1 Introduction
mP_ord^#1 mX_ord^#1
The term binarization
refers to techniques to reformulate mixed-integer linear programs by replacing general integer (bounded) variables by sets of binary variables.
Roy compared several approaches with the goal to separate strong cutting planes in the reformulation and project it back [7]. Recently, Bonami and Margot continued this line of work using simple split disjunctions of ranks 1 and 2 to generate cutting planes [2]. The strongest bounds were obtained by a binarization in which an integer variable is replaced by the sum of binary variables , and symmetry-breaking constraints(1) |
are added. We denote by
the set of ordered binary vectors
satisfying (1) and by their convex hull. Note that Inequalities (1) already describe since the corresponding matrix is totally unimodular (every row has at most one and at most one , see Theorem 19.3 in [8]) and the right-hand side is integral. We call the vectors in ordered binary vectors.mP_even^#1
In this article we consider the strong binarization in combination with parity constraints on the affected integer variables. For this we define the ordered even parity polytope for a vector as
The integer points in are precisely those binary vectors of length whose entry-wise sum is even and which are ordered within each of the groups defined by . In the special case of (the all-ones vector), is the even parity polytope, which has the following outer description [4] (with ):
(2) |
Outline.
We start by describing an extended formulation for for arbitrary , which implies that the associated optimization problem can be solved in polynomial time (in the dimension of ). In Section 3 we state and prove the main result of this paper, a complete outer description for in the original space. After transferring the result to odd parities in Section 4 we derive in Section 5 a simple linear-time algorithm that solves the respective separation problems. Section 6 is dedicated to an application in which binarization is applied to an integer programming model for the graphic traveling salesman problem. During computations for this model we observed that the corresponding parity constraints did not improve the dual relaxation value. In Section 7 we provide a theoretical explanation of this effect.
2 Extended Formulations
δ^out δ^in mmmP_flow^#2-#3(#1)
In this section we review well-known extended formulations of parity polytopes based on dynamic programming, and adapt them to the ordered case. The linear-size formulation for introduced by Carr and Konjevod [3] works as follows. We define a digraph with nodes and arcs . We call the nodes the source and the sink. Clearly, the set of --flows of flow-value in any such acyclic digraph is described by the linear constraints
where we denote by resp. the set of outgoing resp. incoming arcs. Furthermore, again due to the total unimodularity of ’s incidence matrix (see Application 19.2 in [8]), is an integral polytope. We claim that the following map projects onto :
It is not hard to see that the integer points in are precisely the binary vectors with an even number of ’s, since an --path in must use an even number of diagonal arcs. The claim follows because is integral and because maps integral points to integral points. Figure 1 illustrates the constructed network and the projection (denoted by ).
We can now modify the network such that a similar projection yields for arbitrary vectors . The new construction works as follows. The node set of the new digraph is the same as the old digraph for , i.e., the node set of a -grid graph without the lower-left and lower-right corner nodes. Again, and are the left-most and right-most nodes, respectively. The arcs again go from left to right, and the arcs from the grid-layers correspond to the sets of variables for in the sense that the projection of an arc variable only has impact on a subset of the corresponding -variables. For every layer and every node (for if or , and for otherwise) on the left of this layer there are outgoing arcs, each associated to one of the possible values . The arcs with even all go to the same node , and the arcs with odd all go to node .
The projection map is defined by a 0/1-matrix. We describe it column-wise, i.e., we state for each arc in which rows (corresponding to -variables) it contains a -entry. Consider an arc going from node to node that is associated to , and thus satisfies . The column of the projection matrix that corresponds to this arc contains ’s in rows corresponding to the variables . The last part in Figure 1 illustrates a layer for .
The correctness of the modification is justified similarly to the ordinary construction. First, the flow polytope is integral due to total unimodularity of its constraint matrix. Second, the projection matrix is integral. Third, the integral --flows map to the correct vectors . To see this, observe that for every , the corresponding arcs of the network correspond to the vectors in , and that the number of diagonal arcs in such a path must be an even number.
3 Outer Description
We begin with the simple observation that for ordered binary vectors, parity can be measured using a linear function. For we define via . From
we obtain that maps into . Thus, for , the value is binary, and it is easy to check that holds if and only if has an odd number of ’s. Note that we will use for different values of , and assume that it is clear from the context. We can now state our main theorem.
Theorem 1.
Let . The ordered even parity polytope is equal to the set of those points that satisfy the inequalities
(3) |
for all with odd.
In order to prove the theorem we will make use of the following lemma.
Lemma 2.
Let and be finite sets. Then the polytope
is equal to the polytope
Proof of Lemma 2.
The inclusion “” is simple since for , every vector satisfies and .
To see “”, consider a point from the second set, which clearly satisfies . Thus is a convex combination of a point and a point . From the -coordinate we obtain . Similarly, there exist points and such that holds. This implies that holds, which concludes the proof. ∎
In our situation this lemma implies that if we “glue together” arbitrary 0/1-polytopes at a single coordinate, then the outer description of the resulting polytope is obtained by taking the union of the outer descriptions of the two component polytopes. We can now prove Theorem 1.
Proof of Theorem 1.
We start with the validity of Inequalities (3) for . Let be of odd cardinality. Since for every , the inequality can only be violated if all summands are equal to , i.e., if for all and for all . This in turn means that contains an odd number of ’s if and only if holds, contradicting the fact that the overall number of ’s is even because is odd.
We now turn to the proof that Inequalities (3) cut off those points in that have an odd number of ’s. Let be such a point and define to be the set of indices for which has an odd number of ’s. Since has an odd number of ’s, we have that is odd. Furthermore, the left-hand side of (3) is equal to , i.e., violates it.
It remains to prove that Inequalities (3) together with the inequalities that define each for yield an integral polytope. We show this by induction on the number of components of that are strictly greater than . For we have , and is, according to (2), the even parity polytope.
Let . W.l.o.g., we can assume that holds since otherwise we permute the variables suitably. By induction hypothesis we know that
holds, and is a -polytope. Furthermore, the polytope
is affinely isomorphic to . Since the latter is a -polytope and since maps to , is also a -polytope. Applying Lemma 2 to and yields that
is an integral polytope. The orthogonal projection of onto the -variables is again an integral polytope and due to the equation it is equal to the one in question. ∎
4 Odd Parities
mP_odd^#1
In this section we derive the outer description of the ordered odd polytopes. The latter are defined as
again for . The result corresponding to Theorem 1 is the following:
Corollary 3.
The ordered odd parity polytope for is the set of that satisfy Inequalities (3) for all with even.
In principle we could repeat the proof for Theorem 1, and exchange “odd” and “even” suitably. Instead we present a proof that uses the previous result.
Proof.
We claim that is the projection of the face defined by of the polytope onto the variables . This is justified by the fact that fixing the last component to exchanges the meaning of “odd” and “even” for the sum of the (remaining) components.
From Theorem 1 we know the outer description of , so it remains to carry out the projection. The only constraints that involve the last component are the bounds and the Inequalities (3) for odd-cardinality sets . The former need not be considered, and if holds, then the latter inequalities are redundant since they read , and for we know that and are nonnegative. It remains to consider the case of , in which the inequality is , which is equivalent to Inequality (3) for in the projection. Note that has even cardinality. ∎
5 Separation
In this section we show how to solve the separation problem for Inequalities (3).
Theorem 4.
Let and let . We can decide in linear time if holds, and if this is not the case, obtain an odd set whose associated Inequality (3) is violated by .
Δ
Proof.
It is easy to see that it suffices to compute for every and then to minimize over all of odd cardinality. This is done by computing the set . If is odd, then is the minimizer. Otherwise, we let (by we denote the symmetric difference, i.e., ) where is the index for which is minimum.
The computation of can obviously be done in linear time. Furthermore, the construction of and the search for can be carried out in time which results in a linear total running time. ∎
Directly from Corollary 3 we obtain the separation algorithm for the ordered odd parity polytopes. The only difference to the even case is that must be even, i.e., we set (for the same ) if and only if is odd.
Corollary 5.
Let and let . We can decide in linear time if holds, and if this is not the case, obtain an even set whose associated Inequality (3) is violated by .
6 Strengthened Blossom Inequalities for the Graphic TSP
In this section we consider the graphic traveling salesman problem (GTSP), defined as follows. The input consists of an undirected graph , and the goal is to find a minimum-length closed walk in that visits every node at least once. Recently, Sebő and Vygen developed a 7/5-approximation algorithm for this problem [9].
When modeling it as an integer program (IP) one has different options. The first is based on the observation that the problem is equivalent to the traveling salesman problem on the complete graph with nodes and edge weights where is equal to the (combinatorial) distance from to in . This model has binary variables which is much greater than if is sparse.
δ
In order to model the problem with fewer variables we can start with the IP
min | (4) | |||||
s.t. | (5) | |||||
(6) | ||||||
(7) |
where by we denote the cut-sets. The solutions to this problem correspond to the -edge-connected spanning subgraphs of . Since not every such subgraph is a closed walk, this IP does not model the GTSP. The necessary additional requirement is the parity condition for every . Such a constraint cannot be modeled directly, i.e., by adding linear inequalities. To obtain a correct model, one may add integer variables for every . Unfortunately, this does not contribute to the strength of the LP relaxation since relaxing ’s integrality constraints essentially makes them redundant.
It is well-known that an optimum solution will never use an edge more than twice (otherwise we can decrease the value by and obtain a better feasible solution). Hence we can restrict to the set which allows us to perform binarization, i.e., to replace, for every , the variable by with . This again leads to the same strength of the LP relaxation, but now allows us to enforce parity constraints: since every closed walk traverses each cut an even number of times, the following inequalities are valid for every and every with odd.
(8) |
If we identify the variables and with the doubled edges of , then these inequalities are the well-known Blossom Inequalities. These enforce that, for every cut , the projection of onto the variables corresponding to is in the even parity polytope.
We will soon strengthen these inequalities. In order to argue later why the separation problem for the strengthened inequalities can be solved efficiently, we review the separation problem for the unstrengthened ones. The first separation algorithm for Inequalities (8) was developed by Padberg and Rao [6]. The asymptotically fastest separation algorithm is by Letchford et al. [5]. More precisely, they describe how to efficiently solve (a generalization of) the following problem. Given a graph and weight vectors , their algorithm can minimize the objective
(9) |
over all sets and all odd sets . Clearly, by letting be the graph obtained from by doubling every edge and identifying the two edges with the variables and for , respectively, we can solve the separation problem for a point by minimizing with the following weight vectors:
In fact we do not have to double the graph’s edges for the computation: the cases where contains none or both edges contribute the same to the parity constraint on . Hence, we only have to consider the case that has lower weight. Similarly, only one of the two cases of containing precisely one of the two edges needs to be considered. This can be implemented using the weights
for all .
Strengthened Constraints.
To break the symmetry of and we add for each edge . Thus, we can now enforce that, for every cut , the projection of onto the variables corresponding to is in the ordered even parity polytope , where . By Theorem 1, the strengthened version of Constraint (8) reads
(10) |
for every and every with odd. Using almost the same separation procedure we can solve the separation problem by working in the original graph , and defining the weight vectors via
Note that the computational costs for separating Constraints (8) and (10) are almost equal.
7 Strength of the Relaxation
We carried out computational experiments on the GTSP, and it turned out that the lower bound obtained by the linear relaxation (4)–(6) was never improved by binarization of the variables and addition of (strengthened) parity constraints. In fact, in the root node many (strengthened) Blossom Inequalities were added, but with no effect on the dual objective.
We investigated this effect, and observed the following weakness of the approach of binarization and addition of parity constraints. The decisive property of our model is that only parity constraints actually use the binarization variables : all other variables consider the original variables (or, equivalently, the sum of the binarization variables ).
In a more abstract setting we consider arbitrary (integer) variables with domains for all , and apply binarization, i.e., introduce variables and linking constraints for each . We can assume that further (arbitrary) constraints that link the -variables are also present.
Now consider a parity constraint on (a subset of) the -variables, which is of course stated in terms of the corresponding -variables. Suppose we have a certain fractional relaxation solution that may violate this parity constraint, but satisfies all other constraints. If we can modify the -variables such that the linking constraints are still satisfied (i.e., that the sums remain constant) and such that the parity constraint is satisfied, then we obtain a feasible solution of the same value. Unfortunately, such a modification is possible under very mild conditions. In fact, the modifications are very general in the sense that they can be done independently for every set of binarization variables, that is, for all variables for a single . Furthermore, they do neither depend on the parity inequality that is actually violated nor on the parity. Hence, even if there exist many parity constraints whose variable sets overlap, such modifications can still exist, as it is often the case for Blossom Inequalities (note that there actually exist such parity constraints). We will now describe the modifications, and later discuss why the conditions are often satisfied for solutions of the linear relaxation (4)–(6).
Lemma 6.
Let , and . Then the maximum value of over all with is equal to the minimum of , , and .
Proof.
We denote the maximum in the lemma by , and by definition we have . We distinguish three cases, depending on where the minimum of , , and is attained.
Case 1: . Clearly, since holds, we have . This value is obtained by which satisfies and .
Case 2: . This implies and hence for all . For even we obtain . For odd , . Together, this proves . This value is obtained by which satisfies . Depending on the parity of it also satisfies or , i.e., we have .
Case 3: . Since always holds, it remains to construct a solution of that value. We define to be an integer whose distance to is at most , breaking ties such that is not equal to or equal to (but arbitrary otherwise). We furthermore define the values , and . We again consider two cases, depending on :
Case 3 (a): . We consider the vector defined via
and observe that holds, and that the first indices have ’s. Furthermore, we have that and hold.
Case 3 (b): . We consider the vector and observe that holds. Again, we have that and hold. This concludes the proof. ∎
Motivated by Lemma 6 we define for a variable . In the following theorem we binarize a set of variables and consider parity constraints on subsets of them.
Theorem 7.
Let and let . Let be a family of subsets . If every satisfies , then there exist with for all such that for every the vector is contained in the even and odd parity polytopes corresponding to .
Proof.
The theorem essentially states sufficient conditions for the case that after binarization and enforcing of several parity constraints, the values of the original variables remain feasible. Note that in order to satisfy for a constraint on variable set , it suffices that two of the participating variables have a distance to their respective bounds of at least (see Lemma 6), which is not very restrictive for nonbinary variables.
Implications for the Graphic TSP.
We consider an optimum solution of the LP relaxation (4)–(6) of the model introduced in Section 6. The requirements for Theorem 7 are satisfied if and only if for every nontrivial cut (, ), the inequality is satisfied. This is in particular the case if every cut contains two edges with . Note that integrality of the -variables does not play a role here, i.e., even if is integral, and has the wrong parity for some set , then there may exist a (fractional) assignment for -variables that is feasible for Constraints (10). For the IP model this means that the Blossom Inequalities only become useful if relevant -variables are near their bounds or if branching or cutting restricted the -variables further.
Acknowledgements.
We thank the author of the lrs software [1] that we used to compute complete descriptions for ordered parity polytopes of small dimensions. Furthermore, we thank Stefan Weltge and Volker Kaibel for valuable discussions. Finally, we are very grateful to the two referees who found a mistake in the proof of Lemma 6 and whose comments led to improvements in the presentation of the material.
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