Hardness of Approximation

2003 ◽  
pp. 306-333
Author(s):  
Vijay V. Vazirani
Keyword(s):  
Hardness Of Approximation

CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS

2013 ◽  
Vol 05 (04) ◽  
pp. 1350024 ◽  
Author(s):  
B. S. PANDA ◽  
S. PAUL
Keyword(s):  
Dominating Set ◽  
Chordal Graphs ◽  
Hardness Of Approximation ◽  
Induced Subgraph ◽  
Path Graph ◽  
Block Graphs ◽  
Domination Problem ◽  
Np Complete ◽  
Domination In Graphs ◽  
General Graphs

A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs.

scholarly journals Multiwinner Voting with Fairness Constraints

2018 ◽  
Author(s):  
L. Elisa Celis ◽  
Lingxiao Huang ◽  
Nisheeth K. Vishnoi
Keyword(s):  
Group Structure ◽  
Score Function ◽  
Political Orientation ◽  
Hardness Of Approximation ◽  
Score Functions ◽  
Voting Rule ◽  
Representative Subset ◽  
Fairness Constraints ◽  
And Gender ◽  
Ethnicity And Gender

Multiwinner voting rules are used to select a small representative subset of candidates or items from a larger set given the preferences of voters. However, if candidates have sensitive attributes such as gender or ethnicity (when selecting a committee), or specified types such as political leaning (when selecting a subset of news items), an algorithm that chooses a subset by optimizing a multiwinner voting rule may be unbalanced in its selection -- it may under or over represent a particular gender or political orientation in the examples above. We introduce an algorithmic framework for multiwinner voting problems when there is an additional requirement that the selected subset should be ``fair'' with respect to a given set of attributes. Our framework provides the flexibility to (1) specify fairness with respect to multiple, non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score function. We study the computational complexity of this constrained multiwinner voting problem for monotone and submodular score functions and present several approximation algorithms and matching hardness of approximation results for various attribute group structure and types of score functions. We also present simulations that suggest that adding fairness constraints may not affect the scores significantly when compared to the unconstrained case.

scholarly journals Hardness of approximation for quantum problems

2014 ◽  
Vol 14 (5&6) ◽  
pp. 517-540
Author(s):  
Sevag Gharibian ◽  
Julia Kempe
Keyword(s):  
Complexity Theory ◽  
Hardness Of Approximation ◽  
Set Cover ◽  
Polynomial Hierarchy ◽  
Set Cover Problem ◽  
Complete Problems ◽  
Quantum Version ◽  
Classical Quantum ◽  
Cover Problem ◽  
Quantum Ground States

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using the same techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.

scholarly journals On the Optimality of Affine Policies for Budgeted Uncertainty Sets

2021 ◽  
Author(s):  
Omar El Housni ◽  
Vineet Goyal
Keyword(s):  
Robust Optimization ◽  
Structural Properties ◽  
Optimization Problem ◽  
Hardness Of Approximation ◽  
Uncertain Parameters ◽  
Two Stage ◽  
Uncertainty Sets ◽  
Adjustable Robust Optimization ◽  
Robust Optimization Problem ◽  
Budgeted Uncertainty

In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than [Formula: see text] even for budget of uncertainty sets in which [Formula: see text] is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an [Formula: see text]-approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Sign in / Sign up

Export Citation Format

Share Document