グループテスト
統計学や組み合わせ数学において, 何らかの検体を特定する作業を, 検体を個別に調べるのではなく, グループにまとめて調べるという作業に分割するようなやりかたをグループテストとよぶ。1943年にRobert Dorfmanによって最初に研究されて以来, グループテストは比較的新しい応用数学の分野であり、幅広い分野で実用化されており, 現在も活発に研究されている。
グループテストのよく知られている例は、直列に接続されたいくつかの電球の中で, いずれか1つだけが壊れている場合である。目的は、できる限り少ない検査(いくつかの電球を電源に接続すること)で、壊れた電球を見つけることである。単純な方法は、各電球を個別に検査することである。しかし、電球の数が多い場合は、電球をグループにまとめる方がはるかに効率的である。たとえば、電球を2つのグループに分けて, 片方のグループの電球を一度にまとめて接続することにより、壊れた電球がどちらのグループに入っているかを判断でき、たった1回の検査で半数の電球を除外できる。
グループテストを実行するためのスキームには単純なものも複雑なものもあり、ひとつのスキームの中の各段階の検査も異なる場合がある。次の段階の検査法が前の段階の検査結果によって変わるようなスキームは適応的手順(adaptive procedure)と呼ばれる一方, すべての検査法が事前に決まるように設計されたスキームは「非適応的手順」(non-adaptive procedure)と呼ばれる。非適応的手順に含まれる検査のスキームの構造は「プーリング設計」として知られている(訳注: この部分の意味が原文ではわからない)。
グループテストには、統計学、生物学、コンピュータサイエンス、医学、エンジニアリング、サイバーセキュリティなど、多くの応用がある。これらの検査スキームに対する関心は、近年もHuman Genome Projectによって再燃している
。
数学の多くの分野とは異なり、グループテストの起源は、Robert Dorfmanという1人の人物が書いた1つの報告書に遡ることができる。第二次世界大戦中に、アメリカ公衆衛生局とセレクティブ・サービス・システムが, 入隊者の中から梅毒の男性全員を排除する大規模プロジェクトに着手したときにその動機が生じた。梅毒について個人を検査するには、血液サンプルを採取して, それを分析する必要がある。当時、この検査には費用がかかり、すべての兵士を個別に検査することは非常に費用がかかり、非効率的だった。
Supposing there are n soldiers, this method of testing leads to n separate tests. If a large proportion of the people are infected then this method would be reasonable. However, in the more likely case that only a very small proportion of the men are infected, a much more efficient testing scheme can be achieved. The feasibility of a more effective testing scheme hinges on the following property: the soldiers can be pooled into groups, and in each group the blood samples can be combined together. The combined sample can then be tested to check if at least one soldier in the group has syphilis. This is the central idea behind group testing. If one or more of the soldiers in this group has syphilis, then a test is wasted (more tests need to be performed to find which soldier(s) it was). On the other hand, if no one in the pool has syphilis then many tests are saved, since every soldier in that group can be eliminated with just one test.
The items that cause a group to test positive are generally called defective items (these are the broken lightbulbs, syphilitic men, etc.). Often, the total number of items is denoted as n and d represents the number of defectives if it is assumed to be known.
There are two independent classifications for group-testing problems; every group-testing problem is either adaptive or non-adaptive, and either probabilistic or combinatorial.
In probabilistic models, the defective items are assumed to follow some probability distribution and the aim is to minimise the expected number of tests needed to identify the defectiveness of every item. On the other hand, with combinatorial group testing, the goal is to minimise the number of tests needed in a 'worst-case scenario' – that is, create a minmax algorithm – and no knowledge of the distribution of defectives is assumed.
The other classification, adaptivity, concerns what information can be used when choosing which items to group into a test. In general, the choice of which items to test can depend on the results of previous tests, as in the above lightbulb problem. An algorithm that proceeds by performing a test, and then using the result (and all past results) to decide which next test to perform, is called adaptive. Conversely, in non-adaptive algorithms, all tests are decided in advance. This idea can be generalised to multistage algorithms, where tests are divided into stages, and every test in the next stage must be decided in advance, with only the knowledge of the results of tests in previous stages.
Although adaptive algorithms offer much more freedom in design, it is known that adaptive group-testing algorithms do not improve upon non-adaptive ones by more than a constant factor in the number of tests required to identify the set of defective items. In addition to this, non-adaptive methods are often useful in practice because one can proceed with successive tests without first analysing the results of all previous tests, allowing for the effective distribution of the testing process.
There are many ways to extend the problem of group testing. One of the most important is called noisy group testing, and deals with a big assumption of the original problem: that testing is error-free. A group-testing problem is called noisy when there is some chance that the result of a group test is erroneous (e.g. comes out positive when the test contained no defectives). The Bernoulli noise model assumes this probability is some constant, q, but in general it can depend on the true number of defectives in the test and the number of items tested. For example, the effect of dilution can be modelled by saying a positive result is more likely when there are more defectives (or more defectives as a fraction of the number tested), present in the test. A noisy algorithm will always have a non-zero probability of making an error (that is, mislabeling an item).
Group testing can be extended by considering scenarios in which there are more than two possible outcomes of a test. For example, a test may have the outcomes 0, 1 and 2^+, corresponding to there being no defectives, a single defective, or an unknown number of defectives larger than one. More generally, it is possible to consider the outcome-set of a test to be {0, 1, \ldots , k^+} for some k \in \mathbb{N}.
Another extension is to consider geometric restrictions on which sets can be tested. The above lightbulb problem is an example of this kind of restriction: only bulbs that appear consecutively can be tested. Similarly, the items may be arranged in a circle, or in general, a net, where the tests are available paths on the graph. Another kind of geometric restriction would be on the maximum number of items that can be tested in a group,The original problem that Dorfman studied was of this nature (although he did not account for this), since in practice, only a certain number of blood sera could be pooled before the testing procedure became unreliable. This was the main reason that Dorfman's procedure was not applied at the time. or the group sizes might have to be even and so on. In a similar way, it may be useful to consider the restriction that any given item can only appear in a certain number of tests.
There are endless ways to continue remixing the basic formula of group testing. The following elaborations will give an idea of some of the more exotic variants. In the 'good–mediocre–bad' model, each item is one of 'good', 'mediocre' or 'bad', and the result of a test is the type of the 'worst' item in the group. In threshold group testing, the result of a test is positive if the number of defective items in the group is greater than some threshold value or proportion. Group testing with inhibitors is a variant with applications in molecular biology. Here, there is a third class of items called inhibitors, and the result of a test is positive if it contains at least one defective and no inhibitors.
The concept of group testing was first introduced by Robert Dorfman in 1943 in a short report published in the Notes section of Annals of Mathematical Statistics.However, as is often the case in mathematics, group testing has been subsequently re-invented multiple times since then, often in the context of applications. For example, Hayes independently came up with the idea to query groups of users in the context of multiaccess communication protocols in 1978. Dorfman's report – as with all the early work on group testing – focused on the probabilistic problem, and aimed to use the novel idea of group testing to reduce the expected number of tests needed to weed out all syphilitic men in a given pool of soldiers. The method was simple: put the soldiers into groups of a given size, and use individual testing (testing items in groups of size one) on the positive groups to find which were infected. Dorfman tabulated the optimum group sizes for this strategy against the prevalence rate of defectiveness in the population.
After 1943, group testing remained largely untouched for a number of years. Then in 1957, Sterrett produced an improvement on Dorfman's procedure. This newer process starts by again performing individual testing on the positive groups, but stopping as soon as a defective is identified. Then, the remaining items in the group are tested together, since it is very likely that none of them are defective.
The first thorough treatment of group testing was given by Sobel and Groll in their formative 1959 paper on the subject. They described five new procedures – in addition to generalisations for when the prevalence rate is unknown – and for the most optimal one, they provided an explicit formula for the expected number of tests it would use. The paper also made the connection between group testing and information theory for the first time, as well as discussing several generalisations of the group-testing problem and providing some new applications of the theory.
Group testing was first studied in the combinatorial context by Li in 1962, with the introduction of Li's s-stage algorithm. Li proposed an extension of
Dorfman's '2-stage algorithm' to an arbitrary number of stages that required no more
than t = \frac{e}{\log_2(e)}d \log_2(n) tests to be guaranteed to find d or fewer defectives among n items.
The idea was to remove all the items in negative tests, and divide the remaining items into groups as was done with the initial pool. This was to be done s - 1 times before performing individual testing.
Combinatorial group testing in general was later studied more fully by Katona in 1973. Katona introduced the matrix representation of non-adaptive group-testing and produced a procedure for finding the defective in the non-adaptive 1-defective case in no more than t =\lceil \log_2(n) \rceiltests, which he also proved to be optimal.
In general, finding optimal algorithms for adaptive combinatorial group testing is difficult, and although the computational complexity of group testing has not been determined, it is suspected to be hard in some complexity class. However, an important breakthrough occurred in 1972, with the introduction of the generalised binary-splitting algorithm. The generalised binary-splitting algorithm works by performing a binary search on groups that test positive, and is a simple algorithm that finds a single defective in no more than the information-lower-bound number of tests.
In scenarios where there are two or more defectives, the generalised binary-splitting algorithm still produces near-optimal results, requiring at most d - 1 tests above the information lower bound where d is the number of defectives. Considerable improvements to this were made in 2013 by Allemann, getting the required number of tests to less than 0.187d + 0.5\log_2(d) + 5.5 above the information lower bound when n/d \geq 38 and d \geq 10. This was achieved by changing the binary search in the binary-splitting algorithm to a complex set of sub-algorithms with overlapping test groups. As such, the problem of adaptive combinatorial group testing – with a known number or upper bound on the number of defectives – has essentially been solved, with little room for further improvement.
There is an open question as to when individual testing is minmax. Hu, Hwang and Wang showed in 1981 that individual testing is minmax when n \leq  \lfloor (5d + 1)/2 \rfloor, and that it is not minmax when n > 3d. It is currently conjectured that this bound is sharp: that is, individual testing is minmax if and only if n \leq 3d.This is sometimes referred to as the Hu-Hwang-Wang conjecture. Some progress was made in 2000 by Ricccio and Colbourn, who showed that for large n, individual testing is minmax when d \geq n/\log_{3/2}(3) \approx 0.369n.