scholarly journals ADVICE FOR SEMIFEASIBLE SETS AND THE COMPLEXITY-THEORETIC COST(LESSNESS) OF ALGEBRAIC PROPERTIES

2005 ◽  
Vol 16 (05) ◽  
pp. 913-928 ◽  
Author(s):  
PIOTR FALISZEWSKI ◽  
LANE A. HEMASPAANDRA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Worst Case ◽  
Advice Complexity ◽  
The Core ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Algebraic Properties

Informally put, the semifeasible sets are the class of sets having a polynomial-time algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. We provide a tutorial overview of the advice complexity of the semifeasible sets. No previous familiarity with either the semifeasible sets or advice complexity will be assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations. Karp and Lipton introduced advice complexity about a quarter of a century ago.18 Advice complexity asks, for a given power of interpreter, how many bits of "help" suffice to accept a given set. Thus, this is a notion that contains aspects both of descriptional/informational complexity and of computational complexity. We will see that for some powers of interpreter the (worst-case) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic. While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—so-called selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.

COMPUTATIONAL COMPLEXITY OF THE PERFECT MATCHING PROBLEM IN HYPERGRAPHS WITH SUBCRITICAL DENSITY

2010 ◽  
Vol 21 (06) ◽  
pp. 905-924 ◽  
Author(s):  
MAREK KARPIŃSKI ◽  
ANDRZEJ RUCIŃSKI ◽  
EDYTA SZYMAŃSKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Existence Problem ◽  
Matching Problem ◽  
Uniform Hypergraphs ◽  
The Status ◽  
Np Complete

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for [Formula: see text] and trivial for c ≥ ½, leaving the status of the problem with c in the interval [Formula: see text] widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

scholarly journals Polynomial-Time Algorithm for Learning Optimal BFS-Consistent Dynamic Bayesian Networks

Entropy ◽  
2018 ◽  
Vol 20 (4) ◽  
pp. 274 ◽  
Author(s):  
◽  
Keyword(s):  
Bayesian Networks ◽  
Polynomial Time ◽  
Learning Algorithm ◽  
Search Space ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Dynamic Bayesian Networks ◽  
Worst Case ◽  
Current Time ◽  
Transition Networks

Dynamic Bayesian networks (DBN) are powerful probabilistic representations that model stochastic processes. They consist of a prior network, representing the distribution over the initial variables, and a set of transition networks, representing the transition distribution between variables over time. It was shown that learning complex transition networks, considering both intra- and inter-slice connections, is NP-hard. Therefore, the community has searched for the largest subclass of DBNs for which there is an efficient learning algorithm. We introduce a new polynomial-time algorithm for learning optimal DBNs consistent with a breadth-first search (BFS) order, named bcDBN. The proposed algorithm considers the set of networks such that each transition network has a bounded in-degree, allowing for p edges from past time slices (inter-slice connections) and k edges from the current time slice (intra-slice connections) consistent with the BFS order induced by the optimal tree-augmented network (tDBN). This approach increases exponentially, in the number of variables, the search space of the state-of-the-art tDBN algorithm. Concerning worst-case time complexity, given a Markov lag m, a set of n random variables ranging over r values, and a set of observations of N individuals over T time steps, the bcDBN algorithm is linear in N, T and m; polynomial in n and r; and exponential in p and k. We assess the bcDBN algorithm on simulated data against tDBN, revealing that it performs well throughout different experiments.

SINGLE-MACHINE SCHEDULING WITH PROPORTIONALLY DETERIORATING JOBS SUBJECT TO AVAILABILITY CONSTRAINTS

2012 ◽  
Vol 29 (04) ◽  
pp. 1250019 ◽  
Author(s):  
SHISHENG LI ◽  
BAOQIANG FAN
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Deteriorating Jobs ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Total Weighted Completion Time ◽  
Worst Case ◽  
Availability Constraints ◽  
Worst Case Ratio

We address the nonresumable version of the scheduling problem with proportionally deteriorating jobs on a single machine subject to availability constraints. The objective is to minimize the total weighted completion time. We show that there exists no polynomial-time algorithm with a constant worst-case ratio for the problem with two nonavailability intervals unless [Formula: see text]. Furthermore, we propose a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme for the problem with a single nonavailability interval.

scholarly journals Computing the Weighted Isolated Scattering Number of Interval Graphs in Polynomial Time

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Fengwei Li ◽  
Xiaoyan Zhang ◽  
Qingfang Ye ◽  
Yuefang Sun
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interval Graphs ◽  
Perfect Graphs ◽  
Network Vulnerability ◽  
Hamiltonian Properties

The scattering number and isolated scattering number of a graph have been introduced in relation to Hamiltonian properties and network vulnerability, and the isolated scattering number plays an important role in characterizing graphs with a fractional 1-factor. Here we investigate the computational complexity of one variant, namely, the weighted isolated scattering number. We give a polynomial time algorithm to compute this parameter of interval graphs, an important subclass of perfect graphs.

scholarly journals Non-commutative lattice problems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Alexei Myasnikov ◽  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Computational Complexity ◽  
Large Class ◽  
Polynomial Time ◽  
Coxeter Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Group Element ◽  
Surface Groups ◽  
Nontrivial Element

AbstractWe consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest nontrivial element of a subgroup in the case of nilpotent groups, and a large class of surface groups and Coxeter groups. We also provide polynomial time algorithm to compute geodesics in given generators of a subgroup of a free group.

scholarly journals Connected Subgraph Defense Games

Algorithmica ◽  
2021 ◽  
Author(s):  
Eleni C. Akrida ◽  
Argyrios Deligkas ◽  
Themistoklis Melissourgos ◽  
Paul G. Spirakis
Keyword(s):  
Polynomial Time ◽  
Nash Equilibria ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Connected Subgraph ◽  
Expected Number ◽  
Worst Case ◽  
Induced Subgraph ◽  
Other Hand

AbstractWe study a security game over a network played between a defender and kattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $$\lambda $$ λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $$\lambda $$ λ constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is $${\mathtt {NP}}$$ NP -hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any $$\lambda $$ λ ; this is the worst equilibrium defense ratio over all graphs.

scholarly journals Semigroups and one-way functions

2015 ◽  
Vol 25 (01n02) ◽  
pp. 3-36 ◽  
Author(s):  
J. C. Birget
Keyword(s):  
Polynomial Time ◽  
Complexity Classes ◽  
Finitely Generated ◽  
Partial Functions ◽  
Worst Case ◽  
Regular Elements ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
The One

We study the complexity classes 𝖯 and 𝖭𝖯 through a semigroup 𝖿𝖯 ("polynomial-time functions"), consisting of all polynomially balanced polynomial-time computable partial functions. The semigroup 𝖿𝖯 is non-regular if and only if 𝖯 ≠ 𝖭𝖯. The one-way functions considered here are based on worst-case complexity (they are not cryptographic); they are exactly the non-regular elements of 𝖿𝖯. We prove various properties of 𝖿𝖯, e.g. that it is finitely generated. We define reductions with respect to which certain universal one-way functions are 𝖿𝖯-complete.

Unconstrained submodular maximization with modular costs

2021 ◽  
Vol 14 (10) ◽  
pp. 1756-1768
Author(s):  
Tianyuan Jin ◽  
Yu Yang ◽  
Renchi Yang ◽  
Jieming Shi ◽  
Keke Huang ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Profit Maximization ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Viral Marketing ◽  
Trivial Extension ◽  
Maximization Problem ◽  
Solution Quality ◽  
Worst Case ◽  
Submodular Maximization

Given a set V , the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S ⊆ V that maximizes f ( S ) - c ( S ), where f is a non-negative, monotone, and submodular function that gauges the utility of S , and c is a non-negative and modular function that measures the cost of S. This problem finds applications in numerous practical scenarios, such as profit maximization in viral marketing on social media. This paper presents ROI-Greedy, a polynomial time algorithm for USM-MC that returns a solution S satisfying [EQUATION], where S * is the optimal solution to USM-MC. To our knowledge, ROI-Greedy is the first algorithm that provides such a strong approximation guarantee. In addition, we show that this worst-case guarantee is tight , in the sense that no polynomial time algorithm can ensure [EQUATION], for any ϵ > 0. Further, we devise a non-trivial extension of ROI-Greedy to solve the profit maximization problem, where the precise value of f ( S ) for any set S is unknown and can only be approximated via sampling. Extensive experiments on benchmark datasets demonstrate that ROI-Greedy significantly outperforms competing methods in terms of the tradeoff between efficiency and solution quality.

scholarly journals A Maximal Tractable Class of Soft Constraints

2004 ◽  
Vol 22 ◽  
pp. 1-22 ◽  
Author(s):  
D. Cohen ◽  
M. Cooper ◽  
P. Jeavons ◽  
A. Krokhin
Keyword(s):  
Artificial Intelligence ◽  
Computational Complexity ◽  
Polynomial Time ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Soft Constraints ◽  
Soft Constraint ◽  
Binary Constraints ◽  
Binary Constraint

Many researchers in artificial intelligence are beginning to explore the use of soft constraints to express a set of (possibly conflicting) problem requirements. A soft constraint is a function defined on a collection of variables which associates some measure of desirability with each possible combination of values for those variables. However, the crucial question of the computational complexity of finding the optimal solution to a collection of soft constraints has so far received very little attention. In this paper we identify a class of soft binary constraints for which the problem of finding the optimal solution is tractable. In other words, we show that for any given set of such constraints, there exists a polynomial time algorithm to determine the assignment having the best overall combined measure of desirability. This tractable class includes many commonly-occurring soft constraints, such as 'as near as possible' or 'as soon as possible after', as well as crisp constraints such as 'greater than'. Finally, we show that this tractable class is maximal, in the sense that adding any other form of soft binary constraint which is not in the class gives rise to a class of problems which is NP-hard.

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