Simple odd β-cycle inequalities for binary polynomial optimization

We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd β-cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd β-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvátal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs.

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1 Introduction

In binary polynomial optimization

our task is to find a binary vector that maximizes a given multivariate polynomial function. In order to give a mathematical formulation, it is useful to use a hypergraph

, where the node set represents the variables in the polynomial function, and the edge set represents the monomials with nonzero coefficients. In a binary polynomial optimization problem, we are then given a hypergraph , a profit vector , and our goal is to solve the optimization problem

(1)

Using Fortet’s linearization [12, 14], we introduce binary auxiliary variables , for , which are linked to the variables , for , via the linear inequalities equationparentequation

(2a)
(2b)

It is simple to see that

Hence, we can reformulate (1) as the integer linear optimization problem

(3)

We define the multilinear polytope [6], which is the convex hull of the feasible points of (3), and its standard relaxation :

Recently, several classes of inequalities valid for have been introduced, including -link inequalities [4], flower inequalities [7], running intersection inequalities [8], and odd -cycle inequalities [5]. On a theoretical level, these inequalities fully describe the multilinear polytope for several hypergraph instances: flower inequalities for -acyclic hypergraphs, running intersection inequalities for kite-free -acyclic hypergraphs, and flower inequalities together with odd -cycle inequalities for cycle hypergraphs. Furthermore, these cutting planes greatly reduce the integrality gap of (3[8, 5] and their addition leads to a significant reduction of the runtime of the state-of-the-art solver BARON [9]. Unfortunately, we are not able to separate efficiently over most of these inequalities. In fact, while the simplest -link inequalities can be trivially separated in polynomial time, there is no known polynomial-time algorithm to separate the other classes of cutting planes, and it is known that separating flower inequalities is -hard [9].

Contribution.

In this paper we introduce a novel class of cutting planes called simple odd -cycle inequalities. As the name suggests, these inequalities form a subclass of the odd -cycle inequalities introduced in [5]. The main result of this paper is that simple odd -cycle inequalities can be separated in strongly polynomial time. While our inequalities form a subclass of the inequalities introduced in [5], they still inherit the two most interesting properties of the odd -cycle inequalities. First, simple odd -cycle inequalities can have Chvátal rank . To the best of our knowledge, our algorithm is the first known polynomial-time separation algorithm over an exponential class of inequalities with Chvátal rank . Second, simple odd -cycle inequalities, together with standard linearization inequalities and flower inequalities with at most two neighbors, provide a perfect formulation of the multilinear polytope for cycle hypergraphs. Finally, we believe that our separation algorithm could lead to significant speedups in solving several applications that can be formulated as (1) with a hypergraph that contains

-cycles. These applications include the image restoration problem in computer vision

[4, 5], and the low auto-correlation binary sequence problem in theoretical physics [2, 15, 5, 18, 17].

Outline.

We first introduce certain simple inequalities in Section 2 that are then combined to form the simple odd -cycle inequalities in Section 3. Section 4 is dedicated to the polynomial-time separation algorithm. In Section 5 we briefly address the question of redundancy since our inequalities are formally defined for a more general structure than a -cycle. Finally, Section 6 relates the simple odd -cycle inequalities to the general (non-simple) odd -cycle inequalities in [5].

2 Building block inequalities

We consider certain affine linear functions defined as follows.

()
()
()
()

mm mmm mmm mmm In this paper we often refer to , , , as building blocks. Although in these definitions and can be arbitrary subsets of an edge , in the following and will always correspond to the intersection of with another edge. In the next lemma we will show that all building blocks are nonnegative on a relaxation of obtained by adding some flower inequalities [7] to , which we will define now. For ease of notation, in this paper, we denote by the set , for any nonnegative integer .

Let and let , , be a collection of distinct edges in , adjacent to , such that for all with . Then the flower inequality [7, 5] centered at with neighbors , , is defined by

We denote by the polytope obtained from by adding all flower inequalities with at most two neighbors. Clearly is a relaxation of . Furthermore, is defined by a number of inequalities that is bounded by a polynomial in and .

Lemma 1.

Let be a hypergraph and let be one of , , , . Then is valid for . Furthermore, if and , then the implication given in Table 1 holds.

Condition Implication
Table 1: Implications of tight building block inequalities for integer solutions .
Proof.

First, is part of the standard relaxation and the implication is obvious.

Second, is the sum of the following inequalities from the standard relaxation: , for all , and . If and , then each of these inequalities must be tight, thus , for each . The last (tight) inequality yields , i.e., precisely one variable , for , is 0, while all others are 1, which yields the implication from Table 1.

Third, is the sum of the following inequalities: , for all and . The latter is the flower inequality centered at with neighbor . If and , then each of these inequalities must be tight, thus , for each . The last (tight) inequality yields , i.e., either and for exactly one , or and holds for all . Both cases yield the implication from Table 1.

Fourth, we consider . Note that due to , and , the three edges must all be different. Thus, is the sum of , for all and of . The latter is the flower inequality centered at with neighbors and . If and holds, then each of the involved inequalities must be tight, thus and for each . The last (tight) inequality implies , i.e., . ∎

3 Simple odd -cycle inequalities

We will consider signed edges by associating either a “” or a “” with each edge. We denote by the set and by a sign change for . In order to introduce simple odd -cycle inequalities, we first present some more definitions.

Definition 2.

A closed walk in of length is a sequence ----------, where we have as well as and for each , where we denote and for convenience. A signature of is a map . A signed closed walk in is a pair for a closed walk and a signature of . Similarly, we denote , , and . We say that is odd if there is an odd number of indices with ; otherwise we say that is even. Finally, for any signed closed walk in , its length function is the map defined by

where is the set of edge indices for which , and have sign pattern , i.e., .

We remark that the definition of is independent of where the closed walk starts and ends. Namely, if instead of we consider - ---------, and we define accordingly, then we have . Moreover, if or , then is independent of the choice of .

By Lemma 1, the length function of a signed closed walk is nonnegative. We will show that for odd signed closed walks, the length function evaluated in each integer solution is at least . Hence, we define the simple odd -cycle inequality corresponding to the odd signed closed walk as

(4)

We first establish that this inequality is indeed valid for .

Theorem 3.

Simple odd -cycle inequalities (4) are valid for .

Proof.

Let and assume, for the sake of contradiction, that violates inequality (4) for some odd signed closed walk . Since the coefficients of are integer, we obtain . From Lemma 1, we have that holds for all involved functions . Moreover, edge variables for all edges with , node variables for all nodes with , and the expressions for all nodes with are either equal or complementary (see Table 1), where the latter happens if and only if the corresponding edge satisfies . Since the signed closed walk is odd, this yields a contradiction for some edge of or for some node of or for a pair of subsequent edges of . ∎

Next, we provide an example of a simple odd -cycle inequality.

Figure 1: Figure of the closed walk considered in Example 4. The solid edges have sign and the dashed edges have sign .
Example 4.

We consider the closed walk of length given by the sequence -------- with signature depicted in Figure 1. We have , , , , . The corresponding simple odd -cycle inequality is . Using Definition 2, we write in terms of the building blocks as

Using the definition of the building blocks, we obtain

We write the sums explicitly and obtain

Example 4 suggests that, when the function is written explicitly, the coefficients in the function exhibit a certain pattern. This different expression of is formalized in the next lemma. The proof of the lemma can be obtained directly from the definition of by summing up each variable that appears in more than one building block.

Lemma 5.

Given a signed closed walk in with , we have

(5)

Using Lemma 5, we obtain the following result.

Proposition 6.

Simple odd -cycle inequalities are Chvátal-Gomory inequalities for and can be written in the form

(6)
Proof.

Let be an odd signed closed walk in a hypergraph . From Lemma 1 we obtain that holds for each . Lemma 5 reveals that in the inequality , all variables’ coefficients are even integers, while the constant term is an odd integer. Hence, the inequality divided by has integral variable coefficients, and we can obtain the corresponding Chvátal-Gomory inequality by rounding the constant term up. The resulting inequality is the simple odd -cycle inequality (4) scaled by and has the form (6). This shows that simple odd -cycle inequalities are Chvátal-Gomory inequalities for . ∎

It follows from Proposition 6 that, under some conditions on , simple odd -cycle inequalities are in fact -cuts (see [3]) with respect to . Some classes of such cutting planes can be separated in polynomial time, in particular if the involved inequalities only have two odd coefficients. In such a case, these inequalities are patched together such that odd coefficients cancel out and eventually all coefficients are even. We want to emphasize that this generic separation approach does not work in our case since our building block inequalities may have more than 2 odd-degree coefficients. Nevertheless, the separation algorithm presented in the next section is closely related to the idea of cancellation of odd-degree coefficients.

4 Separation algorithm

The main goal of this section is to show that the separation problem over simple odd -cycle inequalities can be solved in strongly polynomial time (Theorem 10). This will be achieved by means of an auxiliary undirected graph in which several shortest-path computations must be carried out. The auxiliary graph is inspired by the one for the separation problem of odd-cycle inequalities for the maximum cut problem [1]. However, to deal with our different problem and the more general hypergraphs we will extend it significantly.

Let be a hypergraph and let . Define to be the set of potential subsequent edge triples. We define the auxiliary graph

and length function as follows.

We point out that the graph can have parallel edges, possibly with different lengths. We immediately obtain the following corollary from Lemma 1.

Corollary 7.

The edge lengths are nonnegative.

We say that two nodes are twins if they only differ in the second component, i.e., the sign. We call a walk in the graph a twin walk if its end nodes are twin nodes. For a walk in , we denote by the total length, i.e., the sum of the edge lengths along the edges in . In the next two lemmas we study the relationship between odd signed closed walks in and twin walks in .

Lemma 8.

For each odd signed closed walk in there exists a twin walk in of length , where is the slack of the simple odd -cycle inequality (4) induced by with respect to . In particular, if the inequality is violated by , then we have .

Proof.

Let be an odd signed closed walk with ----------. For , let be the product of signs of all edges up to . Moreover, define . For each , we determine a walk in of length at most , and construct by going along all these walks in their respective order. The walk depends on , and :

The walks help to understand the meaning of the different node types: the walk starts at a node from if , it starts at a node from if , and it starts at a node from if holds. Similarly, the walk ends at a node from if , it ends at a node from if , and it ends at a node from if holds.

Note that all edges traversed by each are indeed in . It is easily verified that, for each , the walk ends at the same node at which the walk starts. Hence is indeed a walk in . Since holds, is closed and is odd, it can be checked that is a twin walk. Finally, by construction, holds, where the inequality comes from the fact that the minima in the definition of need not be attained by the edges from . By definition of we have , thus . ∎

Lemma 9.

For each twin walk in there exists an odd signed closed walk in whose induced simple odd -cycle inequality (4) has slack with respect to . In particular, if holds, then the inequality is violated by .

Proof.

Let be a twin walk in . We first construct the signed closed walk by processing the edges of in their order. Throughout the construction we maintain the index of the next edge to be constructed, which initially is . Since the construction depends on the type of the current edge (where visits first), we distinguish the relevant cases:

Case 1: and . Hence, and for some and some . We define and continue.

Case 2: and . Hence, and for some and some as well as . We define and . We then increase by and continue.

Case 3: and . Hence, and for some