Constant Unary Constraints and Symmetric Real-Weighted Counting Constraint Satisfaction Problems

Abstract

A unary constraint (on the Boolean domain) is a function from {0,1} to the set of real numbers. A free use of auxiliary unary constraints given besides input instances has proven to be useful in establishing a complete classification of the computational complexity of approximately solving weighted counting Boolean constraint satisfaction problems (or #CSPs). In particular, two special constant unary constraints are a key to an arity reduction of arbitrary constraints, sufficient for the desired classification. In an exact counting model, both constant unary constraints are always assumed to be available since they can be eliminated efficiently using an arbitrary nonempty set of constraints. In contrast, we demonstrate, in an approximate counting model, that at least one of them is efficiently approximated and thus eliminated approximately by a nonempty constraint set. This fact directly leads to an efficient construction of polynomial-time randomized approximation-preserving Turing reductions (or AP-reductions) from #CSPs with designated constraints to any given #CSPs composed of symmetric real-valued constraints of arbitrary arities even in the presence of arbitrary extra unary constraints.

Appendix: Proof of Lemma 2.3

In what follows, we will give the missing proof of Lemma 2.3. For any constraint f of arity k, the notation max|f| indicates the maximum value |f(x)| over all inputs x∈{0,1}^k.

(1)–(2) These properties (reflexivity and transitivity) directly come from the definition of effective T-constructibility.

(3) Let $(\mathcal{H}_{1},\mathcal{H}_{2},\ldots,\mathcal{H}_{n})$ be a generating series of $\mathcal{F}_{1}$ from $\mathcal{F}_{2}$. We need to show that $\#\mathrm{CSP}(\mathcal{H}_{i},\mathcal{G}) \leq_{\mathrm {AP}}\#\mathrm{CSP}(\mathcal{H}_{i+1},\mathcal{G})$ for each adjacent pair $(\mathcal{H}_{i},\mathcal{H}_{i+1})$, where i∈[n−1]. By Lemma 2.1, ≤_AP is transitive; thus, it follows that $\#\mathrm{CSP}(\mathcal{H}_{1},\mathcal{G})\leq _{\mathrm{AP}}\#\mathrm{CSP}(\mathcal{H}_{n},\mathcal{G})$. This is clearly equivalent to $\#\mathrm{CSP}(\mathcal{F}_{1},\mathcal{G})\leq _{\mathrm{AP}}\#\mathrm{CSP}(\mathcal{F}_{2},\mathcal{G})$, as requested.

Taking an arbitrary pair $(\mathcal{H}_{i},\mathcal{H}_{i+1})$ with i∈[n−1], we treat the first case where $(\mathcal{H}_{i},\mathcal{H}_{i+1})$ satisfies Clause (I) of Definition 2.2. Consider a constraint frame $\varOmega=(G,X|\mathcal{H}',\pi)$ with $\mathcal{H}'\subseteq\mathcal{H}_{i}\cup\mathcal{G}$. For convenience, let $\mathcal{H}'=\{f_{1},f_{2},\ldots,f_{d}\}$. Take f _i inductively and consider all subgraphs of G that represents f _i . Choose such subgraphs one by one. Now, let $G_{f_{i}}$ be such a subgraph. By Clause (I), there exists another finite graph G′ that realizes f _i by $\mathcal{H}_{i+1}$. We replace $G_{f_{i}}$ in G by $G'_{f_{i}}$. After all the subgraphs representing f _i are replaced, the obtained graph, say, G′ constitutes a new constraint frame Ω′. It is not difficult to show that csp _Ω′ equals csp _Ω . We continue this replacement process for all f _i ’s. In the end, we conclude that $\#\mathrm{CSP}(\mathcal{H}_{i},\mathcal {G})\leq_{\mathrm{AP}}\#\mathrm{CSP}(\mathcal{H}_{i+1},\mathcal{G})$.

We will examine the second case where $(\mathcal{H}_{i},\mathcal {H}_{i+1})$ satisfies Clause (II) of Definition 2.2. In what follows, for ease of our argument, we assume that $\mathcal{H}_{i}=\{f\}$ and we want to claim that $\#\mathrm{CSP}(f,\mathcal{G})\leq_{\mathrm{AP}}\# \mathrm{CSP}(\mathcal{H}_{i+1},\mathcal{G})$. Take a p-convergence series Λ for f, which is effectively T-constructible from $\mathcal{H}_{i+1}$. Our claim is split into two parts: (a) $\#\mathrm{CSP}(f,\mathcal {G})\leq_{\mathrm{AP}}\#\mathrm{CSP}(\varLambda,\mathcal{G})$ and (b) $\#\mathrm{CSP}(\varLambda,\mathcal{G})\leq_{\mathrm{AP}}\#\mathrm {CSP}(\mathcal{H}_{i+1},\mathcal{G})$. We will prove these parts separately. Since (b) is easy, we start with (b).

(b) We intend to show that $\#\mathrm{CSP}(\varLambda,\mathcal{G})\leq _{\mathrm{AP}}\#\mathrm{CSP}(\mathcal{H}_{i+1},\mathcal{G})$. Let Λ=(f ₁,f ₂,…) and $\mathcal{H}_{i+1}=\{g_{1},g_{2},\ldots ,g_{d}\}$. Now, we take any constraint frame $\varOmega= (G,X|\mathcal {G}',\pi)$ with $\mathcal{G}'\subseteq\varLambda\cup\mathcal{G}$, given to $\#\mathrm{CSP}(\varLambda,\mathcal{G})$. Since the constraint set $\mathcal{G}'$ is finite, for simplicity, we assume that $\mathcal {G}'$ is composed of constraints $h_{1},h_{2},\ldots ,h_{s},f_{i_{1}},f_{i_{2}},\ldots,f_{i_{t}}$, where $s\in\mathbb{N}$, $t\in \mathbb{N}^{+}$, and each constraint h _i belongs to $\mathcal {F}-\varLambda$. For this constraint frame Ω, we will explain how to compute the value csp _Ω . Since Λ is effectively T-constructible from $\mathcal{G}$, there exists a polynomial-time DTM M that, for each index j∈[t], generates an appropriate graph $\tilde{G}_{i_{j}}$ realizing $f_{i_{j}}$ from any graph $G_{i_{j}}$ representing $f_{i_{j}}$.

Each node v labelled $f_{i_{j}}$ (j∈[t]) in G corresponds to a unique subgraph $G_{i_{j}}$, including all dangling edges adjacent to v, that represents $f_{i_{j}}$. By running M on $G_{i_{j}}$, we obtain another subgraph $\tilde {G}_{i_{j}}$ realizing $f_{i_{j}}$, which contains all the dangling edges of $G_{i_{j}}$. It is therefore possible to generate from G another bipartite graph $\tilde{G}$ in which every subgraph $G_{i_{j}}$ of G representing $f_{i_{j}}$ is replaced by its associated subgraph $\tilde {G}_{i_{j}}$ obtained from $G_{i_{j}}$ by M. We denote by Ω′ the constraint frame obtained from Ω by replacing G with $\tilde {G}$ and by modifying π accordingly. The definition of “realizability” implies that csp _Ω′ equals γ⋅csp _Ω for an appropriate number $\gamma\in\mathbb{A}$. Since $\tilde{G}$ contains only constraints in $\mathcal{H}_{i+1}\cup \mathcal{G}$, Ω′ must be a valid input instance to $\#\mathrm {CSP}(\mathcal{H}_{i+1},\mathcal{G})$. As a result, we conclude that $\#\mathrm{CSP}(\varLambda,\mathcal{G})$ is AP-reducible to $\#\mathrm{CSP}(\mathcal{H}_{i+1},\mathcal{G})$.

(a) We want to claim that $\#\mathrm{CSP}(f,\mathcal{G})\leq _{\mathrm{AP}}\#\mathrm{CSP}(\varLambda,\mathcal{G})$. This claim is proven by modifying the proof of [12, Lemma 9.2]. Hereafter, assume that f is of arity k and let Λ=(g ₁,g ₂,…). For convenience, we define AC={x∈{0,1}^k∣f(x)≠0}. By Eq. (3), there exists a constant λ∈(0,1) such that, for every $m\in\mathbb {N}^{+}$ and every x∈{0,1}^k, certain constants c,d∈{±1} satisfies the following condition:

• (*) (1+λ ^m c)g _m (x)≤f(x)≤(1+λ ^m d)g _m (x) for all x∈AC, and |g _m (x)|≤λ ^m for all x∈{0,1}^k−AC.

Without loss of generality, we can assume that γ is an algebraic real number.

Let us take any constraint frame $\varOmega=(G,X|\mathcal{G}',\pi)$ with G=(V ₁|V ₂,E) and $\mathcal{G}'\subseteq\{f\}\cup\mathcal {G}$ given as an input instance to $\#\mathrm{CSP}(f,\mathcal{G})$. It is enough to consider the case where f appears in $\mathcal{G}'$. Let p _f denote the total number of nodes in V ₂ whose labels are f. For simplicity, write L for the set of all 2^k-tuples $\ell=(\ell _{x_{1}},\ell_{x_{2}},\ldots,\ell_{x_{2^{k}}})\in\mathbb{N}^{2^{k}}$ satisfying that $\sum_{i\in[2^{k}]}\ell_{x_{i}}= p_{f}$, where each x _i denotes the lexicographically ith string in {0,1}^k. In addition, we set $L_{f} =\{\ell\in L\mid\forall i\in[2^{k}] [ f(x_{i})=0\rightarrow\ell_{x_{i}}=0\ ]\}$. It is not difficult to show by Eq. (1) that csp _Ω can be expressed in the form $\sum_{\ell\in L_{f}} \alpha_{\ell}(\prod_{x\in AC}f(x)^{\ell_{x}})$ for appropriately chosen numbers $\alpha_{\ell}\in\mathbb{A}$, provided that 0⁰ is treated as 1 for technical reason.

We set a ₀=2^k! 2^4k and $b_{0}= [1+ (2\max|f|)^{|V_{2}|}]\cdot\sum _{\ell\in L-L_{f}}|\alpha_{\ell}|$, which are obviously independent of m. Meanwhile, we arbitrarily fix an integer $m\in\mathbb{N}^{+}$ that satisfies both λ ^m a ₀<1 and λ ^m b ₀<1, and we denote by Ω _m the constraint frame obtained from Ω by replacing every node labeled f with a new node having the label g _m . Concerning this Ω _m , its value $\mathit{csp}_{\varOmega_{m}}$ coincides with the sum Γ _1,m +Γ _2,m , where

$$\varGamma_{1,m} = \sum_{\ell\in L_f} \alpha_{\ell} \prod_{x\in AC}g_m(x)^{\ell_x}\quad\text{and}\quad \varGamma_{2,m} = \sum_{\ell\in L-L_f} \alpha_{\ell} \prod_{x\in\{ 0,1\}^k\wedge\ell_x>0}g_m(x)^{\ell_x}. $$

Next, we will establish a close relationship between csp _Ω and Γ _1,m ; more specifically, we intend to prove the following key claim.

Claim 9

It holds that (1+λ ^m B)Γ _1,m ≤csp _Ω ≤(1+λ ^m B′)Γ _1,m for appropriate numbers $B,B'\in \mathbb{A}$ satisfying |B|,|B′|≤a ₀. Therefore, sgn(csp _Ω )=sgn(Γ _1,m ) holds.

Proof

It is obvious that the second part of the claim follows from the first part, because λ ^m|B|≤λ ^m a ₀<1 and similarly λ ^m|B′|<1 by our choice of m. Henceforth, we aim at proving the first part. Fix ℓ∈L _f arbitrarily. From Condition (*), for appropriate selections of c _ℓ,x ’s and d _ℓ,x ’s in {±1}, we obtain

$$ \prod_{x\in AC}\bigl(1+\lambda^mc_{\ell,x} \bigr)^{\ell_{x}} g_m(x)^{\ell _{x}} \leq\prod _{x\in AC}f(x)^{\ell_{x}}\leq\prod_{x\in AC} \bigl(1+\lambda^md_{\ell,x}\bigr)^{\ell_{x}} g_m(x)^{\ell_{x}}. $$

(5)

Note that, when all elements in $\mathcal{F}'$ are limited to nonnegative constraints, we can always set c _ℓ,x =−1 and d _ℓ,x =1. Equation (5) leads to upper and lower bounds of csp _Ω :

$$ \sum_{\ell\in L_f} \alpha_{\ell}\prod _{x\in AC}\bigl(1+\lambda ^mc_{\ell,x} \bigr)^{\ell_{x}} g_m(x)^{\ell_{x}} \leq \mathit{csp}_{\varOmega} \leq \sum_{\ell\in L_f} \alpha_{\ell}\prod _{x\in AC}\bigl(1+\lambda ^md_{\ell,x} \bigr)^{\ell_{x}} g_m(x)^{\ell_{x}}. $$

(6)

Let us further estimate the first and the last terms in Eq. (6). Let us handle the first term. By considering the binomial expansion of (1+z)ⁿ, it holds that, for any numbers $n\in\mathbb{N}^{+}$ and $z\in\mathbb{R}$ satisfying that −1/n≤z≤2/n, there exists a number e∈{1/2,n} such that 1+nz≤(1+z)ⁿ≤1+enz (more precisely, if z≥0 then e=n; otherwise, e=1/2). Hence, by choosing appropriate numbers e _ℓ,x ∈{±1/2,±ℓ _x }, we obtain

$$\begin{aligned} \prod_{x\in AC}\bigl(1+\lambda^m c_{\ell,x}\bigr)^{\ell_x}g_m(x)^{\ell_x} \geq& \prod _{x\in AC}\bigl(1+\lambda^m \ell_{x}e_{\ell,x}\bigr)g_m(x)^{\ell_x} \\ =& \prod_{x\in AC}\bigl(1+\lambda^m \ell_{x}e_{\ell,x}\bigr)\prod_{x\in AC}g_m(x)^{\ell_x} \end{aligned}$$

since m satisfies that −1<λ ^m ℓ _x e _ℓ,x <1.

For a further estimation, let us focus on the value ∏ _x∈AC (1+λ ^m z _x ) for any series $\{z_{x}\}_{x\in AC}\subseteq [-2^{2k},2^{2k}]_{\mathbb{Z}}$. Since ∏ _x∈AC (1+λ ^m z _x ) has the form $1+\sum_{i=1}^{|AC|}\sum_{y_{1},y_{2},\ldots,y_{i}\in AC}\lambda^{im}z_{y_{1}}z_{y_{2}}\cdots z_{y_{i}}$, where all indices y ₁,y ₂,…,y _i are distinct, if we set $\tilde{B} = \sum _{i=1}^{|AC|}\sum_{y_{1},y_{2},\ldots,y_{i}\in AC}\lambda ^{(i-1)m}|z_{y_{1}}z_{y_{2}}\cdots z_{y_{i}}|$, then we derive that $1-\lambda^{m} \tilde{B}\leq\prod_{x\in AC}(1+\lambda^{m} z_{x}) \leq1+\lambda^{m} \tilde{B}$. Note that |λ ^m z _x |≤λ ^m2^2k≤λ ^m a ₀<1 for any x∈AC since $z_{x}\in[-2^{2k},2^{2k}]_{\mathbb{Z}}$. It therefore follows that

$$\begin{aligned} \sum_{y_1,\ldots,y_i\in AC}\lambda^{(i-1)m}|z_{y_1}\cdots z_{y_i}| \leq&\sum_{y_1\in AC}|z_{y_1}|\sum _{y_2,\ldots,y_i\in AC}1\leq|AC|2^{k}\left( \begin{array}{@{}c@{}} |AC| \\ i \end{array} \right)\\ \leq&|AC|2^{2k}|AC|!. \end{aligned}$$

We then conclude that $\tilde{B}$ satisfies that $|\tilde{B}|\leq\sum_{i=1}^{|AC|} |AC|2^{2k}|AC|! \leq |AC|^{2}2^{2k}|AC|! \leq a_{0}$ since |AC|≤2^k. From this fact, there exists a series $\{B_{\ell}\}_{\ell\in L_{f}}\subseteq\mathbb {A}$ with |B _ℓ |≤a ₀ such that

$$\prod_{x\in AC}\bigl(1+\lambda^{m} \ell_x e_{\ell,x}\bigr) \prod_{x\in AC}g_m(x)^{\ell_x} \geq \bigl(1+\lambda^m B_{\ell}\bigr)\prod _{x\in AC}g_m(x)^{\ell_x}. $$

Finally, we choose an appropriate number $B\in\mathbb{A}$ with |B|≤a ₀ that satisfies

$$\begin{aligned} \sum_{\ell\in L_f}\alpha_{\ell}\bigl(1+ \lambda^mB_{\ell}\bigr)\prod_{x\in AC}g_m(x)^{\ell_x} \geq&\bigl(1 + \lambda^m B\bigr)\sum_{\ell\in L_f} \alpha _{\ell}\prod_{x\in AC}g_m(x)^{\ell_x}\\ =& \bigl(1 + \lambda^m B\bigr)\varGamma_{1,m}. \end{aligned}$$

Concerning the third term in Eq. (6), a similar argument used for the first term shows the existence of an algebraic real number $B'\in\mathbb{A}$ such that |B′|≤a ₀ and

$$ \sum_{\ell\in L_f}\alpha_{\ell}\prod _{x\in AC}\bigl(1+\lambda^m d_{\ell,x} \bigr)^{\ell_x}g_m(x)^{\ell_x} \leq \bigl(1+ \lambda^m B'\bigr)\varGamma_{1,m}. $$

By the selection of B and B′, they certainly satisfy the claim. □

Next, we will give an upper-bound of |Γ _2,m |. Recall that $b_{0}= C \sum_{\ell\in L-L_{f}}|\alpha_{\ell}|$, where $C= 1+ (2\max |f|)^{|V_{2}|}$.

Claim 10

It holds that |Γ _2,m |≤λ ^m b ₀.

Proof

For the time being, we fix a series ℓ∈L−L _f and conduct a basic analysis. For this series ℓ, there exists an element x∈{0,1}^k such that x∉AC and ℓ _x >0. For convenience, we define D={x∈{0,1}^k∣ℓ _x >0} and further partition it into two sets: D ₁={x∈D∣x∈AC} and D ₂={x∈D∣x∉AC}. Notice that D ₂ is nonempty. Since λ<1 and |g _m (x)|≤λ ^m for all x∈D ₂, it follows that

$$ \biggl| \prod_{x\in D_2}g_m(x)^{\ell_x} \biggr| = \prod_{x\in D_2}\bigl|g_m(x)\bigr|^{\ell_x} \leq\prod_{x\in D_2} \lambda^{m\ell_x} \leq\lambda^m. $$

Condition (*) implies that |g _m (x)|≤|f(x)|/min{1+λ ^m c,1+λ ^m d}≤2|f(x)| for any x∈AC because |λ ^m c|,|λ ^m d|<1/2. If max|g _m |≥1, then we obtain

$$\begin{aligned} \biggl| \prod_{x\in D_1}g_m(x)^{\ell_x} \biggr| \leq&\prod_{x\in D_1} \bigl|g_m(x)\bigr|^{\ell_x} \leq\bigl(\max|g_m|\bigr)^{\sum_{x\in D_1} \ell_x} \leq \bigl(\max|g_m|\bigr)^{p_f} \\ \leq&\bigl(2\max|f|\bigr)^{|V_2|} \end{aligned}$$

since $\sum_{x\in D_{1}}\ell_{x} \leq p_{f}\leq|V_{2}|$. When max|g _m |<1, we instead obtain $\prod_{x\in D_{1}}|g_{m}(x)|^{\ell _{x}}\leq1$. Therefore, it holds that

$$ \biggl| \prod_{x\in D}g_m(x)^{\ell_x} \biggr| = \biggl| \prod_{x\in D_2}g_m(x)^{\ell_x} \biggr|\cdot \biggl| \prod_{x\in D_1}g_m(x)^{\ell_x} \biggr| \leq\lambda^m C. $$

The value |Γ _2,m | is upper-bounded by

$$ |\varGamma_{2,m}| \leq\sum_{\ell\in L-L_f} |\alpha_{\ell}| \biggl|\prod_{x\in D}g_m(x)^{\ell_x} \biggr| \leq\lambda^m C \sum_{\ell\in L-L_f} |\alpha_{\ell}| \leq\lambda^m b_0. $$

This completes the proof of the claim. □

To finish the proof of Lemma 2.3, we will present a randomized oracle computation that solves $\#\mathrm{CSP}(f,\mathcal {G})$ with a single query to the oracle $\#\mathrm{CSP}(\varLambda,\mathcal{G})$. First, we want to define a special constant d ₀ corresponding to Ω. The definition of d ₀ requires the following well-known lower bound of the absolute values of polynomials in algebraic real numbers.

Lemma 4.4

[11]

Let $\alpha_{1},\ldots,\alpha_{m}\in\mathbb{A}$ and let c be the degree of $\mathbb{Q}(\alpha_{1},\ldots,\alpha_{m})/\mathbb{Q}$. There exists a constant e>0 that satisfies the following statement for any complex number α of the form $\sum_{k}a_{k}t(\prod _{i=1}^{m}\alpha_{i}^{k_{i}})$, where k=(k ₁,…,k _m ) ranges over [N ₁]×⋯×[N _m ], $(N_{1},\ldots,N_{m})\in\mathbb {N}^{m}$, and $a_{k}\in\mathbb{Z}$. If α≠0, then $|\alpha |\geq(\sum_{k}|a_{k}|)^{1-c}\prod_{i=1}^{m}e^{-cN_{i}}$.

Following the proof of [12, Lemma 9.2] under the assumption that csp _Ω ≠0, it is possible to set values of four series {N _i } _i , {a _k } _k , {α _i } _i , and {k _i } _i appropriately so that Lemma 4.4 provides two constants c,e>0 for which $|\mathit{csp}_{\varOmega}|\geq(\sum _{k}|a_{k}|)^{1-c}\prod_{i}e^{-cN_{i}}$. The desired constant d ₀ is now defined to be $(\sum_{k}|a_{k}|)^{1-c}\prod_{i}e^{-cN_{i}}$. Notice that d ₀ is an algebraic real number.

Let us describe our randomized approximation algorithm.

[Algorithm $\mathcal{M}$] On input instance (Ω,1/ε), set δ=ε/2 and find in polynomial time an integer m≥1 satisfying that λ ^m a ₀<min{1,δ} and λ ^m b ₀<min{d ₀,δ}. Produce another constraint frame Ω _m . make a query with a query word (Ω _m ,1/δ) to the oracle and let w be an answer from the oracle. Notice that w is a random variable since the oracle is a RAS. Compute d ₀ defined above. If |w|<d ₀, then output 0; otherwise, output w.

We want to prove that the above randomized algorithm $\mathcal{M}$ approximately solves $\#\mathrm{CSP}(f,\mathcal{G})$ with high probability. Let us consider two cases separately.

(1) For the first case where csp _Ω =0, we need to prove that M outputs 0 with high probability. Let us evaluate the values Γ _1,m and Γ _2,m . Obviously, Claim 9 implies Γ _1,m =0. By Claim 10 and the choice of m, we derive |Γ _2,m |≤λ ^m b ₀<d ₀. From $\mathit{csp}_{\varOmega_{m}} = \varGamma_{1,m}+\varGamma_{2,m}$, it follows that $|\mathit{csp}_{\varOmega_{m}}| <d_{0}$. This means that $\mathcal{M}$ outputs 0 with high probability.

(2) Next, we consider the case where csp _Ω ≠0. We consider only the case where csp _Ω >0 because the other case csp _Ω <0 can be similarly handled. The choice of d ₀ implies that csp _Ω ≥d ₀. We then choose a number α, not depending on m, for which α(Γ _1,m +sgn(α)b ₀)+b ₀≤BΓ _1,m and |α|≤max{a ₀,b ₀}. For this α, it holds by Claim 10 that

$$\begin{aligned} \bigl(1+\lambda^m\alpha\bigr)\mathit{csp}_{\varOmega_m} =& \bigl(1+ \lambda^m\alpha\bigr) (\varGamma _{1,m}+\varGamma_{2,m}) \\ \leq& \varGamma_{1,m}+\lambda^m\alpha\varGamma_{1,m} + \bigl(1+|\alpha |\bigr)\lambda^mb_0 \\ =& \varGamma_{1,m}+\lambda^m \bigl[\alpha\bigl( \varGamma_{1,m} + sgn(\alpha )b_0\bigr)+b_0\bigr] \\ \leq& \varGamma_{1,m}+\lambda^m B\varGamma_{1,m} = \bigl(1+\lambda ^mB\bigr)\varGamma_{1,m}. \end{aligned}$$

Similarly, we choose α′ with |α′|≤max{a ₀,b ₀} such that $(1+\lambda^{m}B)\varGamma_{1,m}\leq(1+\lambda^{m}\alpha ')(\varGamma_{1,m}+\varGamma_{2,m}) = (1+\lambda^{m}\alpha')\mathit{csp}_{\varOmega_{m}}$.

For simplicity, let γ=max{|α|,|α′|}. Note that δ≥λ ^m γ. Since λ ^m γ<1, it holds that log₂(1+λ ^m γ)≤λ ^m γ≤δ. Thus, we conclude that $1+\lambda^{m}\alpha'\leq2^{\log_{2}(1+\lambda ^{m}\gamma)}\leq2^{\delta}$. Moreover, since log₂(1−λ ^m γ)≥−λ ^m γ, it follows that $1+\lambda^{m}\alpha \geq2^{\log_{2}(1-\lambda^{m}\gamma)}\geq2^{-\delta}$. In conclusion, it holds that $2^{-\delta} \mathit{csp}_{\varOmega} \leq \mathit{csp}_{\varOmega_{m}} \leq2^{\delta} \mathit{csp}_{\varOmega}$. From this follows $\mathit{csp}_{\varOmega_{m}}>0$.

If w is any oracle answer, then it must satisfy that $2^{-\delta }\mathit{csp}_{\varOmega_{m}}\leq w \leq2^{\delta}\mathit{csp}_{\varOmega_{m}}$ because $\mathit{csp}_{\varOmega_{m}}>0$. Therefore, we derive that $w\leq2^{\delta }\mathit{csp}_{\varOmega_{m}}\leq2^{2\delta}\mathit{csp}_{\varOmega}$ and $w\geq2^{-\delta }\mathit{csp}_{\varOmega_{m}}\geq2^{-2\delta}csp_{\varOmega}$. Since ε=2δ, $\mathcal{M}$ outputs a 2^ε-approximate solution using any 2^δ-approximate solution for (Ω _m ,1/δ) as an oracle answer.

This completes the proof of Lemma 2.3.  □