Planar induced subgraphs of sparse graphs

In computer sciencethe clique problem is the computational problem of finding cliques subsets of vertices, all adjacent to each other, also called complete subgraphs in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique a clique with the largest possible number of verticesfinding a maximum weight clique in a weighted graph, listing all maximal cliques cliques that cannot be enlargedand solving the decision problem of testing whether a graph contains a clique larger than a given size.

planar induced subgraphs of sparse graphs

The clique problem arises in the following real-world setting. Consider a social networkwhere the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups of mutual friends. Along with its applications in social networks, the clique problem also has many applications in bioinformaticsand computational chemistry.

Most versions of the clique problem are hard. The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

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Therefore, much of the theory about the clique problem is devoted to identifying special types of graph that admit more efficient algorithms, or to establishing the computational difficulty of the general problem in various models of computation. To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming to be practical for networks comprising more than a few dozen vertices.

Although no polynomial time algorithm is known for this problem, more efficient algorithms than the brute-force search are known. For instance, the Bron—Kerbosch algorithm can be used to list all maximal cliques in worst-case optimal time, and it is also possible to list them in polynomial time per clique. The study of complete subgraphs in mathematics predates the "clique" terminology. But the term "clique" and the problem of algorithmically listing cliques both come from the social sciences, where complete subgraphs are used to model social cliquesgroups of people who all know each other.

They were the first to call complete subgraphs "cliques".

planar induced subgraphs of sparse graphs

Social science researchers have also defined various other types of cliques and maximal cliques in social network, "cohesive subgroups" of people or actors in the network all of whom share one of several different kinds of connectivity relation.

Many of these generalized notions of cliques can also be found by constructing an undirected graph whose edges represent related pairs of actors from the social network, and then applying an algorithm for the clique problem to this graph. Since the work of Harary and Ross, many others have devised algorithms for various versions of the clique problem. Also in the s, beginning with the work of Cook and Karpresearchers began using the theory of NP-completeness and related intractability results to provide a mathematical explanation for the perceived difficulty of the clique problem.

In the s, a breakthrough series of papers beginning with Feige et al. Clique-finding algorithms have been used in chemistryto find chemicals that match a target structure [4] and to model molecular docking and the binding sites of chemical reactions.All were delighted with the experience Martena, United States Express Iceland, July 2015 Larus put together the perfect itinerary, including different kinds of hotels (modern, rustic, farm, historic).

Sarah, United Kingdom Pearls of the South and West, July 2015 I would highly recommend Nordic Visitor to any potential clients. Tracy, United States The Natural Wonders of Iceland, June 2015 This is the first time I booked a trip solely based on internet research and not based on a referral from a friend or other trusted source. Matthew Express Norway, June 2015 We appreciate all the efforts Bjarni went through to make sure that our tour was very enjoyable.

Margery, United States Iceland Grand Tour, June 2015 Overall, we were very happy with Nordic Visitor, especially with the itinerary materials we received on arrival and the seamless experience we had from arrival to departure. Margaret, Canada express iceland, June 2015 Thank you for the good map with all the suggested stops.

Garry, Australia Iceland Full Circle, June 2015 Everything was as expected or better than expected. Derek Bruce, Canada Iceland Complete, June 2015 I found Iceland a most beautiful and exciting country but more important and impressive is the quality of your society as a whole. I thank you at all at Nordic Visitor in particular and all the people of Iceland in general for making my visit so memorable William, United States South Iceland at Leisure, June 2015 Everything was wonderful.

We've already recommended Nordic Visitor to friends. Thanks for a great experience. Darryl, Australia Golden Route of the Nordic Countries, June 2015 We were impressed with how well everything fitted together. Robert, United States Sweden's Top Three by Train, June 2015 Jennie was absolutely TERRIFIC in providing prompt service and information. Kairho1, United States Iceland Grand Tour, June 2015 I used to own a tour company myself so am quite confident that the services we received from Nordic Visitor were not only competent but actually went beyond what the promised.

I cannot recommend NV more highly. Jill, United Kingdom Pearls of the South and West, June 2015 What fantastic service this company provides. Chris, United States Scenic Fjords of Norway, May 2015 The hotels were excellent. Wayne, United States Norway Explorer, May 2015 All accommodations very good. Judith, Canada Iceland Full Circle, May 2015 I enjoyed every place I stayed. Laila, Norway Custom Booking, May 2015 We have already had a presentation of our tour to friends and neighbours.

Marcia, United States Iceland Full Circle, May 2015 The facilities were just great. Charline, Singapore South Iceland at Leisure, May 2015 You guys made my trip seamless, easy and perfect. Henry, United Kingdom Scenic Ring of Iceland, May 2015 All services, from pick up at the Airport to the return bus service and including accommodation and meals, were to a very good standard and problem-free.

Kirk, United States Iceland Full Circle, May 2015 Our trip with Nordic Visitor exceeded our expectations and I doubt we would have been able to book a comparable trip on our own. Dennis, Singapore Iceland Full Circle, April 2015 We came from half-way around the world. Kate, United Kingdom Iceland Winter World, March 2015 My mother and I booked a last minute 4 night holiday to Iceland. Could not be happier. Janis, United Kingdom Iceland Full Circle - Winter, March 2015 Very thoroughly planned with a good variety of accommodation.

Jane, United States Icelandic Winter Highlights, March 2015 Highly recommend taking a guided tour for duration of trip. Deirdre, Australia Northern Lights Circle Tour, March 2015 Hinrik was exceptional in his quality of service. A most enjoyable trip. Helena, United Kingdom Icelandic Winter Highlights, February 2015 We enjoyed the holiday so much that we want to return to Iceland to see more of the wonderful country.

Douglas Howes, United Kingdom Iceland Full Circle, February 2015 Hotels and guest houses an interesting mixture of styles. Ross, United States Northern Lights City Break, February 2015 Our hotels and meals were excellent in each instance. Jimmy, Switzerland Aurora Extravaganza, February 2015 Tour leaders are very experienced and gave us as much to enjoy as they possibly could. Megan, Germany Iceland Winter World, January 2015 I booked everything through Gudrun at Nordic Visitor for Iceland and it was super easy, they were really flexible with me and even when the weather and my schedule changed, it was never an issue.

Christy, United States Icehotel Winter Adventure, January 2015 Lavinia, New Zealand Northern Lights City Break, January 2015 It was an amazing experience and I enjoyed it all. David, United States Golden Circle and South Coast - Winter, December 2014 Great service, great value.Central banks care about the value of their international reserves and open positions in foreign currency as well as about the impact of exchange rate fluctuations on their inflation objectives.

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Governments care about the prices of exports and imports and the domestic currency value of debt payments. No surprise then that forecasting exchange rates has long been at the top of the research agenda in international finance.

Still, most of this literature is characterised by empirical failure. Starting with the seminal contribution of Meese and Rogoff (1983), a vast body of empirical research finds that models which are based on economic fundamentals cannot outperform a naive random walk model (i. In academic jargon, exchange rates are thought to follow a random walk. At first glance, the random walk model makes a lot of sense. The person on the street knows that movements in exchange rates are often hard to explain and is reluctant to believe that fundamental forces are at play.

Exchange rates often swing wildly on a daily basis for reasons that apparently have little connection to economic and financial variables. Even worse, they often move in the opposite direction of differences in short-term interest rates across countries.

Despite its simplicity, therefore, the random walk model remains appealing because it leads to smaller forecasting errors than most other exchange rate models.

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In this race, the random walk always wins. One relationship that does hold in the data is the so-called covered interest parity, which states that the interest rate gap equals the premium on forward contracts.

Indeed, that is basically how banks set forward rates. The Fama regressions put together the uncovered and covered interest parities to check whether the actual exchange rate follows the forward premium.

Decades of research on masses of data by dozens of scholars show that the actual appreciation does not follow the forward rate.

Indeed, it is the currency with the high interest rate that tends to appreciate, not the one with the low interest rate. While troublesome for economic theory, this puzzling behaviour may be valuable to investors. But what happens if we let a new horse enter the race. What happens if we assume that investors ignore the pure theory and instead work off the empirical fact, i. In recent research, we examine whether exchange rate predictability could translate into economic gains for investors using an asset allocation strategy that exploits this predictability (Della Corte, Sarno and Tsiakas, 2007).

In particular, we assess the economic value of the predictive ability of empirical exchange rate models that condition on the forward premium in the context of dynamic asset allocation strategies.

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But statistical evidence of exchange rate predictability in itself does not guarantee that an investor can profit by exploiting this predictability. We therefore evaluate the impact of predictable changes in the conditional FX returns and volatility on the performance of dynamic allocation strategies. Ultimately, we measure how much a risk-averse investor is willing to pay for switching from a dynamic portfolio strategy based on the random walk model to one which conditions on monetary fundamentals, the forward premium or a broader set of variables, including the money supply and income differentials across countries.

Our work suggests that these exchange rate predictions are valuable. In particular, the predictive ability of forward exchange rate premia has substantial economic value in a dynamic allocation strategy. In addition, conditioning on a forecast of future volatility given current information, rather than assuming that volatility in the foreign exchange market is constant, further enhances the predictability of exchange rates and increases risk-adjusted profits.This couldn't have been a better experience :) We highly recommend using NV services.

The little gems along the way were a great touch. We really enjoyed the secret hot springs and homemade ice cream (Efstidalur) that Arnar made notes of on our map. The map with the highlighted routes and stops was a lifesaver.

The planning and prebookings were also a relief so everything was along the way and easy to get through. Overall, we were impressed with how well planned everything was. All the guesthouses we stayed in were very very comfortable.

Much better than we had expected and the owners were all lovely. We always had a good home cooked breakfast and sometimes a dinner was provided at a cost of course. We were so glad we had chosen the "budget" accommodation to experience being in someone's home and meeting others staying there. Would highly recommend this type of accommodation. Found driving around Iceland quite easy. It is the only way to do it rather than be with a tour bus. The GPS helped a lot as did the general maps.

The main map was excellent where our route was plotted out for us. Just loved the whole experience. Each day seemed to be better than the previous such a beautiful country of contrasts. Put plenty of photos on Facebook and so many of our friends did not realise how amazing and picturesque Iceland is.

Clique problem

The day tours were excellent, particularly the ones from Hofn, Isafjordur and Akureyri. We are really grateful to the guides on these tours for making them so interesting and special.

This vacation was our honeymoon. We had just got married in the states a few days prior to the start of our trip. We had the most incredible experience. Everything was just wonderful. Working with our travel agent, Maria, for months prior, could not have been a better experience.Pam, Canada The Natural Wonders of Iceland, April 2016 Accommodation was better than expected and all the provided meals were top quality and representative of the country's cuisine.

Too much to see and not enough time in each destination. From the booking agents help to the overall transparency and guidance in the welcome package, we felt very comfortable and safe in a new country. Marcia, United States Iceland Full Circle - Winter, March 2016 Traditionally we book hotels ourselves when we travel.

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It was so convenient working with Nordic Visitor - having the route planned and the hotels booked. Great to have the Points of Interest marked as well. We found that 10 days was a nice vacation and overview of Iceland - but not enough time to really "get to know" the country and its people. That's why we would like to go back. The hospitality at all locations was amazing. We had no problem with finding vegan food either. Everyone was so very accommodating. Thank you for assisting us in making our trip to Iceland a marvelous time.

Stephen, United Kingdom Iceland Winter World, March 2016 Our whole experience of the holiday was one of relaxed efficiency, at each stage the pick up was there on time, the pack of information provided on our arrival by the delivery driver was thorough and comprehensive. When our northern lights tour was cancelled we were given information via the hotel, a follow up phone call to the NV office was efficiently dealt with. At all stages through the booking process our NV advisor, Audur, was very helpful.

It all made our wedding anniversary trip a very enjoyable relaxed holiday. Liz, United Kingdom South Iceland at Leisure - Winter, February 2016 We had a great time in Iceland and a lot of that was down to the email discussions, planning and preparation done by Sigfus on our behalf before the trip.

Everything ran smoothly and the whole trip was a joy. It was a pleasure to be able to meet him and thank him in person when we dropped our mobile phone off at the Nordic Visitor office at the end of our trip. He was friendly, nice, and funny. My husband and I had a blast riding with him in the van.

My travel consultant, Bjarni is perfect, he answered all my questions in a timely manner and he was able to customized my itinerary to accommodate my stay for an extra night at the hotel. Erica and Christopher, United States Iceland Winter World, January 2016 Prior to the start of the trip, the communication with our booking agent was fantastic- Hilmar is great.

He was able to accommodate our early departure time and rearranged our booking so we could fully enjoy the Blue Lagoon.DABAY first-up after 21 week spell, don't dismiss. TIZA drawn ideally and carrying less weight, in with a chance. Sacred Monarch (1) Scratched 7. Iamthekey (7) Scratched 1. Cries of Joy (12) Looks a toss up between the top two selections.

planar induced subgraphs of sparse graphs

SACRED MONARCH placed last start at Clare when first up and drawn ideally, genuine contender. REDEEKA led throughout for a dominant win last start at Hamilton on a soft track and won't be far away in the run, among the chances.

IAMTHEKEY placed last start at Naracoorte on a heavy track and likely to race just off the speed, the real danger in the race. CRIES OF JOY back from 33 week spell and placed when trialling at Mount Gambier, still in this. Cool Maverick (6) Scratched 9. Normandy Lad (2) 6. Ready for Action (9) COOL MAVERICK short back-up of four days and won last start at Strathalbyn, genuine contender. ZAAZOE has four placings from five runs this prep and placed last start at Clare, dangerous.

NORMANDY LAD 2 wins from three attempts this campaign and two of four wins have come from dry ground, in with a chance. READY FOR ACTION short back-up of six days and came on to finish midfield last start at Mount Gambier on a soft track, place chance.

First Reward (4) 5. Planet Voyage (5) 1. Reef's Revenge (9) FIRST REWARD has good early speed and racing back from the city, a winning chance. PLANET VOYAGE 5 from seven wins have been in the dry and generally races near the speed. EQUIETTO back after 16 week break and resumes well, cannot be ruled out.

planar induced subgraphs of sparse graphs

REEF'S REVENGE resumes from a 16 week spell and expected to settle on speed, place chance. Daffey Dux (3) 2.From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable. Either the true value is or is not within the given interval. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from Bayesian statistics: this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability.

In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the p-value).

A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (statistical significance) and the probability of type II error is the probability that the estimator doesn't belong to the critical region given that the alternative hypothesis is true.

The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false. Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms.

For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably. While in principle the acceptable level of statistical significance may be subject to debate, the p-value is the smallest significance level that allows the test to reject the null hypothesis.

This is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic.

Therefore, the smaller the p-value, the lower the probability of committing type I error. Some problems are usually associated with this framework (See criticism of hypothesis testing):Some well-known statistical tests and procedures are:Misuse of statistics can produce subtle, but serious errors in description and interpretationsubtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors.

For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics. Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the datawhich measures the extent to which a trend could be caused by random variation in the samplemay or may not agree with an intuitive sense of its significance.

The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy. There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.

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In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques used in particular fields are conducted (e. Warne, Lazo, Ramos, and Ritter (2012)). Thus, people may often believe that something is true even if it is not well represented.

Glossary of graph theory terms

Statistical analysis of a data set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable or confounding variable.

For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences.

Its mathematical foundations were laid in the 17th century with the development of the probability theory by Gerolamo Cardano, Blaise Pascal and Pierre de Fermat.


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