In integer and combinatorial optimization, many problems are computationally hard when the dimension is unbounded. In fixed dimensions, the situation is markedly different as many classical problems become tractable. Notably Lenstra’s algorithm forInteger Programming, and Barvinok’s algorithm for counting integer points in finite dimensional rational polytopes are polynomial.
In recent years, there has been a lot of work, including by the authors, to show that many problems in bounded dimension remain computationally hard as soon as one leaves the classical framework (see below). This paper proves hardness of two integer optimizations problems related to translation and expansion of rational polytopes in bounded dimensions.
We then consider the problem of describing Ehrhart quasi-polynomials of rational polytopes. These quasi-polynomials are of fundamental importance in both discrete geometry and integer optimization, yet they remain somewhat mysterious and difficult to study. We apply our result to prove a rather surprising property: that Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations of consecutive values (see below).
1.1. Translation of polytopes
The following problem was considered by Eisenbrand and Hähnle in [EH12].
|Integer Point Minimization (IPM)|
|Input:||, a rational polyhedron , .|
Parametric polytopes were introduced by Kannan [Kan90], who gave a polynomial time algorithm for IPM with and bounded. For larger fixed values , Aliev, De Loera and Louveaux [ADL16] proved that IPM is also polynomial time by employing the short generating functions technique by Barvinok and Woods [BW03] (see also [Bar06b, Bar08]). The following problem is an especially attractive special case:
|Input:||, , , and .|
In terms of parametric polytopes, this asks for a translation of the original polytope by so that it has at most integer points. Polytope Translation is a special case of the Integer Point Minimization problem, when is -dimensional.
Eisenbrand and Hähnle proved that the Polytope Translation is NP-hard for and unbounded:
Theorem 1.1 ([Eh12]).
Given a rational -gon , minimizing over is NP-hard.
Here and everywhere below, denotes the number of integer points in a polytope , and
is the standard first coordinate vector. We prove a similar result forwith a fixed number of vertices.
Given a rational polytope with at most vertices, minimizing over is NP-hard.
This resolves a problem by Eisenbrand.111F. Eisenbrand, personal communication (September 2017). Since the dimension is fixed, the number of facets of is at most an explicit constant. An integer version of this is:
Given a rational polytope with at most vertices and an integer , minimizing over is NP-hard.
While Theorem 1.3 is implied by Theorem 1.2 by a simple argument on rationality, its proof is simpler and will be presented first (cf. Section 3). The technique differs from those in [EH12] and our earlier work on the subject.
To prove Theorem 1.3, we show how to embed a classical NP-hard quadratic optimization problem into Polytope Translation. This is done by viewing each term in the quadratic objective as the integer volume of a separate polygon in , which are then merged in a higher dimension into a single convex polytope (cf. [NP17a, NP17b]). Let us mention that positivity and convexity are major obstacles here, and occupy much of the proof.
1.2. Expansions of polytopes
A quasi-polynomial is an integer function
where , , are periodic with integer period. For a rational polytope , consider the following function:
Ehrhart famously proved that is a quasi-polynomial, called the Ehrhart quasi-polynomial, see e.g. [Bar08, 18]. It is well known and easy to see that .
Many interesting combinatorial problems can be restated in the language of Ehrhart quasi-polynomials. We start with the following classical problem:
|Frobenius Coin Problem|
In other words, this problem asks for the largest integer that cannot be written as a combination of the coins ’s. Such a exists by the condition. Finding is an NP-hard problem when the dimension is not bounded, see [RA96]. For a fixed , Kannan proved that the problem can be solved in polynomial time [Kan92, BW03].
We can restate the Frobenius Coin Problem as follows. Let
Then counts the number of ways to write as an -combination of the ’s. Thus, is the largest , such that . Beck and Robins [BR04] used this setting to consider the following generalization:
|Input:||, , .|
In other words, the problem asks for the largest integer that cannot be represented as a combinations of ’s in different ways. Aliev, De Loera and Louveaux [ADL16] generalized Kannan’s theorem to prove that for fixed and the problem is still in P. Motivated by the above interpretation with the simplex , they also considered the following generalization:
|-Ehrhart Threshold Problem (-ETP)|
|Input:||A rational polytope and .|
For a polytope , this asks for the largest so that contains fewer than integer points. Again, when both and are fixed, it was shown in [ADL16] that this problem is in P. However, for varying we have:
The -ETP is NP-hard for rational polytopes with at most vertices.
It is an open problem whether the -Frobenius Problem is NP-hard when is a part of the input (see 6.1).
1.3. Fluctuations of the Ehrhart quasi-polynomial
It is well known that every quasi-polynomial can be written in the form:
Not all quasi-polynomials arise from polytopes. For instance, cannot be an Ehrhart quasi-polynomial because for all
, yet its leading terms fluctuates between odd and even values of. However, when restricted to finite intervals, every quasi-polynomial can be realized as of a polytope , in the following sense:
Let and be a quasi-polynomial of the form (1.1), with , for and . Then there exists a rational polytope and integers , such that:
Moreover, we have , and polytope has at most vertices. Here the vertices of and the constants can be computed in polynomial time.
Roughly, this theorems say that locally, Ehrhart quasi-polynomials can fluctuate as badly as general quasi-polynomials. In particular, we have:
For every sequence , there exists a polytope and such that:
Moreover, we have and polytope has at most vertices. Here the vertices of and the constants can be computed in polynomial time.
Consider the degree quasi-polynomial
Then for . Now we apply Theorem 1.5 to with . ∎
1.4. Brief historical overview
Integer Programming (IP) asks for given and , to decide whether
Equivalently, the problem ask whether a rational polytope contains an integer point.
When is unbounded, this problem includes Knapsack as a special case, and thus NP-complete (see e.g. [GJ79]). For fixed , the situation is drastically different. Lenstra [Len83] famously showed that IP is in P, even when is unbounded (see also [Sch86]). Barvinok [Bar93] showed that the corresponding counting problem is in FP, pioneering a new technique in this setting (see also [Bar08, Bar17]).
The Parametric Integer Programming (PIP) asks for a given , and , to decide whether
where is a convex polyhedron given by , for some , . Kannan showed that PIP is in P (see also [ES08, NP17b]). In [Kan92], Kannan used the PIP interpretation to show that for a fixed number of coins, the Frobenius Coin Problem is in P. Barvinok and Woods [BW03] showed that the corresponding counting problem is in FP, but only when the dimensions and are fixed (see also [Woo04]).
Although the above list is not exhaustive, most other problems in this area with fixed dimensions are computationally hard, especially in view of our recent works. Let us single out one negative small-dimensional result. We showed in [NP17a] that given two rational polytopes , it is #P-complete to compute
Note that the corresponding decision problem is a special case of PIP, and thus can be decided in polynomial time. This elucidated the limitations of the Barvinok–Woods approach (see also [NP18, NP17b]).
The Frobenius problem and its many variations is thoroughly discussed in [RA05], along with its connections to lattice theory, number theory and convex polyhedra. There are also some efficient practical algorithms for solving it, see [BHNW05]. The -Frobenius Problem, also called the generalized Frobenius problem, has been intensely studied in recent years, see e.g. [AHL13, FS11].
Ehrhart quasi-polynomials become polynomials for integer polytopes, in which case there is a large literature on their structure and properties (see e.g. [Bar08, Bar17] and references therein). We discuss integer polytopes in Section 5. A bounded number of leading coefficients of Ehrhart quasi-polynomials in arbitrary dimensions can be computed in polynomial time [Bar06a] (see also [B+12]). There is also some interesting analysis of the periods of the coefficients , see [BSW08, Woo05]. It seems that fluctuations of Ehrhart quasi-polynomials have not been considered until now.
As mentioned earlier, always denote the number of integer points in a convex polytope . We use to denote translation of by vector . The first coordinate vector is denoted by .
When the ambient space is clear, we use to denote the subspace with specified coordinates . We write for as . Finally, we use and .
2. Proof of Theorem 1.3
2.1. General setup
We start with the following classical QDE problem:
|Quadratic Diophantine Equations|
Manders and Adleman [MA78] proved that this problem is NP-complete (see also [GJ79, 7.2]). Observe that the problem remains NP-complete when we assume , Thus, the problem can be rephrased as the problem of minimizing
where . Indeed, we have if and only if the congruence in QDE is feasible.
Let . The two variables can be encode into a single integer variable by:
It is clear that each pair corresponds to such a unique and vice versa. So we can restate the problem as minimizing over . Now we have:
Here we denote by and the three terms in the above sum. First, we need to convert into a positive term. Fix a large constant , say will suffice for our purposes. We have:
Note that for . Let
We can rephrase the original NP-hard problem as the problem of computing the following minimum:
Note that each function is a product of terms of the form or for some constants . We encode each of these three types of functions as the number of integer points in some translated polytope. From this point on, we assume that , unless stated otherwise.
2.2. Trapezoid constructions
To illustrate the idea, we start with the simplest function with . Let and . Consider the following triangle:
(see Figure 1). Fix a line . It is easy to see that the hypotenuse of intersects at the point . So we have , and thus .
To encode a function with , we take and extend vertically by a distance below the line to make a trapezoid . Similarly, to encode a function with , we translate the hypotenuse of up by , and then extend upward by to get a trapezoid (see Figure 1). Formally, let:
Let us show that these trapezoids encode the function as stated above. For , we have , and thus . For , the hypotenuse of intersects at . Thus, we have , and thus , as desired.
For the function , we can encode it with the following triangle:
(see Figure 2). It is easy to see that the hypotenuse of intersects at the point . So and thus .
By modifying and keeping the same slope , we can encode the functions and with , , by using the following trapezoids:
respectively (see Figure 2).
Let us show that these trapezoids encode the function as stated above. For , we have , and thus . Similarly, for , the hypotenuse of intersects at , and thus . Since , we have , as desired.
Note that the counting function for each constructed trapezoid is periodic modulo . In other words, for every , and the same result holds for . From this point on, we let take values over in place of our earlier restriction .
2.3. The product construction
The next step is to construct polytopes that encode products functions of the form and .
Consider any functions of these forms. We take the trapezoids whose counting functions encode ’s. Each is described by a system:
We embed into the 2-dimensional subspace spanned by coordinates inside (with coordinates ). Then define:
It is clear that for every
and every vertical hyperplanein , we have .222Note that each intersects exactly one such hyperplane with . Therefore, we have
So the -dimensional polytope encodes the product . Note that is combinatorially a cube, which means it has facets and vertices.
2.4. Putting it all together
Now we embed them into as follows:
Note that are all disjoint. Define the polytope
Because of the way are embedded in , for every we have:
Thus, for every , we have:
By (2.4), we conclude that computing the following minimum is NP-hard:
Note that the polytopes have vertices, respectively. Thus, polytope has in total vertices, as desired.
3. Proof of Theorem 1.2
We modify the construction in the proof of Theorem 1.3 by perturbing all its ingredients to ensure that the desired minimum coincides with the one in the integer case. This construction is rather technical and assumes the reader is familiar with details in the proof above.
Recall that , , and . We perturb all constructed trapezoids as follows. Denote by the maximum slope over all hypotenuses of all constructed trapezoids. By a quick inspection of the terms in (2.2) and (2.3), one can see that . Take much smaller than and . For example, works. Now translate each constructed trapezoid by a distance horizontally in . Let be such a translated copy of some .333Recall that each encodes some function as for every . Then it is not hard to see that for all . In fact, due to the perturbation, we have:
for every and . This can be checked directly for all the trapezoid of types constructed in the proof of Theorem 1.3. Define the real set
For , denote by the (unique) integer such that . By the above observations, we have for every . Now we take these perturbed trapezoids and construct as similar to above, using the same product construction (see (2.5)). Note that and by (2.6), for every we have:
We need to “patch up” to make it the whole real line . Let
(see Figure 4).
We have: if , and otherwise.
Denote by the horizontal slice of at height . Then for the bottom edge , we have if , and otherwise. In other words, if and only if lies in some -th segment of (). Also every next slice is translated by , i.e., . There are in total non-empty slices, which implies the claim. ∎
We also embed into as:
Now let . By the above embeddings, we have:
If , we have:
We conclude that the following minimum is NP-hard to compute:
Note that the polytopes have vertices, respectively. Thus, polytope has in total vertices. This completes the proof of Theorem 1.2.
4.1. Proof of Theorem 1.4
Recall the polytope from Theorem 1.3 with 60 vertices and the translation vector . From the construction in Section 2, it is clear that has at least one integer point, which we call . We translate by so that , meanwhile still keeping the same for every .
Consider a very large multiple of (quantified later). Then for every , the two polytopes
satisfy , even though is just slightly larger. Since both polytopes are closed, if they differ by very little, we should have . To ensure this for all , it is enough to pick so that , where:
Here denotes the shortest distance between sets. Both and are polynomially bounded in and the largest over all vertex coordinates of (see [Sch86, Ch.10]). So only needs to be polynomially large in and the coordinates of .
Now we have for every . Let , then . Thus, for every . Recall that