPT), i.e., these problems can be solved by algorithms of runtime.

Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

It is shown that if G is a twin-free cubic graph on n vertices, then $$Gamma(G) \leq 3n/4}$$ .