scholarly journals A Tight Lower Bound for Planar Multiway Cut with Fixed Number of Terminals

2012 ◽  
pp. 677-688 ◽  
Author(s):  
Dániel Marx
Keyword(s):  
Lower Bound ◽  
Fixed Number ◽  
Multiway Cut

A LOW OVERHEAD SCHEDULE FOR A 3D-GRID GRAPH

1992 ◽  
Vol 02 (04) ◽  
pp. 363-372 ◽  
Author(s):  
E. BAMPIS ◽  
J.-C. KÖNIG ◽  
D. TRYSTRAM
Keyword(s):  
Lower Bound ◽  
Scheduling Algorithm ◽  
Three Dimensional ◽  
Fixed Number ◽  
Communication Overhead ◽  
Grid Graph ◽  
Precedence Graph

We propose a new scheduling algorithm for a three-dimensional grid precedence graph of size n, and we prove that the communication overhead is just in Θ( log log n). Finally, we show that a lower bound for the overhead of any schedule with a fixed number of processors (p ≥ 3) is Ω( log log n).

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

scholarly journals Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

2021 ◽  
Vol 68 (4) ◽  
pp. 1-26
Author(s):  
Vincent Cohen-Addad ◽  
Éric Colin De Verdière ◽  
Dániel Marx ◽  
Arnaud De Mesmay
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Undirected Graph ◽  
Time Algorithm ◽  
Planar Case ◽  
Embedded Graphs ◽  
Multiway Cut ◽  
Hard Cutting ◽  
Cutting Problems ◽  
Unweighted Graph

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph  G embedded on a surface S is a subgraph of  G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus  G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph  G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph  G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus  g ≥ 0 of the graph and the number of terminals  t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value  G of the genus.

scholarly journals Guessing with a Bit of Help

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 39 ◽  
Author(s):  
Nir Weinberger ◽  
Ofer Shayevitz
Keyword(s):  
Lower Bound ◽  
Data Processing ◽  
Maximum Entropy ◽  
Binary Vector ◽  
Upper Bounds ◽  
Fixed Number ◽  
Binary Symmetric Channel ◽  
Symmetric Channel ◽  
Data Processing Inequality ◽  
Binary Erasure Channel

What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice’s guessing-moments with and without observing Bob’s bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general k ≥ 1 ) and majority functions (for k = 1 ). We further provide a lower bound via maximum entropy (for general k ≥ 1 ) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for k = 1 ). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel and conjecture that greedy dictator functions achieve the optimal guessing ratio.

scholarly journals On the Convergence of the Benjamini–Hochberg Procedure

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2154
Author(s):  
Dean Palejev ◽  
Mladen Savov
Keyword(s):  
Lower Bound ◽  
Exponential Type ◽  
Alternative Hypothesis ◽  
Fixed Number ◽  
Multiple Comparison ◽  
Asymptotic Convergence ◽  
Test Statistic ◽  
Scientific Methods

The Benjamini–Hochberg procedure is one of the most used scientific methods up to date. It is widely used in the field of genetics and other areas where the problem of multiple comparison arises frequently. In this paper we show that under fairly general assumptions for the distribution of the test statistic under the alternative hypothesis, when increasing the number of tests, the power of the Benjamini–Hochberg procedure has an exponential type of asymptotic convergence to a previously shown limit of the power. We give a theoretical lower bound for the probability that for a fixed number of tests the power is within a given interval around its limit together with a software routine that calculates these values. This result is important when planning costly experiments and estimating the achieved power after performing them.

scholarly journals Graph Homomorphisms between Trees

10.37236/4096 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Péter Csikvári ◽  
Zhicong Lin
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Extremal Problems ◽  
Fixed Number ◽  
Star Graph ◽  
Spectral Entropy ◽  
Pendant Vertex ◽  
Path Graph ◽  
Graph Homomorphisms ◽  
Fixed Tree

In this paper we study several problems concerning the number of homomorphisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization and a dual of Bollobás and Tyomkyn's result concerning the number of walks in trees.Some other main results of the paper are the following. Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$. For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$. As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$. We also show that if $T_m$ is any fixed tree and$$\hom(T_m,P_n)>\hom(T_m,T_n),$$for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$.All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most.

scholarly journals On the evaluation of ‘saving-consumption’ plans

2005 ◽  
Vol 4 (1) ◽  
pp. 17-30 ◽  
Author(s):  
STEVEN VANDUFFEL ◽  
JAN DHAENE ◽  
MARC GOOVAERTS
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Risk Measures ◽  
Fixed Number ◽  
Consumption Pattern ◽  
Withdrawal Period

Knowledge of the distribution function of the stochastically compounded value of a series of future (positive and/or negative) payments is needed for solving several problems in an insurance or finance environment, see e.g. Dhaene et al. (2002a,b). In Kaas et al. (2000), convex lower-bound approximations for such a sum have been proposed. In case of changing signs of the payments however, the distribution function or the quantiles of the lower bound are not easy to determine, as the approximation for the random compounded value of the payments will in general not be a comonotonic sum.In this paper, we present a method for determining accurate and easy computable approximations for risk measures of such a sum, in case one first has positive payments (savings), followed by negative ones (consumptions).This particular cashflow pattern is observed in ‘saving–consumption’ plans. In such plans, a person saves money on a regular basis for a certain number of years. The amount available at the end of this period is then used to generate a yearly pension for a fixed number of years. Using the results of this paper we can find accurate and easy to compute answers to questions such as: ‘What is the minimal required yearly savings effort, α, during a fixed number of years, such that we will be able to meet, with a probability of at least (1−ε), a given consumption pattern during the withdrawal period?’

Interactive multiobjective DEA target setting using lexicographic DDF

2020 ◽  
Vol 54 (6) ◽  
pp. 1703-1722 ◽  
Author(s):  
Narges Soltani ◽  
Sebastián Lozano
Keyword(s):  
Efficient Frontier ◽  
Directional Distance Function ◽  
Fixed Number ◽  
Optimization Approach ◽  
Target Setting ◽  
Interactive Process ◽  
Decision Making Unit ◽  
Setting Approach ◽  
Promising Area ◽  
Current Region

In this paper, a new interactive multiobjective target setting approach based on lexicographic directional distance function (DDF) method is proposed. Lexicographic DDF computes efficient targets along a specified directional vector. The interactive multiobjective optimization approach consists in several iteration cycles in each of which the Decision Making Unit (DMU) is presented a fixed number of efficient targets computed corresponding to different directional vectors. If the DMU finds one of them promising, the directional vectors tried in the next iteration are generated close to the promising one, thus focusing the exploration of the efficient frontier on the promising area. In any iteration the DMU may choose to finish the exploration of the current region and restart the process to probe a new region. The interactive process ends when the DMU finds its most preferred solution (MPS).

Phospholipid-Protein Interactions in the Formation of Prothrombin Activator

1965 ◽  
Vol 14 (03/04) ◽  
pp. 431-444 ◽  
Author(s):  
E. R Cole ◽  
J. L Koppel ◽  
J. H Olwin
Keyword(s):  
Complex Formation ◽  
Protein Interactions ◽  
Calcium Ions ◽  
Fixed Number ◽  
Factor V ◽  
Clotting Factors ◽  
Number Of Binding Sites ◽  
C Complex ◽  
Autoprothrombin C

SummarySince Ac-globulin (factor V) is involved in the formation of prothrombin activator, its ability to complex with phospholipids was studied. Purified bovine Ac-globulin was complexed to asolectin, there being presumably a fixed number of binding sites on the phospholipid micelle for Ac-globulin. In contrast to the requirement for calcium ions in the formation of complexes between asolectin and autoprothrombin C, calcium ions were not required for complex formation between asolectin and Ac-globulin to occur ; in fact, the presence of calcium prevented complex formation occurring, the degree of inhibition being dependent on the calcium concentration. By treating isolated, pre-formed aso- lectin-Ac-globulin complexes with calcium chloride solutions, Ac-globulin could be recovered in a much higher state of purity and essentially free of asolectin.Complete activators were formed by first preparing the asolectin-calcium- autoprothrombin C complex and then reacting the complex with Ac-globulin. A small amount of this product was very effective as an activator of purified prothrombin without further addition of calcium or any other cofactor. If the autoprothrombin C preparation used to prepare the complex was free of traces of prothrombin, the complete activator was stable for several hours at room temperature. Stable preparations of the complete activator were centrifuged, resulting in the sedimentation of most of the activity. Experimental evidence also indicated that activator activity was highest when autoprothrombin C and Ac-globulin were complexed to the same phospholipid micelle, rather than when the two clotting factors were complexed to separate micelles. These data suggested that the in vivo prothrombin activator may be a sedimentable complex composed of a thromboplastic enzyme, calcium, Ac-globulin and phospholipid.

Sign in / Sign up

Export Citation Format

Share Document