scholarly journals Online Learning over a Finite Action Set with Limited Switching

2020 ◽  
Author(s):  
Jason M. Altschuler ◽  
Kunal Talwar
Keyword(s):  
Phase Transition ◽  
Combinatorial Optimization ◽  
Open Problem ◽  
High Probability ◽  
Fixed Number ◽  
Optimal Order ◽  
Minimax Regret ◽  
Minimax Rates ◽  
Arbitrary Switching ◽  
Finite Action

This paper studies the value of switching actions in the Prediction From Experts problem (PFE) and Adversarial Multiarmed Bandits problem (MAB). First, we revisit the well-studied and practically motivated setting of PFE with switching costs. Many algorithms achieve the minimax optimal order for both regret and switches in expectation; however, high probability guarantees are an open problem. We present the first algorithms that achieve this optimal order for both quantities with high probability. This also implies the first high probability guarantees for several other problems, and, in particular, is efficiently adaptable to online combinatorial optimization with limited switching. Next, to investigate the value of switching actions more granularly, we introduce the switching budget setting, which limits algorithms to a fixed number of (costless) switches. Using this result and several reductions, we unify previous work and completely characterize the complexity of this switching budget setting up to small polylogarithmic factors: for both PFE and MAB, for all switching budgets, and for both expectation and high probability guarantees. Interestingly, as the switching budget decreases, the minimax regret rate admits a phase transition for PFE but not for MAB. These results recover and generalize the known minimax rates for the (arbitrary) switching cost setting.

scholarly journals When Do Envy-Free Allocations Exist?

2019 ◽  
Vol 33 ◽  
pp. 2109-2116 ◽  
Author(s):  
Pasin Manurangsi ◽  
Warut Suksompong
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Fair Division ◽  
The Other ◽  
Other Hand ◽  
Additive Utilities ◽  
The One

We consider a fair division setting in which m indivisible items are to be allocated among n agents, where the agents have additive utilities and the agents’ utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m = Ω (n log n) but not when m = n + o (n), and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition— instead, the transition is governed by the divisibility relation between m and n. On the one hand, if m is divisible by n, an envy-free allocation exists with high probability as long as m ≥ 2n. On the other hand, if m is not “almost” divisible by , an envy-free allocation is unlikely to exist even when m = Θ(n log n)/log log n).

ON THE EXISTENCE OF LOOKAHEAD DELEGATORS FOR NFA

2007 ◽  
Vol 18 (05) ◽  
pp. 949-973 ◽  
Author(s):  
BALA RAVIKUMAR ◽  
NICOLAE SANTEAN
Keyword(s):  
Open Problem ◽  
Finite Automata ◽  
Fixed Number ◽  
Buffer Size ◽  
Membership Problem ◽  
Web Services Composition ◽  
Deterministic Finite Automata ◽  
Practical Applications ◽  
Services Composition ◽  
The Right

We investigate deterministically simulating (i.e., solving the membership problem for) nondeterministic finite automata (NFA), relying solely on the NFA's resources (states and transitions). Unlike the standard NFA simulation, involving an algorithm which stores at each step all the states reached nondeterministically while reading the input, we consider deterministic finite automata (DFA) with lookahead, which choose the “right” NFA transitions based on a fixed number of input symbols read ahead. This concept, known as lookahead delegation, arose in a formal study of web services composition and its subsequent practical applications. Here we answer several related questions, such as “when is lookahead delegation possible?” and “how hard is it to find a delegator with a given lookahead buffer size?”. In particular, we show that only finite languages have the property that all their NFA have delegators. This implies, among others, that delegation is a machine property, rather than a language property. We also prove that the existence of lookahead delegators for unambiguous NFA is decidable, thus partially solving an open problem. Finally, we show that finding delegators (even for a given buffer size) is hard in general, and is more efficient for unambiguous NFA, and we give an algorithm and a compact characterization for NFA delegation in general.

scholarly journals AN IMPROVED LOWER BOUND FOR THE CRITICAL PARAMETER OF STAVSKAYA’S PROCESS

2020 ◽  
Vol 102 (3) ◽  
pp. 517-524
Author(s):  
ALEX D. RAMOS ◽  
CALITÉIA S. SOUSA ◽  
PABLO M. RODRIGUEZ ◽  
PAULA CADAVID
Keyword(s):  
Phase Transition ◽  
Lower Bound ◽  
Open Problem ◽  
Critical Parameter ◽  
Critical Value ◽  
The State ◽  
Multicomponent Systems ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton

We consider Stavskaya’s process, which is a two-state probabilistic cellular automaton defined on a one-dimensional lattice. The state of any vertex depends only on itself and on the state of its right-adjacent neighbour. This process was one of the first multicomponent systems with local interaction for which the existence of a kind of phase transition has been rigorously proved. However, the exact localisation of its critical value remains as an open problem. We provide a new lower bound for the critical value.

scholarly journals The Phase Transition Analysis for the Random Regular Exact 2-(d,k)-SAT Problem

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1231
Author(s):  
Guoxia Nie ◽  
Daoyun Xu ◽  
Xiaofeng Wang ◽  
Xi Wang
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Solution Space ◽  
Regular Structure ◽  
Generation Model ◽  
Transition Analysis ◽  
Sat Problem ◽  
Second Moments ◽  
Instance Generation ◽  
Theoretical Results

In a regular (d,k)-CNF formula, each clause has length k and each variable appears d times. A regular structure such as this is symmetric, and the satisfiability problem of this symmetric structure is called the (d,k)-SAT problem for short. The regular exact 2-(d,k)-SAT problem is that for a (d,k)-CNF formula F, if there is a truth assignment T, then exactly two literals of each clause in F are true. If the formula F contains only positive or negative literals, then there is a satisfiable assignment T with a size of 2n/k such that F is 2-exactly satisfiable. This paper introduces the (d,k)-SAT instance generation model, constructs the solution space, and employs the method of the first and second moments to present the phase transition point d* of the 2-(d,k)-SAT instance with only positive literals. When d<d*, the 2-(d,k)-SAT instance can be satisfied with high probability. When d>d*, the 2-(d,k)-SAT instance can not be satisfied with high probability. Finally, the verification results demonstrate that the theoretical results are consistent with the experimental results.

scholarly journals A Multiple Pheromone Table Based Ant Colony Optimization for Clustering

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Kai-Cheng Hu ◽  
Chun-Wei Tsai ◽  
Ming-Chao Chiang ◽  
Chu-Sing Yang
Keyword(s):  
Approximate Solution ◽  
Combinatorial Optimization ◽  
Ant Colony Optimization ◽  
High Probability ◽  
Search Strategy ◽  
Optimization Problems ◽  
Clustering Algorithms ◽  
Ant Colony ◽  
Combinatorial Optimization Problems ◽  
Search Directions

Ant colony optimization (ACO) is an efficient heuristic algorithm for combinatorial optimization problems, such as clustering. Because the search strategy of ACO is similar to those of other well-known heuristics, the probability of searching particular regions will be increased if better results are found and kept. Although this kind of search strategy may find a better approximate solution, it also has a high probability of losing the potential search directions. To prevent the ACO from losing too many potential search directions at the early iterations, a novel pheromone updating strategy is presented in this paper. In addition to the “original” pheromone table used to keep track of thepromisinginformation, a second pheromone table is added to the proposed algorithm to keep track of theunpromisinginformation so as to increase the probability of searching directions worse than the current solutions. Several well-known clustering datasets are used to evaluate the performance of the proposed method in this paper. The experimental results show that the proposed method can provide better results than ACO and other clustering algorithms in terms of quality.

scholarly journals Phase transition of multivariate polynomial systems

2009 ◽  
Vol 19 (1) ◽  
pp. 9-23 ◽  
Author(s):  
GIORDANO FUSCO ◽  
ERIC BACH
Keyword(s):  
Phase Transition ◽  
Finite Field ◽  
Unique Solution ◽  
High Probability ◽  
Polynomial System ◽  
Random Systems ◽  
Polynomial Systems ◽  
The Other ◽  
Multivariate Polynomial ◽  
Prime Field

A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand, if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular, the limiting probability for no solution is 1/e at the phase transition, over a prime field.We also study the probability of having exactly s solutions, with s ≥ 1. In particular, the probability of a unique solution is asymptotically 1/e if the number of equations equals the number of variables. The probability decreases very rapidly if the number of equations increases or decreases.Our motivation is that many cryptographic systems can be expressed as large multivariate polynomial systems (usually quadratic) over a finite field. Since decoding is unique, the solution of the system must also be unique. Knowing the probability of having exactly one solution may help us to understand more about these cryptographic systems. For example, whether attacks should be evaluated by trying them against random systems depends very much on the likelihood of a unique solution.

scholarly journals A Third Strike Against Perfect Phylogeny

2019 ◽  
Vol 68 (5) ◽  
pp. 814-827 ◽  
Author(s):  
Leo Van Iersel ◽  
Mark Jones ◽  
Steven Kelk
Keyword(s):  
Open Problem ◽  
Efficient Algorithm ◽  
Classical Result ◽  
Input Data ◽  
Fixed Number ◽  
Evolutionary Trees ◽  
Perfect Phylogeny ◽  
Fixed Size ◽  
Algorithm Development ◽  
Negative Results

Abstract Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam’s Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result states that 2-state characters permit a perfect phylogeny precisely if each subset of 2 characters permits one. More recently, it was shown that for 3-state characters the same property holds but for size-3 subsets. A long-standing open problem asked whether such a constant exists for each number of states. More precisely, it has been conjectured that for any fixed number of states $r$ there exists a constant $f(r)$ such that a set of $r$-state characters $C$ has a perfect phylogeny if and only if every subset of at most $f(r)$ characters has a perfect phylogeny. Informally, the conjecture states that checking fixed-size subsets of characters is enough to correctly determine whether input data permits a perfect phylogeny, irrespective of the number of characters in the input. In this article, we show that this conjecture is false. In particular, we show that for any constant $t$, there exists a set $C$ of $8$-state characters such that $C$ has no perfect phylogeny, but there exists a perfect phylogeny for every subset of at most $t$ characters. Moreover, there already exists a perfect phylogeny when ignoring just one of the characters, independent of which character you ignore. This negative result complements the two negative results (“strikes”) of Bodlaender et al. (1992,2000). We reflect on the consequences of this third strike, pointing out that while it does close off some routes for efficient algorithm development, many others remain open.

scholarly journals Algorithms for Manipulating Sequential Allocation

2020 ◽  
Vol 34 (02) ◽  
pp. 2302-2309
Author(s):  
Mingyu Xiao ◽  
Jiaxing Ling
Keyword(s):  
Computational Complexity ◽  
Open Problem ◽  
Arbitrary Number ◽  
Fixed Number ◽  
Preference Ranking ◽  
Best Response ◽  
Sequential Allocation ◽  
Np Complete ◽  
Polynomially Solvable ◽  
Novel Algorithm

Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most preferred item among all items having not been allocated yet. This problem is well-known to be not strategyproof, i.e., an agent may get more utility by reporting an untruthful preference ranking of items. It arises the problem: how to find the best response of an agent? It is known that this problem is polynomially solvable for only two agents and NP-complete for an arbitrary number of agents. The computational complexity of this problem with three agents was left as an open problem. In this paper, we give a novel algorithm that solves the problem in polynomial time for each fixed number of agents. We also show that an agent can always get at least half of its optimal utility by simply using its truthful preference as the response.

scholarly journals Quantum phase transitions of strongly correlated metals

2020 ◽  
pp. 6-13
Author(s):  
Martha Yolima Suárez Villagrán ◽  
Nikolaos Mitsakos ◽  
John H. Miller Jr
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Critical Temperature ◽  
Quantum Phase Transition ◽  
Open Problem ◽  
Quantum Phase Transitions ◽  
Quantum Phase ◽  
Strongly Correlated ◽  
Made In

In this article, we discuss several aspects of the quantum phase transition, with special emphasis on the metalinsulator transition. We start with a review of key experimental and theoretical works and then discuss how doping a system reduces the critical temperature of the overall phase transition. Although many aspects of the quantum phase transition still remain an open problem, onsiderable progress has been made in revealing the underlying physics, both theoretically and experimentally.

scholarly journals The Conduciveness of CA-Rule Graphs

2013 ◽  
Vol 19 (2) ◽  
pp. 255-266
Author(s):  
Valmir C. Barbosa
Keyword(s):  
Combinatorial Optimization ◽  
Directed Graph ◽  
Hamming Distance ◽  
Fixed Number ◽  
Neighborhood Size ◽  
Analytical Expressions ◽  
Insight Into

Given two subsets A and B of nodes in a directed graph, the conduciveness of the graph from A to B is the ratio representing how many of the edges outgoing from nodes in A are incoming to nodes in B. When the graph's nodes stand for the possible solutions to certain problems of combinatorial optimization, choosing its edges appropriately has been shown to lead to conduciveness properties that provide useful insight into the performance of algorithms to solve those problems. Here we study the conduciveness of CA-rule graphs, that is, graphs whose node set is the set of all CA rules given a cell's number of possible states and neighborhood size. We consider several different edge sets interconnecting these nodes, both deterministic and random ones, and derive analytical expressions for the resulting graph's conduciveness toward rules having a fixed number of non-quiescent entries. We demonstrate that one of the random edge sets, characterized by allowing nodes to be sparsely interconnected across any Hamming distance between the corresponding rules, has the potential of providing reasonable conduciveness toward the desired rules. We conjecture that this may lie at the bottom of the best strategies known to date for discovering complex rules to solve specific problems, all of an evolutionary nature.

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