Erratum to “Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks”

1. Related Works

This erratum is devoted to give some precisions on the related works in stand of view complexity and approximation given in [1].

1.1. Complexity Results

Table 1 gives the previous complexity results.

Table 1 

1.2. Approximation Results

If the network is a ring, there are two approximation results for Rayward-Smith’s [2] algorithm as follows.(i)In the general case, the performance ratio is upper bounded by 𝑚/2+3/2−1/𝑚, and there exists an instance for which the performance ratio is equal to 𝑚/8+1/2 [3].(ii) If the number of processors is even, the upper-bound can be improved to 1+(3/8)𝑚−1/2𝑚, and there exists an instance such that the performance ratio is equal to ⌈√𝑚⌉ [4].

The first two corollaries of the paper must be replaced by the two following corollaries.

Previous complexity results on the processors network model.

Corollary 1.1. The problem of deciding whether an instance of (𝑃,𝐺^∗)∣𝛽; 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘); 𝑝_𝑖=1, dup∣𝐶_max with 𝐺^∗∈𝒢 has a schedule of length at most three is 𝒩𝒫-complete with 𝛽∈{prec,bipartiteofdepthtwo}.

Proof. See [3].

Moreover, nonapproximability results can be deduced.

Corollary 1.2. No polynomial-time algorithm exists with a performance bound less than 4/3 unless 𝒫=𝒩𝒫 for the problems (𝑃,𝐺^∗)∣𝛽; 𝑐_𝑖𝑗=𝑑(𝜋^𝑙,𝜋^𝑘); 𝑝_𝑖=1∣𝐶_max and (𝑃,𝐺^∗)∣𝛽; 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘); 𝑝_𝑖=1, dup∣𝐶_max𝛽∈{prec,bipartiteofdepthtwo} with 𝐺^∗∈𝒢.

Proof. See [3].

The rest of the paper is devoted to extend the result to the bipartite of depth one and the main complexity result is given in the following theorem and two corollaries.

Theorem 1.3. The problem of deciding whether an instance of (𝑃,𝐺^∗)∣bipartiteofdepthone, 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘)=1, 𝑝_𝑖=1∣𝐶_max has a schedule of length at most three is 𝒩𝒫-complete.

Corollary 1.4. The problem of deciding whether an instance of (𝑃,𝐺^∗)∣bipartiteofdepthone; 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘); 𝑝_𝑖=1, dup∣𝐶_max with 𝐺^∗∈𝒢 has a schedule of length at most three is 𝒩𝒫-complete.

Corollary 1.5. No polynomial-time algorithm exists with a performance bound less than 4/3 unless 𝒫=𝒩𝒫 for the problems (𝑃,𝐺^∗)∣bipartiteofdepthone; 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘); 𝑝_𝑖=1∣𝐶_max and (𝑃,𝐺^∗)∣bipartiteofdepthone; 𝑐_𝑖𝑗=𝑑(𝜋^ℓ,𝜋^𝑘); 𝑝_𝑖=1, dup∣𝐶_max with 𝐺^∗∈𝒢.