In this paper we consider the decision version of Integer Linear Program (ILP) in standard form. Here, given a matrix with rows (constraints) and
columns and vectorsand the task is to decide whether the set
is non-empty. We are going to study structural properties of the incidence graph of the matrix . An integer program (IP) is a standard IP (SIP) if its set of solutions is described by (SSol), that is, if it is of the form
where is the objective function; in case is a linear function the above SIP is said to be a linear SIP. Before we go into more details we first review some recent development concerning algorithms for solving (linear) SIPs in variable dimension with the matrix admitting a certain decomposition.
Let be a block matrix, that is, , where are integral matrices. We define an -fold 4-block product of for a positive integer as the following block matrix
where is a matrix containing only zeros (of appropriate size). One can ask whether replacing in the definition of the set of feasible solutions (SSol) can give us an algorithmic advantage leading to an efficient algorithm for solving such SIPs. We call such an SIP an -fold 4-block IP. We derive two special cases of the -fold 4-block IP with respect to special cases for the matrix (see monographs [4, 21] for more information). If both and are void (not present at all), then the result of replacing with in (SIP) yields the -fold IP. Similarly, if and are void, we obtain the 2-stage stochastic IP.
The first, up to our knowledge, pioneering algorithmic work on -fold 4-block IPs is due to Hemmecke et al. . They gave an algorithm that given , the block matrix , and vectors finds an integral vector with minimizing . The algorithm of Hemmecke et al.  runs in time , where is the number of rows of , is the number of columns of , is the size of the input, and is a computable function. Thus, from the parameterized complexity viewpoint this is an algorithm for parameters . This algorithm has been recently improved by Chen et al.  who give better bounds on the function ; it is worth noting that Chen et al.  study also the special case where
is a zero matrix and even in that case present analgorithm. Since the work of Hemmecke et al.  the question of whether it is possible to improve the algorithm to run in time or not has become a major open question in the area of mathematical programming.
Of course, the complexity of the two aforementioned special cases of -fold 4-block IP are extensively studied as well. The first algorithm111That is, an algorithm running in time . for the -fold IPs (for parameters ) is due to Hemmecke et al. . Their algorithm has been subsequently improved [18, 9]. Altmanová et al.  implemented the algorithm of Hemmecke et al.  and improved the polynomial factor (achieving the same running time as Eisenbrand et al. ) the above algorithms (from cubic dependence to ). The best running time of an algorithm solving -fold IP is due to Jansen et al.  and runs in nearly linear time in terms of .
Last but not least, there is an algorithm for solving the 2-stage stochastic IP due to Hemmecke and Schultz . This algorithm is, however, based on a well quasi ordering argument yielding a bound on the size of the Graver basis for these IPs. Very recently Klein  presented a constructive approach using Steinitz lemma and give the first explicit (and seemingly optimal) bound on the size of the Graver basis for 2-stage (and multistage) IPs. It is worth noting that possible applications of 2-stage stochastic IP are much less understood than those of its counterpart -fold IP.
In the past few years, algorithmic research in this area has been mainly application-driven. Substantial effort has been taken in order to find the right formalism that is easier to understand and yields algorithms having the best possible ratio between their generality and the achieved running time. It turned out that the right formalism is connected with variants of the Gaifman graph (see e.g. ) of the matrix (for the definitions see the Preliminaries section).
In this paper we focus on the incidence (Gaifman) graph. We investigate the (negative) effect of the treedepth of the incidence Gaifman graph on tractability of ILP feasibility.
Given a matrix and vectors . Deciding whether the set defined by (SSol) is non-empty is -hard even if and .
For integers by we denote the set and is a shorthand for . We use bold face letters for vectors and normal font when referring to their components, that is, is a vector and is its third component. For vectors of vectors we first use superscripts to access the “inner vectors”, that is, is a vector of vectors and is the third vector in this collection.
From Matrices to Graphs.
Let be an integer matrix. The incidence Gaifman graph of is the bipartite graph , where contains one vertex for each row of and contains one vertex for each column of . There is an edge between the vertex and if , that is, if row contains a nonzero coefficient in column . The primal Gaifman graph of is the graph , where is the set of columns of and whenever there exists a row of with a nonzero coefficient in both columns and . The dual Gaifman graph of is the graph , where is the set of rows of and whenever there exists a column of with a nonzero coefficient in both rows and .
Undoubtedly, the most celebrated structural parameter for graphs is treewidth, however, in the case of ILPs bounding treewidth of any of the graphs defined above does not lead to tractability (even if the largest coefficient in is bounded as well see e.g. [18, Lemma 18]). Treedepth is a structural parameter which is useful in the theory of so-called sparse graph classes, see e.g. . Let be a graph. The treedepth of , denoted , is defined by the following recursive formula:
Let be an integer matrix. The incidence treedepth of , denoted , is the treedepth of its incidence Gaifman graph . The dual treedepth of , denoted , is the treedepth of its dual Gaifman graph . The primal treedepth is defined similarly.
The following two well-known theorems will be used in the proof of theorem 1.
Theorem 2 (Chinese Remainder Theorem).
Let be pairwise co-prime integers greater than and let be integers such that for all it holds . Then there exists exactly one integer such that
Theorem 3 (Prime Number Theorem).
Let denote the number of primes in , then .
It is worth pointing out that, given a positive integer encoded in unary, it is possible to the -th prime in polynomial time.
2 Proof of theorem 1
Before we proceed to the proof of theorem 1 we include a brief sketch of its idea. To prove -hardness, we will give a polynomial time reduction from 3-SAT which is well known to be -complete . The proof is inspired by the -hardness proof for ILPs given by a set of inequalities, where the primal graph is a star, of Eiben et. al .
Let be a 3-CNF formula. We encode an assignment into a variable . With every variable of the formula we associate a prime number . We make be the boolean value of the variable ; i.e., using auxiliary gadgets we force to always be in . Further, if for a clause by we denote the product of all of the primes associated with the variables occurring in , then, by Chinese Remainder Theorem, there is a single value in , associated with the assignment that falsifies , which we have to forbid for . We use the box constraints, i.e., the vectors , for an auxiliary variable taking the value to achieve this. For example let and let the primes associated with the three variables be and , respectively. Then we have and, since and is the only assignment falsifying this clause, we have that is the forbidden value for . Finally, the (SIP) constructed from is feasible if and only if there is a satisfying assignment for .
Proof (of theorem 1).
Let be a 3-CNF formula with variables and clauses (an instance of 3-SAT). Note that we can assume that none of the clauses in contains a variable along with its negation. We will define an SIP, that is, vectors , and a matrix with rows and columns, whose solution set is non-empty if and only if a satisfying assignment exists for . Furthermore, we present a decomposition of the incidence graph of the constructed SIP proving that its treedepth is at most 5. We naturally split the vector of the SIP into subvectors associated with the sought satisfying assignment, variables, and clauses of , that is, we have . Throughout the proof denotes the -th prime number.
We associate the part of with the variable and bind the assignment of to . We add the following constraints
and box constraints
to the SIP constructed so far.
Proof of Claim.
Note that by (the proof of) the above claim the conditions (1) and (2) essentially replace the large coefficient () used in the condition (5). This is an efficient trade-off between large coefficients and incidence treedepth which we are going to exploit once more when designing the clause gadget.
By the above claim we get an immediate correspondence between and truth assignments for . For an integer and a variable we define the following mapping
Notice that (4) implies that the mapping for . We straightforwardly extend the mapping for tuples of variables as follows. For a tuple of length , the value of is and we say that is defined if all of its components are defined.
Let be a clause with variables . We define as the product of the primes associated with the variables occurring in , that is, . We associate the part of with the clause . Let be the unique integer in for which is defined and gives the falsifying assignment for . The existence and uniqueness of follows directly from the Chinese Remainder Theorem. We add the following constraints
and box constraints
to the SIP constructed so far.
Proof of Claim.
Let be the SIP with constraints (1), (2), (6), and (7) and box constraints given by (3), (4), (8), (9), and . By the creftype 1, constraints (1), (2), (3), (4), are equivalent to the assertion that is defined. Then by the creftype 2, constraints (6), (7), (8), (9) are equivalent to checking that every clause in is satisfied by . This finishes the reduction and the proof of its correctness.
In order to finish the proof we have to bound the number of variables and constraints in the presented SIP and to bound the incidence treedepth of . It follows from the Prime Number Theorem that . Hence, the number of rows and columns of is at most .
It holds that .
Proof of Claim.
Let be the incidence graph of the matrix . It is easy to verify that is a cut-vertex in . Observe that each component of is now either a variable gadget for with (we call such a component a variable component) or a clause gadget for with (we call such a component a clause component). Let be the variable component (of ) containing variables and be the clause component containing variables . Let and . It follows that .
Refer to fig. 1. Observe that if we delete the variable together with the constraint (2) from , then each component in the resulting graph contains at most two vertices. Each of these components contains either
a variable and an appropriate constraint (1) (the one containing and ) for some or
the variable .
Since treedepth of an edge is 2 and treedepth of the one vertex graph is 1, we have that .
The bound on follows the same lines as for , since indeed the two gadgets have the same structure. Now, after deleting and (7) in we arrive to a graph with treedepth of all of its components again bounded by two (in fact, none of its components contain more than two vertices). Thus, and the claim follows.
3 Complementary Tractability Results
Treedepth and Degree Restrictions
It is worth noting that the proof of theorem 1 crucially relies on having variables as well as constraints which have high degree in the incidence graph. Thus, it is natural to ask whether this is necessary or, equivalently, whether bounding the degree of variables, constraints, or both leads to tractability. It is well known that if a graph has bounded degree and treedepth, then it is of bounded size, since indeed the underlying decomposition tree has bounded height and degree and thus bounded number of vertices. Let (SIP) with variables be given. Let denote the maximum arity of a constraint in its constraint matrix and let denote the maximum occurrence of a variable in constraints of . In other words, denotes the maximum number of nonzeros in a row of and denotes the maximum number of nonzeros in a column of . Now, we get that ILP can be solved in time , where is some computable function and is the length of the encoding of the given ILP thanks to Lenstra’s algorithm .
The above observation can in fact be strengthened—namely, if the arity of all the constraints or the number of occurences of all the variables in the given SIP is bounded, then we obtain a bound on either primal or dual treedepth. This is formalized by the following lemma.
For every (SIP) we have
The proof idea is to investigate the definition of the incidence treedepth of , which essentially boils down to recursively eliminating either a row, or a column, or decomposing a block-decomposable matrix into its blocks. Then, say for the second inequality above, eliminating a column can be replaced by eliminating all the at most rows that contain non-zero entries in this column.
We now proceed to the proof itself—in particular, we prove only the second inequality, as the first one is completely symmetric. The proof is uses induction with respect to the total number of rows and columns of the matrix . The base of the induction, when has one row and one column, is trivial, so we proceed to the induction step.
Observe that is disconnected if and only if is disconnected if and only if is a block-decomposable matrix. Moreover, the incidence treedepth of is the maximum incidence treedepth among the blocks of , and the same also holds for the dual treedepth. Hence, in this case we may apply the induction hypothesis to every block of and combine the results in a straightforward manner.
Assume then that is connected. Then
Let be the vertex for which the minimum on the right hand side is attained. We consider two cases: either is a row of or a column of .
Suppose first that is a row of . Then we have
as required, where the second inequality follows from applying the induction assumption to with the row removed.
Finally, suppose that is a column of . Let be the set of rows of that contain non-zero entries in column ; then and is non-empty, because is connected. If we denote by the matrix obtained from by removing column , then we have
as required. Here, in the second inequality we used the fact that is a subgraph of , while in the third inequality we used the induction assumption for the matrix .
Vertex Cover Number
It is natural to ask, whether there are other (more restrictive) structural parameters than treedepth that allow for polynomial-time or even fixed-parameter tractability for (SIP). Indeed, one such parameter is the (mixed) fracture number of , which was introduced in  and is defined as the minimum integer such that has a deletion set of size at most ensuring that every component of has size at most . It is easy to see that the treedepth of a graph is upper bounded by twice its fracture number. It has been shown in [6, Corollary 8] that (SIP) becomes solvable in polynomial-time if both the fracture number of and are bounded by a constant. Moreover the question whether this result can be improved to fixed-parameter tractability is known to be equivalent to the corresponding and long-standing open questions for 4-block -fold ILPs . Though we are not able to resolve this question, we can at least show fixed-parameter tractability for a slightly more restrictive parameter than fracture number, namely, the vertex cover number of . Towards this result, we need the following auxiliary corollary, which follows easily from [11, Theorem 4.1] and shows that (SIP) is fixed-parameter tractable parameterized by both the number of rows in and .
(SIP) can be solved in time , where and are the number of rows respectively columns of .
Eisenbrand and Weismantel recently proved that the corollary holds if all variables in the given (SIP) have a lower bound of , see [11, Theorem 4.1]. Since one can transform any (SIP) into an (SIP), where all variables have a lower bound of , by replacing any variable , where , with , where is a new variable with bounds , and subtracting (where denotes the -th column of ) from , we obtain that [11, Theorem 4.1] holds for general (SIP).
(SIP) can be solved in time , where is the size of a minimum vertex cover for .
Let be an instance of (SIP) with matrix . It is well-known, see e.g. [3, Chapter 1], that a minimum vertex cover of an -vertex graph can be found in time , where is its size. Hence, we may assume that we are given a vertex cover of of size . Let be the set of all constraints that correspond to vertices in . Because is a vertex cover, we obtain that the constraints in can only contain the at most variables in . Moreover, since we can assume that all rows of are linear independent, we obtain that . Hence and the theorem now follows from corollary 5.
We have shown that, unlike the primal and the dual treedepth, the incidence treedepth of a constraint matrix of (SIP) does not (together with the largest coefficient) provide a way to tractability. This shows that our current understanding of the structure of the incidence Gaifman graph is not sufficient. Furthermore, it is not hard to see that the matrix in our hardness result (cf. fig. 1) has topological length (and height ). Topological length is a newly introduced parameter ([10, Definition 18]) that allows to contract vertices of degree two in the tree witnessing bounded treedepth (i.e., in the tree in whose closure the incidence Gaifman graph emerges as a subgraph). It is worth pointing out that in our reduction we have topological height and constant height while the -fold -block IP structure implies topological height and the height of the two levels is an additional parameter. This further stimulates the question of whether an algorithm for -fold -block IP exists or not. Thus, the effect on tractability of some other “classical” graph parameters shall be investigated.
Namely, whether ILP parameterized by the largest coefficient and treewidth and the maximum degree of the incidence Gaifman graph is in or not. Furthermore, one may also ask about parameterization by the largest coefficient and the feedback vertex number of the incidence Gaifman graph.
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