scholarly journals Statistically Efficient, Polynomial-Time Algorithms for Combinatorial Semi-Bandits

2021 ◽  
Vol 5 (1) ◽  
pp. 1-31
Author(s):  
Thibaut Cuvelier ◽  
Richard Combes ◽  
Eric Gourdin
Keyword(s):  
Computational Complexity ◽  
Linear Function ◽  
Polynomial Time ◽  
Budget Constraint ◽  
Approximation Ratio ◽  
Polynomial Time Algorithms ◽  
Asymptotic Complexity ◽  
Regret Bound ◽  
Approximate Version

We consider combinatorial semi-bandits over a set of arms X \subset \0,1\ ^d where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound R(T) = O( d (łn m)^2 (łn T) / Δ_\min ) after T rounds, where m = \max_x \in X 1^\top x. However, ESCB it has computational complexity O(|X|), which is typically exponential in d, and cannot be used in large dimensions. We propose the first algorithm that is both computationally and statistically efficient for this problem with regret R(T) = O( d (łn m)^2 (łn T) / Δ_\min ) and computational asymptotic complexity O(δ_T^-1 poly(d)), where δ_T is a function which vanishes arbitrarily slowly. Our approach involves carefully designing AESCB, an approximate version of ESCB with the same regret guarantees. We show that, whenever budgeted linear maximization over X can be solved up to a given approximation ratio, AESCB is implementable in polynomial time O(δ_T^-1 poly(d)) by repeatedly maximizing a linear function over X subject to a linear budget constraint, and showing how to solve these maximization problems efficiently.

TERM EQUATION SATISFIABILITY OVER FINITE ALGEBRAS

2010 ◽  
Vol 20 (08) ◽  
pp. 1001-1020 ◽  
Author(s):  
TOMASZ A. GORAZD ◽  
JACEK KRZACZKOWSKI
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Finite Algebra ◽  
Satisfiability Problem ◽  
Finite Algebras ◽  
Maximal Clones ◽  
Polynomial Time Algorithms ◽  
Np Complete

We study the computational complexity of the satisfiability problem of an equation between terms over a finite algebra (TERM-SAT). We describe many classes of algebras where the complexity of TERM-SAT is determined by the clone of term operations. We classify the complexity for algebras generating maximal clones. Using this classification we describe many of algebras where TERM-SAT is NP-complete. We classify the situation for clones which are generated by an order or a permutation relation. We introduce the concept of semiaffine algebras and show polynomial-time algorithms which solve the satisfiability problem for them.

On the Complexity of Deadlock Recovery

1986 ◽  
Vol 9 (3) ◽  
pp. 323-342
Author(s):  
Joseph Y.-T. Leung ◽  
Burkhard Monien
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Arbitrary Number ◽  
The Other ◽  
Np Hard ◽  
Total Cost ◽  
Other Hand ◽  
Input Parameters ◽  
Resource Type

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.

scholarly journals An Overview of Algorithms for Network Survivability

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
F. A. Kuipers
Keyword(s):  
Polynomial Time ◽  
Network Connectivity ◽  
Network Survivability ◽  
Disjoint Paths ◽  
Polynomial Time Algorithms ◽  
Concise Overview ◽  
Main Components

Network survivability—the ability to maintain operation when one or a few network components fail—is indispensable for present-day networks. In this paper, we characterize three main components in establishing network survivability for an existing network, namely, (1) determining network connectivity, (2) augmenting the network, and (3) finding disjoint paths. We present a concise overview of network survivability algorithms, where we focus on presenting a few polynomial-time algorithms that could be implemented by practitioners and give references to more involved algorithms.

scholarly journals Reconstruction of score sets

2014 ◽  
Vol 6 (2) ◽  
pp. 210-229
Author(s):  
Antal Iványi
Keyword(s):  
Polynomial Time ◽  
Constructive Proof ◽  
Construction Algorithm ◽  
Polynomial Time Algorithms ◽  
General Polynomial ◽  
Degree Set ◽  
New Algorithms

Abstract The score set of a tournament is defined as the set of its different outdegrees. In 1978 Reid [15] published the conjecture that for any set of nonnegative integers D there exists a tournament T whose degree set is D. Reid proved the conjecture for tournaments containing n = 1, 2, and 3 vertices. In 1986 Hager [4] published a constructive proof of the conjecture for n = 4 and 5 vertices. In 1989 Yao [18] presented an arithmetical proof of the conjecture, but general polynomial construction algorithm is not known. In [6] we described polynomial time algorithms which reconstruct the score sets containing only elements less than 7. In [5] we improved this bound to 9. In this paper we present and analyze new algorithms Hole-Map, Hole-Pairs, Hole-Max, Hole-Shift, Fill-All, Prefix-Deletion, and using them improve the above bound to 12, giving a constructive partial proof of Reid’s conjecture.

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