## A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi’s conjecture.

Two days ago Nati Linial sent me an email entitled “A sensation in the morning news”. The link was to a new arXived paper by Yaroslav Shitov: Counterexamples to Hedetniemi’s conjecture.

Hedetniemi’s 1966 conjecture asserts that if $G$ and $H$ are two graphs, then the chromatic number of their tensor product $G \times H$ equals the minimum of their individual chromatic numbers.  Here, the vertex set of $G \times H$ is the Cartesian product of $V(G)$ and $V(H)$ and two vertices $(g_1,h_1)$ and $(g_2,h_2)$ are adjacent if $g_1$ is adjacent to $g_2$ and  $h_1$ is adjacent to $h_2$. (mistake corrected.) Every coloring of $G$ induces a coloring  of $G \times H$, and so is every coloring of $H$.  Therefore, $\chi (G \times H) \le \min (\chi (G), \chi (H))$. Hedetniemi conjectured that equality always hold and this is now refuted by  by Yaroslav Shitov.

The example and the entire proof are quite short (the entire paper is less than 3 pages; It is a bit densely-written).

Yaroslav Shitov

To tell you what the construction is, I need two important definitions.  The first  is the notion of the exponential graph ${\cal E}_c(H)$.

The exponential graph  ${\cal E}_c(H)$ arose in the study of Hedetniemi’s conjecture in a 1985 paper by El-Zahar and Sauer. The vertices of ${\cal E}_c(H)$ are all maps from $V(H)$ to $\{1,2,\dots,c\}$. Two maps $\phi, \psi$ are adjacent if whenever $v,u$ are adjacent vertices of $H$ then $\phi(u) \ne \psi (v)$El-Zahar and Sauer showed that importance of the case  that $H$ is a graph and $G$ is an exponential graph of $H$ for Hedetniemi’s conjecture. (The entire conjecture reduces to this case.) It is thus crucial to study  coloring of exponential graphs which is the subject of the three claims of Section 1 of Shitov’s paper.

The second definition is another important notion of product of graphs: The strong product G ⊠ H of two graphs $H$ and $G$. The set of vertices is again the Cartesian product of the two sets of vertices. This time,  $(g_1,h_1)$ and $(g_2,h_2)$ are adjacent in G ⊠ H if either

(a) $g_1$ is adjacent to $g_2$ and  $h_1$ is adjacent to $h_2$

OR

(b) $g_1$ is adjacent to $g_2$ and  $h_1 = h_2$ or $g_1 =g_2$ and  $h_1$ is adjacent to $h_2$

(The edges of condition (a) are the edges of the tensor product of the two graphs and the edges of condition (b) are the edges of the Cartesian product of the two graphs.)

For Shitov’s counterexample given in Section 2 of his paper, $H$ is the strong product of a graph $L$ with girth at least 10 and fractional chromatic number at least 4.1 with a large clique of size $q$. The second graph $G$ is the exponential graph ${\cal E}_c(H)$. Put $c =\lceil 4.1q \rceil$.  Shitov shows that when $c$ is sufficiently large then  the chromatic number of both $H,G$ is $c$,  but the chromatic number of their tensor product is smaller than $c$.

(Have a look also at Yaroslav’s other arXived papers! )

### Finite and infinite combinatorics

Let me make one more remark. (See the Wikipedea article.) The infinite version of Hedetniemi’s conjecture was known to be false.  Hajnal (1985) gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors. Rinot (2013) proved that in the constructible universe, for every infinite cardinal , there exist a pair of graphs of chromatic number greater than , such that their product can still be colored with only countably many colors. (Here is the paper.) Is there a relation between the finite case and the infinite case? (Both theories are quite exciting but direct connections are rare. A rare statement where the same proof applies for the finite and infinite case is the inequality $2^n>n$.

Here is a link to a survey article by Claude Tardif, (2008), “Hedetniemi’s conjecture, 40 years later” .

A few more thing worth knowing:

1) The weak version of the conjecture that asserts that If $\chi (G)= \chi (H))=n$, then $\chi (G \times H) \ge f(n)$ where $f(n)$ tends to infinity with $n$ is still open.

2)  Xuding Zhu proved in 2011 that the fractional version of the conjecture is correct,

3) The directed version of the conjecture was known to be false (Poljak and Rodl, 1981).

4) The conjecture is part of a rich and beautiful theory of graph homomorphisms (and the category of graphs) that I hope to come back to in another post.

## Answer to TYI 37: Arithmetic Progressions in 3D Brownian Motion

Consider a Brownian motion in three dimensional space. We asked (TYI 37) What is the largest number of points on the path described by the motion which form an arithmetic progression? (Namely, $x_1,x_2, x_t$, so that all $x_{i+1}-x_i$ are equal.)

Here is what you voted for

TYI37 poll: Final-results

Analysis of the poll results:  Almost surely 2 is the winner with 30.14% of the 209 votes, and almost surely infinity (28.71%) comes close at second place. In the  third place is  almost surely 3 (14.83%),  and then comes positive probability for each integer (13.4%), almost surely 5 (5.26%),  almost surely 6 (2.87%), and  almost surely 4 (2.39%).

## Test your political intuition: which coalition is going to be formed?

Almost surely 2 (briefly AS2) and almost surely infinity (ASI) can form a government  with no need for a larger coalition. But they represent two political extremes. Is AS3 politically closer to AS2 or to ASI? “k with probability p_k for every k>2” (briefly, COM) represent a complicated political massage. Is it closer to AS2 or to ASI? (See the old posts on which coalition will be formed.)

TYI37 poll: Partial results. It was exciting to see how the standing of the answers changed in the process of counting the votes.

Posted in Combinatorics, Open discussion, Probability | | 1 Comment

## The last paper of Catherine Rényi and Alfréd Rényi: Counting k-Trees

A k-tree is a graph obtained as follows: A clique with k vertices is a k-tree. A k-tree with n+1 vertices is obtained from a k-tree with n-vertices by adding a new vertex and connecting it to all vertices of a  k-clique. There is a beautiful formula by Beineke and Pippert (1969) for the number of k-trees with n labelled vertices. Their number is

${{n} \choose {k}}(k(n-k)+1)^{n-k-2}.$

If we count rooted k-trees where the root is a k-clique the formula becomes somewhat simpler.

$(k(n-k)+1)^{n-k-1}.$

In 1972, when I was a teenage undergraduate student I was very interested in various extensions of Cayley’s formula for counting labeled trees. I thought about the question of finding a Prüfer code for k-trees and  how to extend the results by Beineke and  Pippert when  for every clique of size $k-1$ in the k-tree we specify its “degree”, namely, the number of k-cliques containing it. (I will come back to the mathematics at the end of the post.) I thank Miki Simonovits for the photos and description and very helpful comments.

Above, Kató Renyi, Paul Turan, Vera Sós, and Paul Erdős ; below Kató, Vera, and Lea Schönheim. Pictures: Jochanan (Janos) Schönheim.

From right, Rényi, Turán and Erdős and Grätzer.

While I was working on enumeration of $k$-trees I came across  a paper by Catherine Rényi and Alfréd Rényi that did everything I intended to do and quite a bit more.

What caught my eye was a heartbreaking footnote: when the paper was completed Catherine Rényi was no longer alive.

The proceedings where the paper appeared were of a conference in combinatorics in Hungary in 1969. This was the first international conference in combinatorics that took place in Hungary.  The list of speakers consists of the best combinatorialists in the world and many young people including Laci Lovasz, Laci Babai, Endre Szemeredi, and many more who since then have become world-class  scientists.

Years later Vera Sós told me the story of Alfréd Rényi’s lecture at this conference, the first international conference in combinatorics that took place in Hungary:  “Kató died on August 23, on the day of arrival of the conference on “Combinatorial Theory and its Applications” (Balatonfured, August 24-29). Alfréd Renyi gave his talk (with the same title as the paper) on August 27 and his talk was longer than initially scheduled.  They proved the results in the paper just the week before the conference. The paper appeared in the proceedings  of the conference.”

Alfréd Renyi was one of the organizers of the conference and also served as one of the editors of the proceedings of the conference, which appeared in 1970. A few months after the conference, on February 1, 1970 Alfréd Rényi  died of a violent illness. The proceedings are dedicated to the memory of Catherine Rényi and Alfréd Rényi.

Two pictures showing Alfréd and Catherine Rényi and a picture of Alfred Rényi and Paul Erdős.

Repeating a picture from last-week post. From left: Sándor Szalai,  Catherine Rényi, Alfréd Rényi, András Hajnal and Paul Erdős (Matrahaza)

Going back to my story. I was 17 at the time and naturally I wondered if counting trees and similar things is what I want to do in my life. Shortly afterwards I went to the army. Without belittling the excitement of the army I quickly reached the conclusion that I prefer to count trees and to do similar things. My first result as a PhD student was another high dimensional extension of Cayley’s formula (mentioned in this post and a few subsequent posts).  The question of how to generalize both formulas for $k$-trees and for my hypertrees is still an open problem. We know the objects we want to count,  we know what the outcome should be, and we know that we can cheat and use weighted counting, but still I don’t know how to make it work.

1. Regarding the degree sequences for k-trees. You cannot specify the actual (k-2)-faces because those (in fact just the graph) determines the k-tree completely. So you need to count rooted k-trees and to specify the (k-2)-faces in terms of how they “grew” from the root.
2. The case that all degrees are 1 and 2 that correspond to paths for ordinary trees and to triangulating polygons with diagonals for 2-trees are precisely the stacked (k-1)-dimensional polytopes. This is a special case of the Renyi & Renyi formula that was also  found, with a different proof, by Beineke and Pippert.
3. It is  unlikely that there would be a matrix-tree formula for k-trees since telling  if a graph contains a 2-tree  is known to be NP complete. See this MO question. Maybe some matrix-tree formulas are available when we start with special classes of graphs.
4. Regarding the general objects – those are simplicial complexes that are Cohen-Macaulay and their dual (blocker) is also Cohen-Macaulay.

This post is just about a single paper of Catherine Rényi and Alfréd Rényi mainly through my eyes from 45 years ago. Catherine Rényi’s  main interest originally was  Number theory, she was a student of Turàn, and soon she became  interested in the theory of Complex Analytic Functions. Alfréd Rényi was a student of Frigyes Riesz and he is known for many contributions in number theory, graph theory and combinatorics and primarily in probability theory.  Alfréd Rényi wrote several papers about enumeration of trees, and this joint paper was Catherine Rényi ‘s first paper on this topic.

Posted in Combinatorics, People | | 7 Comments

## Are Natural Mathematical Problems Bad Problems?

One unique aspect of the conference “Visions in Mathematics Towards 2000” (see the previous post) was that there were several discussion sessions where speakers and other participants presented some thoughts about mathematics (or some specific areas), discussed and argued.  In the lectures themselves you could also see a large amount of audience participation and discussions which was very nice.

Let me draw your attention to  one question raised and discussed in one of the discussion sessions.

### 3.4 Discussion on Geometry with introduction by M. Gromov

Now, lets skip a lot of interesting staff and move to minute 23:20 where Noga Alon asked Misha Gromov to elaborate a statement from his opening lecture of the conference that  the densest packing problem in $R^3$ is not interesting.  In what follows Misha Gromov passionately argued that natural problems are bad problems (or are even stupid questions), and a lovely discussion emerged (in 25:00 Yuval Neeman commented about cosmology in response to Connes’s earlier remarks but then around 27:00 Vitali asked Misha to name some bad problems in geometry and the discussion resumed.) Misha made several lovely provocative further comments: he rejected the claim that this is a matter of taste, and argued that people make conjectures when they absolutely have no right to do so.

Misha argues passionately that natural problems are stupid problems

Actually one problem that Misha mentioned in his lecture as interesting (see also Gromov’s proceedings paper Spaces and questions), and that was raised both by him and by me is to prove an exponential upper bound for the number of simplicial 3-spheres with n facets. I remember that we talked about it in the conference and Misha was certain that the problem could be solved for shellable spheres while I was confident that the case of shellable spheres would be as hard as the general case.  He was right! This goes back to works of physicists Durhuus and Jonsson see this paper On locally constructible spheres and balls by Bruno Benedetti and  Günter M. Ziegler.

##### (Disclaimer: I asked quite a few questions that were both unnatural and stupid and made several conjectures when I had no right to do so.)
| Tagged | 1 Comment

## An Invitation to a Conference: Visions in Mathematics towards 2000

Let me invite you to a conference. The conference took place in 1999 but only recently the 57 videos of the lectures and the discussion sessions are publicly available. (I thank Vitali Milman for telling me about it.) One novel idea of Vitali Milman was to hold discussion sessions and they were quite interesting. (But, I am biased, I like discussions.) I will invite you to one of the heated discussions in the next post. There were very many nice talks and very many nice visions. And it is fun to watch the videos and judge the ideas in the perspective of time.

The proceedings appeared as GAFA special volumes, but alas the articles are not electronically available even to GAFA’s subscribers. Let me encourage both Birkhauser and the contributors to make them more available. My talk was: An invitation to Tverberg’s theorem, and my own contribution to the Proceedings Combinatorics with a Geometric Flavor is probably the widest scope survey article I ever wrote.  At the end of each section I added a brief philosophical thought about mathematics and those are collected in the post “about mathematics“.

Two more things: The conference (and few others organized by Vitali) was  in Tel Aviv with a few days at the dead see and this worked very nicely.  Vitali also organized in the mid 90s another very successful geometry conference unofficially celebrating Gromov’s 50th birthday with, among others, a very nice lecture by Gregory Perelman. If videos will become available I will be delighted to invite you to that conference as well. Update from Vitali: It was also the week of Jeff Cheeger’s 50th birthday which was also celebrated. Grisha Perelman gave an absolutely excellent talk  talk on works of Cheeger.  Lectures were not videotaped.

Avi Wigderson’s lecture

Are so called “natural questions” good for mathematics. Specifically is Kepler’s questions about the densest packing of unit balls in 3-space interesting? Watch a discussion of Misha Gromov, Noga Alon, Laci Lovasz and others. (next post)

We were all so much younger!  (And in that old millennium,  we were also all men 😦 )

Misha Gromov argues passionately that natural problems are bad problems (see next post)

Pictures of most participants ad two slides are below

## The (Random) Matrix and more

Three pictures, and a few related links.

### Van Vu

Spoiler: In one of the most intense scenes, the protagonist, with his bare hands and against all odds, took care of the mighty Wigner semi-circle law in two different ways. (From VV’s FB)

More information on Van Vu’s series of lectures. Van Vu’s home page; Related posts: did physicists really just prove that the universe is not a computer simulation—that we can’t be living in the Matrix? (Shtetl-Optimized); A related 2012 post on What’s New;

Two more pictures the first also from FB Continue reading

Posted in Combinatorics, People, What is Mathematics | | 1 Comment

## Gothenburg, Stockholm, Lancaster, Mitzpe Ramon, and Israeli Election Day 2019

Lancaster – Watching the outcomes of the Israeli elections (photo: Andrey Kupavskii)

### Sweden

I just came back from a trip to Sweden and the U.K. I was invited to Gothenburg to be the opponent for a Ph. D. Candidate  Malin Palö Forsström (by now Dr. Malin Palö Forsström),  who wrote her excellent Ph. D. thesis under the supervision of Jeff Steif in Chalmers University. We also used the opportunity for a lovely mini-mini-workshop

From Gothenburg I took the train to Stockholm to spend the weekend with Anders Björner and we talked about some old projects regarding algebraic shifting.  We had dinner with several colleagues including Svante Linusson who is a candidate for the European parliament!

Stockholm: With Anders and Cristins in the late 80s (left, I think this was also when I was an opponent), Svante Linusson ten days ago (right)

### The United Kingdom

The British Mathematical Colloquium at Lancaster was a lovely 4-day general meeting, an opportunity to meet some old and new friends (and Internet MO friend Yemon Choi in real life), and to learn about various new developments. I am aware of the fact that my list of unfulfilled promises is longer than those of most politicians, but I do hope to come back to some mathematics from this trip to Sweden and to Lancaster.

### Election Day

Last week’s Tuesday was election day in Israel,  and as much as I like to participate (and to devote a post to election day here on the blog – in 2009, 2012, and 2015) I had to miss the election, for the first time since 1985. (I still tried to follow the outcomes in real time.)

### The Negev, Israel

And we are now spending a three-day vacation and doing some mild hiking in Mitzpe Ramon, in the Negev, the Israeli desert.  The view around here is spectacular. I first fell in love with the sights of the Negev when I spent six months here when I was 19 (in the army). Since then we have been caming here many times over the years, and in 2002 the annual meeting of the Israeli Mathematical Union took place here, in the same hotel.

Ein Ovdat (left). The 2002 Annual meeting of the IMU (right). A large number of Israeli mathematicians come to a substantial fraction of these annual events.

### The stance of the main Israeli parties on quantum computing

One anecdote about the Israeli election is that both major political parties of Israel, the Likud, led by Benjamin (Bibi) Netanyahu that won 35 seats in the parliament and will probably lead the coalition, and the newly formed “Blue-and-White” party, led by Benny (Benjamin) Gantz that also won 35 seats and will probably lead the opposition, stand behind quantum computing! 🙂

Left – A paragraph from “Blue and White’s” charter with a pledge to quantum computing (I thank Noam Lifshitz for telling me about it). Right –  a news item (click for the article) about the quantum computing vision of Netanyahu and the Likud party.

### Update April 2, 2019: the links below are not working anymore.

Google Plus is a nice social platform with tens of millions participants. I found it especially nice for scientific posts, e.g. by John Baez, Moshe Vardi, or about symplectic geometry,  about Majorana Fermions, and with a discussion about What is combinatorics. A few months ago Google announced  that Google+ will be closed. This is going to happen today (March 31, 2019) in a few hours. For example, I am not even sure if the above valuable links will continue to operate. (Every individual user can get his own stuff.)

My question is different. Is it legitimate for Google to close Google+? Is it legitimate and is it ethical for Google to eliminate existing content from the public domain?

### Let me state clearly my opinion:

The fact that huge content that millions of people spent time and effort creating and that people spent time and effort to search and find, is deleted — by a huge and successful company — is very problematic

(Related post: What do firms want .)

Posted in Combinatorics, Economics, Open discussion, Rationality | Tagged | 9 Comments

## 10 Milestones in the History of Mathematics according to Nati and Me

Breaking news: David Harvey and Joris Van Der Hoeven. Integer multiplication in time O(nlogn). 2019. (I heard about it from Yoni Rozenshein on FB (חפירות על מתמטיקה); update GLL post. )

_____

Update: There were many interesting comments here and on FB. Itay Ben-Dan wrote a very interesting : Alternative to Gil Kalai & Nati Linial 10 Milestones in the History of Mathematics.

In 2006, the popular science magazine “Galileo” prepared a special issue devoted to milestones in the History of several areas of science and Nati Linial and me wrote the article about mathematics Ten milestones in the history of mathematics (in Hebrew). Our article had 10 sections highlighting one or two discoveries in each section.

Here are our choices. What would you add? what would you delete?

# The list

### 1) Numbers and Number Systems – The Irrationality of the square root of 2

Discovery No.1: the square root of 2 is not a rational number.

### 2) Geometry, the Discovery of Non-Euclidean Geometry, and Topology

Discovery no.2(A): Euclidean Geometry

Discovery no.2(B): Non-Euclidean Geometry

### 3) Algebra, Equations and Mathematical Formulas. Galois Theory.

Discovery no.3:  Abel-Galois Theorem: there is no solution with radicals to the general equation of the fifth degree and above.

### 4) Analysis and the Connection to Physics

Discovery no. 4(A): Differential and integral calculus (Isaac Newton, Gottfried Leibniz, 17th Century).

Discovery no. 4(B): The analysis of complex functions (Augustin-Louis Cauchy, Bernhard Riemann, 19th century).

### 5) Proofs and their Limitations: Logic, Set Theory, the Infinity, and Gödel’s Incompleteness Theorem.

Discovery no. 5(A): There are various kinds of infinity. For example, there are more real numbers than natural numbers.

Posted in Open discussion, What is Mathematics | Tagged | 38 Comments

## Short Presburger arithmetic is hard!

This is a belated report on a remarkable breakthrough from 2017. The paper is Short Presburger arithmetic is hard, by Nguyen and Pak.

Danny Nguyen

### Integer programming in bounded dimension: Lenstra’s Theorem

Algorithmic tasks are often intractable. But there are a few miracles where efficient algorithms exist: Solving systems of linear equations, linear programming, testing primality,  and solving integer programming problems when the number of variables is bounded. The last miracle is a historic 1983 theorem of Hendrik Lenstra (Here is the paper) and it is the starting point of this post.

Lensra’s theorem: Consider a system of linear inequalities

### Ax ≤ b

where $x=(x_1,x_2,\dots x_k)$ is a vector of $k$ variables, A is an integral k by n matrix and b is an integral vector of length n.

Let $k$ be a fixed integer. There is a polynomial time algorithm to determine if the system has an integral solution.

Of course, the full Integer Programming problem when $k$ is part of the input, is NP-complete. This problem came already (second in Karp’s list!) in the Mayflower of NP-complete problems –  Karp’s paper.

Sasha Barvinok famously showed in 1993 that even counting the number of solutions is in P. (Barvinok utilized the short generating function approach pioneered by Brion, Vergne and others.)

### Kannan’s theorem

Next,  I want to describe an amazing 1990 theorem of Ravi Kannan,

Kannan’s theorem considers formulas with one quantifier alternation in the Presburger arithmetic and it asserts that when the number of variables is fixed,  there is a polynomial time algorithm to decide if the formula is satisfiable.

(Here is a free version of Kannan’s paper.) Also here the counting problems were tackled with great success. Barvinok and Kevin Woods remarkably showed how to count projections of integer points in a (single) polytope in polynomial time, and subsequently Woods extended this approach to general Presburger expressions Φ with a fixed number of inequalities!

An important strengthening was achieved by Friedrich Eisenbrand and  Gennady Shmonin in the 2008 paper Parametric integer programming in fixed dimension. See also the survey chapter by Barvinok Lattice points and lattice polytopes.

You can find the formulation of Kannan’s theorem in full generality a little further but let me present now a special case related to the famous Frobenius coin problem. (See this post on GLL for more on Presburger arithmetic)

### Frobenius coin problem

Given k coins with integral values, the Frobinius coin problem is to determine the largest integer that cannot be described as positive integer combinations of the values of the coins. (See also this post on GLL.)

Theorem (Kannan): There is a polynomial time algorithm to solve the Frobenius coin problem for every fixed number of coins.

The issue of the way theory meets practice for the problems discussed in this post is very interesting but we will not discuss it. Let me remark that Herb Scarf (who along with Kannan played a role in B-L (Before Lenstra) developments) offered another approach for the solution of the Frobenius coin problem and related IP (Integer Programming)  problems based on his theory of maximal lattice-free convex bodies. See this related post.

### More than one quantifier

Given the result of Kannan and later that of Barvinok and Woods, many people expected that also for two alternations, or even for any other fixed number of alternations, Presburger arithmetic would be in polynomial time. Nguyen and Pak proved that the problem is NP-complete already for two quantifier alternations! Here is the link to the paper Short Presburger arithmetic is hard. Igor Pak’s homepage has a few other related papers.

Let me bring here Sasha Barvinok’s MathSciNet featured review of Nguyen and Pak’s paper which tells the story better than I could.

## Barvinok’s featured review to Nguyen and Pak’s paper

Presburger arithmetic allows integer variables, integer constants, Boolean operations (&, ∧, ¬), quantifiers (∃, ∀), equations and inequalities (=, <, >, ≤, ≥), addition and subtraction (+, −) and multiplication by integer constants. It does not allow multiplication of variables (if we allow multiplication of variables, we get Peano arithmetic).

Geometrically, a quantifier-free formula of Presburger arithmetic describes the set of integer points in a Boolean combination of rational polyhedra (that is, in the set obtained from finitely many rational polyhedra by taking unions, intersections and complements). Similarly, a formula of Presburger arithmetic with existential quantifiers only describes the set of integer points obtained from the set of integer points in a Boolean combination of polyhedra by a projection along some coordinates.

Unlike Peano arithmetic, Presburger arithmetic is decidable. Here the authors zoom in on the computational complexity of Presburger arithmetic, once the combinatorial complexity of the formula is bounded in advance. If we fix the number of variables, the validity of a formula with no quantifier alternations (that is, of the type ∃$x_1$ . . . ∃$x_k$Φ($x_1, \dots , latex x_k$) or of the type ∀$x_1$ . . . ∀$x_k$Φ($x_1, \dots , x_k$)) can be established in polynomial time by Lenstra’s integer programming algorithm [see H. W. Lenstra Jr., Math. Oper. Res. 8 (1983), no. 4, 538–548; MR0727410].

For a fixed number of variables, formulas with one quantifier alternation (∃$x_1$ . . . ∃$x_k$$y_1$ . . . ∀$y_m$Φ($x_1, \dots , x_k, y_1, \dots , y_m$)) can also be solved in polynomial time, as shown by R. Kannan [in Polyhedral combinatorics (Morristown, NJ, 1989), 39–47, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 1, Amer. Math. Soc., Providence, RI, 1990; MR1105115]. The decision procedure can be characterized as a polynomial time algorithm for parametric integer programming.

Suppose now that we fix the number of variables and the number of Boolean operations in advance (and hence get what is called a short formula of Presburger arithmetic). Thus the only parameters of the formula are the numerical values of the constants in the formula. The authors show that deciding validity becomes NP-complete if one allows
two quantifier alternations. Remarkably, they present an example of a formula

∃z ∈ $\mathbb Z$ ∀y ∈ $\mathbb Z^2$ ∃x ∈ $\mathbb Z^2$ Φ(x, y, z)

with an NP-complete decision problem, even though Φ contains at most 10 inequalities.
Another remarkable example is an NP-complete decision problem for a formula of the
type

∃z ∈ $\mathbb Z$ ∀y ∈ $\mathbb Z^2$ ∃x ∈ $\mathbb Z^6$: Ax + By + Cz ≤ b,

with at most 24 inequalities.

As the number of quantifier alternations is allowed to increase, the computational complexity in the polynomial hierarchy also moves up. The authors also describe the computational complexity of corresponding counting problems.

The proof is very clever; it uses the continued fraction expansion of a rational number to encode a growing family of intervals, with the help of which the authors build an NP-complete problem.