scholarly journals Approximation and Parameterized Complexity of Minimax Approval Voting

2018 ◽  
Vol 63 ◽  
pp. 495-513
Author(s):  
Marek Cygan ◽  
Łukasz Kowalik ◽  
Arkadiusz Socała ◽  
Krzysztof Sornat
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Approximation Scheme ◽  
Hamming Distance ◽  
Approval Voting ◽  
Exponential Time ◽  
Running Time ◽  
Exponential Time Hypothesis

We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance d from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time O*(2o(d log d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O*(d2d) algorithm of Misra, Nabeel and Singh is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O*((3/ε)2d), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time nO(1/ε2⋅log(1/ε))⋅poly(m), where n is a number of voters and m is a number of alternatives. It almost matches the running time of the fastest known PTAS for Closest String due to Ma and Sun.

scholarly journals Matching Cut in Graphs with Large Minimum Degree

Algorithmica ◽  
2020 ◽  
Author(s):  
Chi-Yeh Chen ◽  
Sun-Yuan Hsieh ◽  
Hoang-Oanh Le ◽  
Van Bang Le ◽  
Sheng-Lung Peng
Keyword(s):  
Polynomial Time ◽  
Minimum Degree ◽  
Positive Root ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Running Time ◽  
Subexponential Time ◽  
Exponential Time Hypothesis ◽  
Exponential Time Algorithm

AbstractIn a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be $${\mathsf {NP}}$$ NP -complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is $${\mathsf {NP}}$$ NP -complete on graphs with minimum degree two. In this paper, we show that, for any given constant $$c>1$$ c > 1 , Matching Cut is $${\mathsf {NP}}$$ NP -complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant $$\epsilon >0$$ ϵ > 0 , Matching Cut remains $${\mathsf {NP}}$$ NP -complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree $$\delta >n^{1-\epsilon }$$ δ > n 1 - ϵ . We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree $$\delta \ge 3$$ δ ≥ 3 in $$O^*(\lambda ^n)$$ O ∗ ( λ n ) time, where $$\lambda$$ λ is the positive root of the polynomial $$x^{\delta +1}-x^{\delta }-1$$ x δ + 1 - x δ - 1 . Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is $$O^*(1.0099^n)$$ O ∗ ( 1 . 0099 n ) on graphs with minimum degree $$\delta \ge 469$$ δ ≥ 469 . Complementing our hardness results, we show that, for any two fixed constants $$1< c <4$$ 1 < c < 4 and $$c^{\prime }\ge 0$$ c ′ ≥ 0 , Matching Cut is solvable in polynomial time for graphs with large minimum degree $$\delta \ge \frac{1}{c}n-c^{\prime }$$ δ ≥ 1 c n - c ′ .

Chapter 16. Worst-Case Upper Bounds

2021 ◽  
Author(s):  
Evgeny Dantsin ◽  
Edward A. Hirsch
Keyword(s):  
Structural Complexity ◽  
Upper Bounds ◽  
Worst Case ◽  
Exponential Time ◽  
Running Time ◽  
Dichotomy Theorem ◽  
Exponential Time Hypothesis ◽  
Satisfiability Algorithms

The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related structural-complexity topics such as Schaefer’s dichotomy theorem, reductions between various restricted cases of SAT, the exponential time hypothesis, etc.

scholarly journals The Complexity of Malign Ensembles

1990 ◽  
Vol 19 (335) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Average Case ◽  
Exponential Time ◽  
Running Time ◽  
Exponential Time Algorithms

We analyze the concept of <em> malignness</em>, which is the property of probability ensembles of making the average case running time equal to the worst case running time for a class of algorithms. We derive lower and upper bounds on the complexity of malign ensembles, which are tight for exponential time algorithms, and which show that no polynomial time computable malign ensemble exists for the class of superlinear algorithms. Furthermore, we show that for no class of superlinear algorithms a polynomial time computable malign ensemble exists, unless every language in P has an expected polynomial time constructor.

POLYNOMIAL APPROXIMATION SCHEMES FOR THE MAX-MIN ALLOCATION PROBLEM UNDER A GRADE OF SERVICE PROVISION

2009 ◽  
Vol 01 (03) ◽  
pp. 355-368 ◽  
Author(s):  
JIANPING LI ◽  
WEIDONG LI ◽  
JIANBO LI
Keyword(s):  
Polynomial Time ◽  
Service Provision ◽  
Processing Time ◽  
Approximation Scheme ◽  
Allocation Problem ◽  
Approximation Schemes ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Running Time ◽  
Grade Of Service

The max-min allocation problem under a grade of service provision is defined in the following model: given a set [Formula: see text] of m parallel machines and a set [Formula: see text] of n jobs, where machines and jobs are all entitled to different levels of grade of service (GoS), each job [Formula: see text] has its processing time pj and it can only be allocated to a machine Mi whose GoS level is no more than the GoS level the job Jj has. The goal is to allocate all jobs to m machines to maximize the minimum machine load among these m machines, where the machine load of Mi is the sum of the processing time of jobs executed on Mi. The best known approximation algorithm [4] to solve this problem produces an allocation in which the minimum machine completion time is at least Ω ( log log log m/ log log m) of the optimal value. In this paper, we respectively present four approximation schemes to solve this problem and its two special versions: (1) a polynomial time approximation scheme (PTAS) with a running time [Formula: see text] for the general version, where ϵ > 0; (2) a PTAS and a fully polynomial time approximation scheme (FPTAS) with the running time O(n) for the version where the number m of machines is fixed; (3) a PTAS with the running time O(n) for the version where the number of GoS levels is bounded by k.

scholarly journals A Scalable Scheme for Counting Linear Extensions

2018 ◽  
Author(s):  
Topi Talvitie ◽  
Kustaa Kangas ◽  
Teppo Niinimäki ◽  
Mikko Koivisto
Keyword(s):  
Artificial Intelligence ◽  
Partial Order ◽  
Polynomial Time ◽  
State Of The Art ◽  
Hard Problem ◽  
Approximate Counting ◽  
Linear Extensions ◽  
Exponential Time ◽  
Running Time ◽  
Cnf Formulas

Counting the linear extensions of a given partial order not only has several applications in artificial intelligence but also represents a hard problem that challenges modern paradigms for approximate counting. Recently, Talvitie et al. (AAAI 2018) showed that an exponential time scheme beats the fastest known polynomial time schemes in practice, even if allowing hours of running time. Here, we present a novel scheme, relaxation Tootsie Pop, which in our experiments exhibits polynomial characteristics and significantly outperforms previous schemes. We also instantiate state-of-the-art model counters for CNF formulas; two natural encodings yield schemes that, however, are inferior to the more specialized schemes.

TWO APPROXIMATION SCHEMES FOR SCHEDULING ON PARALLEL MACHINES UNDER A GRADE OF SERVICE PROVISION

2012 ◽  
Vol 29 (05) ◽  
pp. 1250029 ◽  
Author(s):  
WEIDONG LI ◽  
JIANPING LI ◽  
TONGQUAN ZHANG
Keyword(s):  
Polynomial Time ◽  
Parallel Machines ◽  
Approximation Scheme ◽  
Approximation Schemes ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Running Time ◽  
Grade Of Service ◽  
Special Case ◽  
Polynomial Time Approximation

We consider the offline scheduling problem to minimize the makespan on m parallel and identical machines with certain features. Each job and machine are labeled with the grade of service (GoS) levels, and each job can only be executed on the machine whose GoS level is no more than that of the job. In this paper, we present an efficient polynomial-time approximation scheme (EPTAS) with running time O(n log n) for the special case where the GoS level is either 1 or 2. This partially solves an open problem posed in (Ou et al., 2008). We also present a simpler full polynomial-time approximation scheme (FPTAS) with running time O(n) for the case where the number of machines is fixed.

A QPTAS for ɛ-Envy-Free Profit-Maximizing Pricing on Line Graphs

2021 ◽  
Vol 9 (3) ◽  
pp. 1-31
Author(s):  
Khaled Elbassioni
Keyword(s):  
Approximation Scheme ◽  
Line Graph ◽  
Hierarchical Decomposition ◽  
Running Time ◽  
The Best Approximation ◽  
Maximum Price ◽  
Unlimited Supply ◽  
On Line ◽  
Doubling Metrics ◽  
Limited Supply

We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set E of n edges with a limited supply for each edge, and a set of m clients, where each client specifies one interval of E she is interested in and a budget B j which is the maximum price she is willing to pay for that interval. An envy-free pricing is one in which every customer is allocated an (possibly empty) interval maximizing her utility. Grandoni and Rothvoss (SIAM J. Comput. 2016) proposed a polynomial-time approximation scheme ( PTAS ) for the unlimited supply case with running time ( nm ) O ((1/ɛ) 1/ɛ ) , which was extended to the limited supply case by Grandoni and Wiese (ESA 2019). By utilizing the known hierarchical decomposition of doubling metrics , we give a PTAS with running time ( nm ) O (1/ ɛ 2 ) for the unlimited supply case. We then consider the limited supply case, and the notion of ɛ-envy-free pricing in which a customer gets an allocation maximizing her utility within an additive error of ɛ. For this case, we develop an approximation scheme with running time ( nm ) O (log 5/2 max e H e /ɛ 3 ) , where H e = B max ( e )/ B min ( e ) is the maximum ratio of the budgets of any two customers demanding edge e . This yields a PTAS in the uniform budget case, and a quasi-PTAS for the general case. The best approximation known, in both cases, for the exact envy-free pricing version is O (log c max ), where c max is the maximum item supply. Our method is based on the known hierarchical decomposition of doubling metrics, and can be applied to other problems, such as the maximum feasible subsystem problem with interval matrices.

scholarly journals Finding Points in General Position

2017 ◽  
Vol 27 (04) ◽  
pp. 277-296 ◽  
Author(s):  
Vincent Froese ◽  
Iyad Kanj ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Subset Selection ◽  
Selection Problem ◽  
Fixed Parameter Tractability ◽  
Maximum Cardinality ◽  
Exponential Time ◽  
Running Time ◽  
Fixed Parameter ◽  
Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

Sign in / Sign up

Export Citation Format

Share Document