Analyzing Quadratic Unconstrained Binary Optimization Problems Via Multicommodity Flows

Abstract

Quadratic Unconstrained Binary Optimization (QUBO) problems concern the minimization of quadratic polynomials in n {0, 1}-valued variables. These problems are NP-complete, but prior work has identified a sequence of polynomial-time computable lower bounds on the minimum value, denoted by C₂, C₃, C₄,…. It is known that C₂ can be computed by solving a maximum-flow problem, whereas the only previously known algorithms for computing C_k (k > 2) require solving a linear program. In this paper we prove that C₃ can be computed by solving a maximum multicommodity flow problem in a graph constructed from the quadratic function. In addition to providing a lower bound on the minimum value of the quadratic function on {0, 1}ⁿ, this multicommodity flow problem also provides some information about the coordinates of the point where this minimum is achieved. By looking at the edges that are never saturated in any maximum multicommodity flow, we can identify relational persistencies: pairs of variables that must have the same or different values in any minimizing assignment. We furthermore show that all of these persistencies can be detected by solving single-commodity flow problems in the same network.

1 Introduction

In this paper, we study the problem of Quadratic Unconstrained Binary Optimization (QUBO), which determines the minimum over {0, 1}ⁿ of quadratic pseudo-Boolean functions of the form

$$f(x_1,\dots ,x_n)=c_0+\sum _{j=1}^n{c}_i{x}_i+\sum _{1\le i<j\le n}{c}_{ij}{x}_i{x}_j$$

Many problems in the area of computer vision involve minimizing energy functions, and quadratic pseudo-Boolean functions are a common class of energy functions, which allows us to phrase vision problems as QUBO problems naturally. For a thorough survey of the applications of QUBO in computer vision, see [7]. Even in cases when the QUBO problem cannot be fully solved by an efficient algorithm, one can often extract useful information for the computer vision applications by discovering persistencies, i.e. partial assignments of the variables that must occur in an assignment that minimizes the function.

Although the QUBO problem is NP-hard, several approaches have been proposed to give good lower bounds on the minimum value. Many of these approaches involve rewriting the quadratic pseudo-Boolean function using posiforms, which we now define.

Let V = {x₁, x₂,…, x_n} be the set of variables, and let V̄ = {x̄₁, x̄₂,…, x̄_n} denote the set of negated variables, where x̄_i = 1−x_i. The elements of L = V ∪V̄ are called literals.

We can represent non-negative pseudo-Boolean functions as posiforms, i.e. multilinear polynomial expressions over all the literals with positive term coefficients

$$\phi (x_1,\dots ,x_n,{\overline{x}}_1,\dots ,{\overline{x}}_n)=\sum _{T\in \mathrm{\Omega }}{a}_T\prod _{u\in T}u$$

where Ω is a collection of subsets of L, and a_T > 0 for all T ∈ Ω. Every posiform φ defines a unique pseudo-Boolean function f by substituting x̄_i = 1 −x_i, but a pseudo-Boolean function can have multiple posiform representations.

A particular lower bound called roof-dual, denoted by C₂, is introduced in [8]. If f is a quadratic pseudo-Boolean function, then C₂ is the maximum c such that f = c + φ, where φ is a posiform of degree two. It is shown in [5] that we can compute C₂ by solving a maximum flow problem on a special graph called an implication network. The roof-dual value is then generalized to a family of lower bounds C_k. (See [2] for definitions.)

In this paper we will focus on the bound C₃. Let be the cone of nonnegative quadratic pseudo-Boolean functions having a posiform of degree three. Then as defined in [3]

$$C_3=max\{c\in \mathcal{R}\mid f=c+\phi ,\phi \in {\mathcal{P}}_3\}.$$

It is proved in [3] that C₃ can be computed by solving an odd cycle packing problem, but due to the fact that C₂ can be computed using maximum flow, we are interested in formulating C₃ as the solution of a flow problem.

Among the set of all possible posiforms of a quadratic pseudo-Boolean function, we will work with the bi-form in particular, which is introduced in [6]. For variables x and y, we call xȳ + x̄y a positive bi-term, and $xy+\overline{xy}$ a negative bi-term. If E denotes a collection of bi-terms such that no pair of variables is involved in more than one element of E, and α_e > 0 for all e ∈ E, then we call φ = Σ_e∈E α_ee a bi-form. By introducing variable x₀, we can write any quadratic psudo-Boolean function f in variables x₁,…, x_n uniquely as f(x₁,…, x_n) = c_f + φ_f (1, x₁,…, x_n), where φ_f is bi-form in the variables x₀, x₁,…, x_n. (See [4].)

Given a quadratic pseudo-Boolean function f = c_f +φ_f, where φ_f is a bi-form, we define an undirected graph G_φf = (V_φf, E_φf) as follows. V_φf = L, where L is the set of literals of f. We associate two edges to each bi-term. For positive bi-term c(x_ix̄_j + x̄_ix_j), we have edges (x_i, x_j) and (x̄_i, x̄_j), with weight $\frac{c}{2}$. For negative bi-term c(x_ix_j +x̄_ix̄_j), we have edges (x_i, x̄_j) and (x̄_i, x_j), with weight $\frac{c}{2}$. Conversely, edge (u, v) with capacity δ in G_φf corresponds to the bi-term δ(uv̄ + ūv).

We prove in Section 3 that for any quadratic pseudo-Boolean function f, C₃ is the sum of c_f and the maximum multicommodity flow value of network G_φf with source-sink pairs {(x₀, x̄₀), (x₁, x̄₁)…, (x_n, x̄_n)}. (See Section 2 for definitions related to multicommodity flow.)

More than the numerical lower bound on the minimum value, it is also useful to obtain partial optimal assignments. The concept of persistency and autarky are defined in [5], and we can generalize the idea of persistency and autarky to relational persistency and relational autarky. Call a set of literals valid if it contains at most one of x_i and x̄_i for all i. Consider a valid subset U ∈ L. We will say U is a strong (respectively, weak) relational persistency if literals in U have the same value in all (respectively, some) optimal assignment.

For any assignment x = (x₁,…, x_n) and Boolean value b ∈ {0, 1}, let x[U → b] denote the assignment y = (y₁,…, y_n) specified by

$$y_j=\{\begin{array}{ll}b\hfill & \text{if}{x}_j\in U\hfill \\ 1-b\hfill & \text{if}{\overline{x}}_j\in U\hfill \\ x_j\hfill & \text{otherwise}.\hfill \end{array}$$

We will call U a strong (respectively, weak) relational autarky if for all Boolean assignments x, there exists a value b ∈ {0, 1} such that f(x[U → b]) < f(x) (respectively, f(x[U → b]) ≤ f(x)), or x[U → b] = x. Relational persistencies and relational autarkies allow us to reduce the size of the optimization problem, since we can rewrite a posiform with a smaller variable set, while maintaining the same minimum value.

In Section 4, we focus on finding strong relational persistencies. When we push flow along a path in G_φf from x_i to x̄_i, it is equivalent to rewriting the bi-terms associated with edges on the path into a cubic posiform, and it turns out we can derive strong relational persistencies from the residual graph. In Theorem 2, we prove that if an edge (u, v) can never be saturated in a solution of our maximum multicommodity flow problem, literals u and v must have the same value in any minimizer of f. Then in Theorem 3, we consider the maximum flow problem on G_φf with single source-sink pair (x_i, x̄_i), and we show that in the residual graph, the set of literals in the connected component containing x_i is a strong relational persistency. Furthermore, we show that if U is a strong relational persistency, and x_i ∈ U, then there must be a maximum flow assignment for source-sink pair (x_i, x̄_i) such that U is a connected component in the residual graph.

Unsurprisingly, many techniques used in studying C₃ generalize those used in studying C₂. We introduce the variable x₀ simply to make the posiform homogeneously quadratic, and we don’t treat x₀ differently in computing C₃ and deriving relational persistencies. Various known facts about C₂ and its associated persistencies can be interpreted as a special case of the facts derived here, when we enforce the constraint x₀ = 1. For example, [5] obtains strong relational autarkies from the residual graph of running a maximum flow computation for the source-sink pair (x₀, x̄₀). In Theorem 3, we use the same technique over all source-sink pairs (x_i, x̄_i).

2 Definitions and notations

In this section, we summarize some definitions and notations involving multicommodity flow problems and their relationship to quadratic Boolean formulas.

2.1 Weighted Signed Graphs, Odd Cycles

A weighted signed graph G is an undirected graph (V, E) together with a set of positive weights {α_e|e ∈ E} and a partition {E⁺, E⁻} of the edge-set E. The edges in E⁺ are called positive, and those in E⁻ are called negative. A cycle of G is called odd if it contains an odd number of negative edges. The set of odd cycles is denoted by .

2.2 Odd cycle Packing

Given weighted signed graph G, the odd cycle packing problem (CP) is:

$$\begin{array}{l}\text{maximize}\sum _{C\in \mathcal{C}}{\xi }_C\\ \text{s}.\text{t}.\sum _{C\ni e}{\xi }_C\le {\alpha }_e\forall e\in E(G)\\ {\xi }_C\ge 0\forall C\in \mathcal{C}\end{array}$$ (1)

2.3 Multicommodity Flow

Given undirected graph G = (V, E) with edge capacities c_e ≥ 0, and source-sink pairs (s₁, t₁), · · ·, (s_k, t_k), let P_i = {paths from s_i to t_i}.¹ The maximum multicommodity flow problem on G is:

$$\begin{array}{l}\text{maximize}\sum _{i=1}^k\sum _{p\in P_i}{\zeta }_p\\ \text{s}.\text{t}.\sum _{i=1}^k\sum _{p\in P_i\mid P\ni e}{\zeta }_p\le c_e\forall e\in E\\ {\zeta }_p\ge 0\forall p\in \underset{i=1}{\overset{k}{\cup }}{P}_i\end{array}$$ (2)

Any vector (ζ_p) satisfying the constraints in (2) is called a multicommodity flow, or simply a flow, regardless of whether it maximizes the objective function. For a thorough introduction to multicommodity flow theory, we refer the reader to [1].

2.4 The graphs G_f and G_φ

Given quadratic pseudo-boolean function f, we can write f uniquely as the sum of a constant and a bi-form:

$$f=c_f+{\phi }_f.$$

We can construct a weighted signed graph G_f = (V_f, E_f) corresponding to bi-form φ_f as follows. The vertex set V_f is equal to {x₀,…, x_n}, the set of variables of φ_f. We associate an edge to each bi-term in φ_f. Edges have the same sign as their associated bi-terms, and the weights α_e are the same as the coefficients of their associated bi-terms.

Recall that there is another graph associated to φ_f, namely the graph G_φf with vertex set {x₁,…, x_n, x̄₁,…, x̄_n} defined in Section 1. Henceforth we will use the notation G_φ rather than G_φf for convenience. In Section 3 we will relate the maximum multicommodity flow problem in G_φ to the odd cycle packing problem in G_f.

2.5 Symmetric flow

The graph G_φ has a two-fold symmetry given by a permutation ι: V_φ → V_φ. This permutation is defined by setting ι (x_i) = x̄_i and ι (x̄_i) = x_i. We can extend ι to edges in the obvious way: if u, v are literals and e = (u, v) is an edge of G_φ, then the definition of G_φ ensures that ē = (ū, v̄) is also an edge of G_φ and we can define ι (e) = ē. Similarly we can extend ι to sets of vertices — ι (S) = {u|ι (u) ∈ S} — and to paths: for a path p in G_φ, if p = v₁ → · · · → v_k then ι(p) = ι(v_k) → ··· → ι (v₁). (Note that ι reverses the direction of the path.) Finally we can extend ι to flows: if ζ is any flow, then ι (ζ) is defined by setting ι (ζ)_p = ζ_ι(p) for all paths p. Note that ι(ζ) is also a flow because the edge capacities are preserved by ι. A flow ζ is called symmetric if it satisfies ι (ζ) = ζ. Note that if ζ is any maximum multicommodity flow, then $\frac{1}{2}(\zeta +\iota (\zeta ))$ is also a maximum multicommodity flow, and it is symmetric. Thus, there is always at least one symmetric flow that optimizes (2).

3 Computing C₃ reduces to multicommodity flow

Give quadratic pseudo-boolean function f, we can find its unique biform representation, f = c_f + φ_f, then construct G_f and G_φ according to φ_f. Denote by v(CP) the optimal value of the cycle packing problem on G_f, and v(PP) the optimal value of the multicommodity flow problem on G_φ with source-sink pairs (x₀, x̄₀), (x₁, x̄₁),…, (x_n, x̄_n).

Lemma 1

For any quadratic pseudo-boolean function f, C₃ = c_f + v(CP).

Proof

See [3].

Lemma 2

For any quadratic pseudo-boolean function f, v(PP) = v(CP).

Proof

To prove Lemma 2, we will show that for every feasible solution ξ of the cycle packing problem, we can construct a feasible symmetric solution ζ of the multicommodity flow problem with the same objective function value, and vice versa.

ξ → ζ:

Given odd cycle C ∈ , starting from the variable with the lowest index, we can derive two paths p_C and $p_C^{\prime }$ in G_f where

$$\begin{array}{l}{p}_C=x_{c_0}\to x_{c_1}\to \dots \to x_{c_k}\to x_{c_0}\text{and}\\ p_C^{\prime }=x_{c_0}\to x_{c_k}\to \dots \to x_{c_1}\to x_{c_0}.\end{array}$$

For every path p in G_f, there is a unique path λ(p) = u₀ → … → u_k+1 in G_φ satisfying u₀ = x_c0, and u_i ∈ {x_ci, x̄_ci} for all i > 0. In fact λ(p) can be defined by letting p[1..i] denote the initial segment of p starting at x_c0 and ending at x_ci, and setting

$$u_i=\{\begin{array}{ll}{x}_i\hfill & \text{if}p[\mathrm{1..}i]\text{contains}\text{an}\text{even}\text{number}\text{of}\text{negative}\text{edges}\hfill \\ {\overline{x}}_i\hfill & \text{if}p[\mathrm{1..}i]\text{contains}\text{an}\text{odd}\text{number}\text{of}\text{negative}\text{edges.}\hfill \end{array}$$

Note that u_k+1 = x̄_c0 when p is an odd cycle.

Given feasible CP solution ξ, we can construct solution ζ of the multicommodity flow problem:

$$\forall C\in \mathcal{C}{\zeta }_{\lambda (p_C)}={\zeta }_{\lambda (p_C^{\prime })}=\frac{{\xi }_C}{2}$$

It is clear to see ξ and ζ have the same objective function value, and ζ is symmetric. The flow ζ is also feasible, because if ζ violates the capacity constraint of edge (u, v) in PP, ξ must violate the corresponding constraint of the edge (var(u), var(v)) in CP. Here and henceforth, var denotes the function from literals to variables defined by var(x_i) = var(x̄_i) = x_i.

ζ → ξ:

Give path p ∈ P_i in G_φ, we can write p as

$$p:x_i\to u_1\to \dots \to u_k\to {\overline{x}}_i$$

Extending the function var to paths, we get paths in G_f

$$\mathit{var}(p):\mathit{var}(x_i)=x_i\to \mathit{var}(u_1)\to \dots \to \mathit{var}(u_k)\to \mathit{var}({\overline{x}}_i)=x_i$$

Consider V_φ as the union of two sets V_f and V̄_f. An easy observation is that if an edge’s two endpoints are in the same set, its associated bi-term is positive, and if its endpoints are in different sets, it is associated to a negative bi-term. Since p goes from x_i to x̄_i, we know there are odd number of edges in p that are associated to negative bi-terms, so var(p) = C ∈ . Given a feasible solution ζ of the multicommodity flow problem, we can construct a solution ξ of the cycle packing problem:

$$\forall C\in \mathcal{C}{\xi }_C=\sum _i\sum _{\underset{\mathit{var}(p)=C}{p\in P_i}}{\zeta }_p$$

ξ has the same objective function value as ζ. We can prove ξ is feasible by contradiction.

Suppose ξ violates the constraint

$$\sum _{C\ni e_0}{\xi }_C\le {\alpha }_{e_0}{e}_0=(x_i,x_j)$$

Each edge in E_f is associated with a bi-term, which correpsonds to two edges in E_φ, so for edge e₀ there are two corresponding edges e₁, e₂ ∈ E_φ, where

$$e_1=\{\begin{array}{ll}(x_i,x_j)\hfill & \text{if}{e}_0\text{positive}\hfill \\ (x_i,{\overline{x}}_j)\hfill & \text{if}{e}_0\text{negative}\hfill \end{array}{e}_2=\{\begin{array}{ll}({\overline{x}}_i,{\overline{x}}_j)\hfill & \text{if}{e}_0\text{positive}\hfill \\ ({\overline{x}}_i,x_j)\hfill & \text{if}{e}_0\text{negative}\hfill \end{array}$$

From the correspondence between paths in G_φ and odd cycles in G_f, we know every path p with var(p) = C must use one of e₁ and e₂, so

$$\begin{array}{l}\sum _{C\ni e_0}{\xi }_C>{\alpha }_{e_0}{e}_0=(x_i,x_j)\\ \sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_1\\ \mathit{var}(p)=C\end{array}}{\zeta }_p+\sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_2\\ \mathit{var}(p)=C\end{array}}{\zeta }_p>{\alpha }_{e_0}\\ \sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_1\\ \mathit{var}(p)=C\end{array}}{\zeta }_p+\sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_2\\ \mathit{var}(p)=C\end{array}}{\zeta }_p>2c_{e_1}=2c_{e_2}=c_{e_1}+c_{e_2}\end{array}$$

We know at least one of the two constraints in multicommodity flow is violated, i.e.:

$$\begin{array}{l}\sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_1\\ \mathit{var}(p)=C\end{array}}{\zeta }_p>c_{e_1}\\ \text{or}\sum _i\sum _{\begin{array}{c}p\in P_i,p\ni e_2\\ \mathit{var}(p)=C\end{array}}{\zeta }_p>c_{e_2}\end{array}$$

which contradicts the feasibility of ζ. Thus we know ξ must be feasible.

Since for every feasible solution ξ of cycle packing problem, we can construct a feasible solution ζ of the multicommodity flow problem with the same objective function value, and vice versa, v(CP) = v(PP).

Theorem 1

For any quadratic pseudo-boolean function f, C₃ = c_f + v(PP).

Proof

Theorem 1 is a direct result of Lemma 1 and Lemma 2.

4 Persistency and autarky results

Theorem 2

If e = (u, v) is an edge of G_φ such that e is not saturated in any symmetric maximum multicommodity flow, then {u, v} is a strong relational persistency for the corresponding pseudo-Boolean function f.

Proof

Notice that given a feasible solution ζ for the multicommodity flow problem on G_φ, we can express the graph’s edge capacity vector as the sum of two non-negative vectors, one for the flow value on edges, and the other for the remaining edge capacities. We refer to these vectors as “the flow layer” and “the remaining graph layer.” The flow layer can be directly derived from the flow assignments, and the remaining graph layer, G_φ,ζ = (V_φ,ζ, E_φ,ζ), where

• V_φ,ζ = V_φ

• E_φ,ζ = {e ∈ E_φ | c_e > Σ_iΣ_p∈Pi, _Pe ζ_p}

• For all e ∈ E_φ,ζ, the capacity is c_e,ζ = c_e − Σ_iΣ_p∈Pi|Pe ζ_p.

We will make use of the following lemma from [3].

Lemma 3

$\begin{array}{l}{\sum }_{i=1}^{l-1}{u}_i{\overline{u}}_{i+1}+u_l{u}_1+{\sum }_{i=1}^{l-1}{\overline{u}}_i{u}_{i+1}+{\overline{u}}_l{\overline{u}}_1=1+2u_1({\overline{u}}_1{u}_2+\dots +\\ {\overline{u}}_{l-1}{u}_l)+2{\overline{u}}_1(u_1{\overline{u}}_2+\dots +u_{l-1}{\overline{u}}_l)\end{array}$

Proof

Lemma 3 can be easily proven by induction. See [3] for a proof.

Using Lemma 3, we can transform a flow along path p: x_i → u₁ → u₂ → … → u_k → x̄_i with value δ to the sum of a constant and a cubic posiform:

$$\delta +2\delta x_i({\overline{u}}_1{u}_2+\dots +{\overline{u}}_{k-1}{u}_k)+2\delta {\overline{x}}_i(u_1{\overline{u}}_2+\dots +u_{k-1}{\overline{u}}_k)$$

It follows that the flow layer can be written as the sum of a constant — which is equal to the flow value — and a homogeneous cubic Boolean function, while the remaining graph layer corresponds to a homogeneous quadratic Boolean function.

Suppose e = (u, v) is never saturated in any optimal symmetric multicommodity flow solution. Among all optimal solutions, consider those with the maximum number of vertices in the component containing v in the graph G_φ,ζ, and among these solutions, let ζ be the one with the minimum c_{e, ζ}. By our assumption on ζ, if we maintain the same total flow value, it will be impossible to expand the connected component containing v, and it is also impossible to decrease c_{e, ζ} without making the component containing v smaller. Suppose there exists an optimal assignment x^* with u ≠ v. Without loss of generality, we can assume that x^* assigns u = 0 and v = 1.

Define S to be the set of vertices of the connected component containing v in the subgraph induced by vertices with value 1 according to x^* in G_{φ, ζ}. We will show that changing the values of literals in S to 0, and their negations to 1, will give an assignment x′ such that f(x′) < f(x^*), thus contradicting the hypothesis that x^* is an optimal assignment.

Write the pseudo-Boolean function f as the sum of constant, cubic terms corresponding to the flow solution ζ, and quadratic terms corresponding to G_{φ, ζ}. An easy observation is that setting the two endpoints of an edge to the same value will give the corresponding bi-term value 0. It is then easy to see x′ will decrease the value of quadratic terms by at least c_{e, ζ}. Now if we can show changing from x^* to x′ will not increase the value of cubic terms, then we know f(x′) will be strictly less than f(x^*).

Lemma 4

Given symmetric flow assignment ζ₁, for path p from x_i to x̄_i with flow value (ζ₁)_p > 0, and vertex x_j ≠ x_i, if x_j or x̄_j is on p, or there exists a simple path p′ in G_{φ, ζ1} from x_j or x̄_j to any vertex on p, we can get a flow assignment ζ₂ such that:

1. ζ₁ and ζ₂ have the same flow value

2. E_{φ, ζ1} ⊆ E_{φ, ζ2}

3. if e′ = (u′, v′) belongs to p′ but not p, ζ₂ puts more flow on e′ than ζ₁ does, and c_{e′, ζ 2} < c_e′,ζ1

4. if e′ = (u′, v′) is not in p, c_{e′, ζ 2} ≤ c_{e′, ζ 1}

Proof

Since ζ₁ is symmetric, a flow along path p = x_i → u₁ → … → u_k → x̄_i implies a flow along path ι(p) = x_i → ū_k → … → ū₁ → x̄_i with the same value. The two paths together will then form a generalized cycle, which we define to be a walk that starts and ends at the same vertex, but may traverse an edge more than once. If x_j or x̄_j is on p, we can cut the cycle into two possibly non-simple paths at x_j and x̄_j, then take some shortcuts to make the two paths simple. For example, consider

$$\begin{array}{l}p=x_i\to u_1\to u_2\to \dots \to u_{k-1}\to {\overline{u}}_1\to {\overline{x}}_i\\ \iota (p)=x_i\to u_1\to {\overline{u}}_{k-1}\to \dots \to {\overline{u}}_2\to {\overline{u}}_1\to {\overline{x}}_i\end{array}$$

then redistributing the flow as from u₂ to ū₂ will give

$$\begin{array}{l}{u}_2\to \dots \to u_{k-1}\to {\overline{u}}_1\to {\overline{u}}_2\\ u_2\to u_1\to {\overline{u}}_{k-1}\to \dots \to {\overline{u}}_2\end{array}$$

Notice that the redistribution operation may increase some edges’ remaining capacities. In the above example, the remaining capacities of c_{(xi,u1), ζ} and c_{(x̄i,ū1), ζ} are increased. If an edge is not on p, then its remaining capacity will not be increased.

If there exists a simple path from x_j or x̄_j to some v′ on p, we only need to consider the first case, since by symmetry the second case will be equivalent to the first one if we switch p and ι(p). Without loss of generality, assume the path p_e from x_j to v′ uses no edge on p or ι(p); otherwise we can take a different v′ and get a shorter p_e. We can first redistribute the flow as from v′ to v̄′ without increasing the remaining capacity of any edge on p_e, then extend a small portion of the flow δ′ as from x_j to x̄_j using p_e and ι(p_e). If we make δ′ small enough, the redistribution will not saturate any edge on p_e and ι(p_e)

Lemma 5

All paths involving vertices in S with positive flow value must use the edge (u, v) or (ū, v̄).

Proof

Suppose there is flow along p passing v′ ∈ S without using (u, v) and (ū, v̄). Since there exist paths u → v → v′ and v̄′ → v̄ → ū in G_{φ, ζ}, we know from Lemma 4 that we can get another optimal flow assignment without shrinking the connected component containing v, but decrease c_{(u,v), ζ}, which contradicts our assumption about ζ.

Lemma 5 implies that all flow involving the literals with different values in x′ and x^* are along paths using the (undirected) edge (u, v) or (ū, v̄). Since p and ι(p) yield the same function terms, and p uses (ū, v̄) implies ι(p) uses (u, v), we only need to prove x′ will not increase the value of the cubic terms corresponding to flow along paths using (u, v). Since all such paths use vertex u, we can redistribute the flows as from u to ū. We will no longer assume the minimality of c_{(u,v), ζ}, since the redistribution may increase c_{(u,v), ζ}.

By Lemma 3, we know the cubic terms corresponding to these flow paths all have the form:

$$\delta u(\overline{u}{v}_1+{\overline{v}}_1{v}_2+\dots )+\delta \overline{u}(u{\overline{v}}_1+v_1{\overline{v}}_2+\dots )$$

which can be further reduced to quadratic terms by the assumption u = 0:

$$\delta (v_1{\overline{v}}_2+v_2{\overline{v}}_3+\dots )$$

Consider any specific flow path. When we change x^* to x′, there are only two cases that the value of the above expression can be increased:

The first case is v_ℓ = 1 in x′, and v_ℓ+1 ∈ S. Then the term v_ℓv̄_ℓ+1 has value 0 according to x^*, but value 1 according to x′. The second case is the symmetric counterpart of the first case, namely v_ℓ ∈ ι(S), and v_ℓ+1 = 0 in x′. If we can prove the first case is impossible, by symmetry the second case is also impossible.

Suppose we have the first case in the flow along path p. By the fact that v_ℓ = 1 in x′, we know v_ℓ ∉ S. If v_ℓ ∈ ι(S), the edge (v_ℓ, v_ℓ+1) must be saturated, since otherwise there exists a path from u to ū in G_{φ, ζ}, contradicting our assumption that ζ is a maximum flow. If v_ℓ ∉ ι(S), v_ℓ = 1 in x^*, and by the definition of S, v_ℓ ∉ S implies the edge (v_ℓ, v_ℓ+1) is saturated. Either way, (v_ℓ, v_ℓ+1) is saturated. Furthermore, since v_ℓ+1 is in S, there exists a path from v to v_ℓ+1 in G_{φ, ζ} that does not use edge (v_ℓ, v_ℓ+1). Hence we can reroute some small enough portion of the original flow to use the alternate path from v to v_ℓ+1, thus making the edge (v_ℓ, v_ℓ+1) unsaturated without changing the value of the flow solution. This will enlarge S to S ∪ {v_ℓ}, contradicting the maximality of S.

By the above discussion, we know x′ won’t increase the value of the cubic terms, and will decrease the value of quadratic terms by at least c_{(u,v), ζ}. This contradicts the optimality of x^*, so u and v must have the same value in any optimal assignment.

4.1 Persistencies and autarkies from single-commodity flows

Lemma 6

Any subset of a strong relational autarky is a strong relational persistency.

Proof

Suppose S is a subset of some strong relational autarky S′. Consider any optimal assignment x. Since it is not possible to have f(x[S′ → b]) < f(x), it must be the case that x assigns the same value to all literals in S′. Since S ⊆ S′, all literals in S have the same value in all optimal assignments, which makes S a strong relational persistency.

Consider any symmetric optimal solution ζ of the single commodity maximum flow problem with source x_s and sink x̄_s on G_φ. Using the same definition of G_{φ, ζ} as in the proof of Theorem 2, let S denote the connected component containing x_s in G_{φ, ζ}.

Theorem 3

S is a strong relational persistency of the corresponding pseudo-Boolean function f. Conversely, if U is a strong relational persistency containing x_i, then there exists a symmetric optimal assignment ζ of the single commodity flow problem with source-sink pair (x_i, x̄_i), such that U is completely contained in the connected component containing x_i in G_φ,ζ.

Proof

We will start with the first part of the theorem. Notice that S contains at most one of x_i and x̄_i for all i, since otherwise symmetry implies a path from x_s to x̄_s in G_{φ, ζ}.

First consider the case when ζ is a maximum flow assignment such that the set S is maximal. If S has at most one element, then S is a strong relational autarky by definition. Otherwise, consider any assignment x, and assume that two literals in S have different value according to x. Let b = 0. We shall prove f(x[S → b]) < f(x), from which it follows that S is a strong relational autarky.

Let K = S ∪ ι(S). Write f as the sum of a constant, cubic terms corresponding to flow solution ζ, and quadratic terms corresponding to G_{φ, ζ}. We will show that in this expression for f, all terms involving literals in K will have value 0 according to x[S → b]. Consider any quadratic term involving literals in K. The term must correspond to edges in G_{φ, ζ} whose endpoints are both in S or both in ι(S), since there is no edge between S and V_{φ, ζ} \S. The term will vanish since the endpoints of its corrsponding edge are set to the same value in x[S → b].

For the cubic terms, since all flows are from x_s to x̄_s, using Lemma 3, we know the cubic terms all have the form:

$$\delta x_s({\overline{x}}_s{v}_1+{\overline{v}}_1{v}_2+\dots )+\delta {\overline{x}}_s(x_s{\overline{v}}_1+v_1{\overline{v}}_2+\dots )$$

We have x_s = b = 0 in x[S → b], so the cubic terms can be reduced to:

$$v_1{\overline{v}}_2+v_2{\overline{v}}_3+\dots$$

Consider any term in the above formula involving variables in K. Similiar to the proof of Theorem 2, there are only two cases such that the term will not vanish. The first case is v_ℓ = 1, and v_ℓ+1 ∈ S. Then the term v_ℓv̄_ℓ+1 has value 1 according to x[S → b]. The second case is the symmetric counterpart of the first case, namely v_ℓ ∈ ι(S), and v_ℓ+1 = 0. If we can prove the first case is impossible, by symmetry the second case is also impossible. Using the same argument in proof of Theorem 2, if the first case happens, we can expand S without changing the total flow value, thus contradicting the maximality of S. So all terms involving literals in K will vanish.

S is a strong relational autarky for the following reason. Terms not using literals in K will have the same value with respect to x and x[S → b], and all terms using literals in K will vanish under x[S → b]. Moreover, since there exist two literals in S that are assigned different values by x, and the two literals are connected in G_{φ, ζ}, at least one term using literals in K will not vanish under x, implying f(x[S → b]) < f(x).

Now consider an arbitrary maximum flow solution ζ without assuming S to be maximal. The only difference from the above case is that we can expand S by rerouting flows. It is clear to see that the new value of S after the expansion will be a superset of the former value of S. Thus starting from an arbitrary ζ and S, we can always get another maximum flow assignment ζ′ with a maximal S′, and S ⊆ S′, where S′ is a strong relational autarky. By Lemma 6, we know S is a strong relational persistency.

Now we prove the second part of the theorem. Let ζ be the maximum single commodity flow assignment with a maximal connected component containing x_i, and denote the connected component by S. Recall that we assume U is a strong relational persistency containing x_i.

The proof is by contradiction. Assuming U ⊈ S, we get a partition {U₁,U₂} of U where U₂ = U\S, and U₁ = U∩S. U₂ is nonempty since U ⊈ S, and U₁ is also nonempty since x_i ∈ U₁. By definition of strong relational persistency, literals in U₁ and U₂ must have the same value in all optimal assignments. Consider an optimal assignment x^*. We can assume all literals in U₁ and U₂ have value 1 in x^*, since the bi-form will have the same function value if we negate all variables’ values. By the discussion from the first part of the proof, we know x^* [S → 0] must also be an optimal assignment, where literals in U₁ have value 0, and literals in U₂ have value 1. The optimality of x^* [S → 0] contradicts the assumption that U is a strong relational persistency, thus U ⊆S.

Notice if U is a strong relational persistency, then ι(U) is also a strong relational persistency, and they are semantically equivalent, so a strong relational persistency containing x̄_i implies an equivalent strong relational persistency containing x_i. Theorem 3 then suggests that we can derive all strong relational persistencies by computing single commodity maximum flows in G_φ. The reverse direction of Theorem 3 also suggests that if S = {u, v} is derived from Theorem 2 as a strong relational persistency, then it can also be derived from Theorem 3 as a subset of a strong relational autarky.

Remark

The forward direction of Theorem 3 follows from remarks in [6], and the preprocessing code in [5] also applied the idea. The reverse direction is new, and guarantees that we can derive all strong relational persistencies by single commodity maximum flows.