In this section, we show how we proved the theorems stated in the Results section.

### Proof of Theorem 1

We first present an upper bound on the size of an optimal solution of ChrP to show that ChrP is in NP. Then, we prove that ChrP is NP-hard.

**Lemma 1** *Let G* = (*V*, *E*) *be a chromosome graph. Also*, *let C be a multi-set of chromosomes on G that minimizes W*(*C*) *such that* |*c*_{
i
}| ≤ *λ*_{
i
} *for c*_{
i
} ∈ *C. Then*, *C has at most U*(4|*V*| + 1)(|*E*| + 1) *edges*.

*Proof* Let *c* ∈ *C* be a chromosome in *C*. We define an edge *e* in *c* as *non-excessive* if *e* ∈ *E*_{
S
} and *m*(*C*, *e*) ≤ *n*(*e*), and *excessive* otherwise. Let *t*_{
c
} be the number of non-excessive edges visited by *c*. If *t*_{
c
} *>*0, *c* can be written as *c* = *p*_{1}*e*_{1}*p*_{2}*e*_{2} ... *e*_{
t
}c*p*_{
t
}c_{+1}, where *e*_{
k
} (1 ≤ *k* ≤ *t*_{
c
}) is a non-excessive edge and *p*_{
k
} (1 ≤ *k* ≤ *t*_{
c
}+1) is a possibly empty path that contains only excessive edges (Figure 6). If *p*_{
k
} contains a cycle as its subpath, the cycle can be removed to decrease *W*(*C*), a contradiction. Accordingly, *p*_{
k
} does not contain a cycle. This implies that *p*_{
k
} visits at most 2|*V*| vertices and, thus, 2|*V*| edges. Therefore, at most, 4|*V* | excessive edges are visited for each non-excessive edge. Note that a non-excessive edge *e* can be visited, at most, *n*(*e*)-times. Therefore, {\sum}_{c\in C}{t}_{c}\le {\sum}_{e\in Es}n\left(e\right).

Chromosomes such that *t*_{
c
} = 0 can exist only if they contribute to the decrease of the first or the second term of *W*(*C*) defined by (1). Accordingly, the number of such chromosomes is, at most, *n*_{
N
} +*n*_{
T
}. In addition, a chromosome *c*, such that *t*_{
c
} = 0, does not contain any cycles because such a cycle can be removed to decrease *W*(*C*). Therefore, at most, *c* visits 2|*V*| vertices and, thus, 2|*V*| edges.

Consequently, *C* contains, at most, 2|*V*|(*n*_{
N
} +*n*_{
T
} ) + (4|*V*| +1) P_{
e
}***_{
E
}S *n*(*e*) ≤ *U*(4|*V* |+1)(|*E*|+1) edges.

**Lemma 2** *The problem ChrP is in NP*.

*Proof* Once an optimal solution *C* is given, whether or not *W*(*C*) is greater than a given constant can be determined in *O*(|*V* ||*E*|) time by Lemma 1. □

**Lemma 3** *The problem ChrP is NP-hard*.

*Proof* The *Hamiltonian Cycle problem (HC)* is a problem of finding a cycle that visits each vertex of a graph exactly once, and is a well-known NP-complete problem [34]. Here, we reduce HC to ChrP. Consider HC on a directed graph *H* = (*V'*, *E'*), where {V}^{\prime}=\left\{{v}_{1}^{\prime},{v}_{2}^{\prime},\dots ,{v}_{\left|{V}^{\prime}\right|}^{\prime}\right\} is a set of vertices and *E'* is a set of edges. We construct a chromosome graph *G* = (*V*, *E*) from *H* (Figure 7), where

V={\displaystyle \bigcup _{1\le i\le \left|{V}^{\prime}\right|}}\left\{{v}_{i,0}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,1}^{+},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,1}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,2}^{+},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,2}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,3}^{+}\right\}

is a set of vertices, and *E* = *E*_{
S
} ∪ *E*_{
L
} ∪ *E*_{
R
} is a set of edges. Here, *E*_{
S
} consists of

\begin{array}{c}{e}_{1,0}=\u27e8-{v}_{1,0}^{-},+{v}_{1,1}^{+},1,1\u27e9,\\ {e}_{1,1}=\u27e8-{v}_{1,1}^{-},+{v}_{1,2}^{+},2,1\u27e9,\\ {e}_{1,2}=\u27e8-{v}_{1,2},+{v}_{1,3}^{+},1,1\u27e9,\\ {e}_{i,0}=\u27e8-{v}_{i,0},+{v}_{i1}^{+},0,1\u27e9\phantom{\rule{1em}{0ex}}\left(2\le i\le \left|{V}^{\prime}\right|\right),\\ {e}_{i,1}=\u27e8-{v}_{i,1}^{-},+{v}_{i2}^{\u2021},1,1\u27e9\phantom{\rule{1em}{0ex}}\left(2\le i\le \left|{V}^{\prime}\right|\right),\\ {e}_{i,2}=\u27e8-{v}_{i,2}^{-},+{v}_{i,3}^{\u2021},0,1\u27e9\phantom{\rule{1em}{0ex}}\left(2\le i\le \left|{V}^{\prime}\right|\right).\end{array}

*E*_{
R
} consists of

\begin{array}{c}{\widehat{e}}_{i,1}=\u27e8-{v}_{i,1}^{+},+{v}_{i,1}^{-},0,0\u27e9\left(1\phantom{\rule{2.77695pt}{0ex}}\le i\le \left|{V}^{\prime}\right|\right),\\ {\widehat{e}}_{i,2}=\u27e8-{v}_{i,2}^{+},+{v}_{i,2}^{-},0,0\u27e9\left(1\phantom{\rule{2.77695pt}{0ex}}\le i\le \left|{V}^{\prime}\right|\right).\end{array}

*E*_{
L
} consists of

{e}_{{i}^{\prime}:i}=\u27e8-{v}_{i,2}^{+},+{v}_{i,1}^{-},0,0\u27e9\phantom{\rule{1em}{0ex}}\left(\left({v}_{i}^{\prime},\phantom{\rule{2.77695pt}{0ex}}{v}_{i}^{\prime}\right)\in {E}^{\prime}\right).

In addition, we set *n*_{
N
} = 1, *n*_{
T
} = 0, and *λ*_{
i
} = |*V'*| + 3 for any *i*. Then, we prove that *H* has a Hamiltonian cycle if, and only if, ChrP on *G* has a solution *C* such that *W*(*C*) = 0. Suppose that *h* is a Hamiltonian cycle on *H*. Let *c* be a chromosome that begins with {e}_{1,0}{\widehat{e}}_{1,1}{e}_{1,1} and then visits {e}_{{i}^{\prime}:i}{e}_{i,1} in the order that edges \left({v}_{{i}^{\prime}},{v}_{i}\right) appear in *h* from *i'* = 1, and finally ends with {e}_{1,1}{\widehat{e}}_{1,2}{e}_{1,2}. Then, a set of a single chromosome *C* = {*c*} satisfies *W*(*C*) = 0 and \left|c\right|=\left|{V}^{\prime}\right|+3\le {\lambda}_{1}.

Conversely, let *C* be a solution of ChrP that satisfies *W*(*C*) = 0. Because (2) holds, |*C*| = 1, Tr(*C*) = 0, and *m*(*C*, *e*) = *n*(*e*). Let *c* be the only chromosome in *C*. Because *n*(*e*_{1,1}) = 2 and *n*(*e*_{i,1}) = 1 for 2 ≤ *i* ≤ |*V'*|, a path that visits vertices {v}_{i}^{\prime}\in {V}^{\prime} in the order that *e*_{i,1 }appears in *c* is a Hamiltonian cycle on *H*. □

Theorem 1 directly follows Lemma 2 and 3.

### Proof of Theorem 2

#### Circulation on a bidirected graph

Let *G* = (*V*, *E*) be a bidirected graph, and *a*_{
v,e
} for *v* ∈ *V* and *e* ∈ *E* be an integer such that

{a}_{v,e}=\left\{\begin{array}{c}2\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}e{\mathsf{\text{hastwo}}}^{\prime}{+}^{\prime}\mathsf{\text{-endsat}}v,\hfill \\ 1\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}e{\mathsf{\text{hasonlyone}}}^{\prime}{+}^{\prime}\mathsf{\text{-endat}}v,\hfill \\ -1\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}e{\mathsf{\text{hasonlyone}}}^{\prime}{-}^{\prime}\mathsf{\text{-endat}}v,\hfill \\ -2\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}e{\mathsf{\text{hastwo}}}^{\prime}{-}^{\prime}\mathsf{\text{-endsat}}v,\hfill \\ 0\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}e\mathsf{\text{isnotconnectedto}}v.\hfill \end{array}\right.

Also let *b*_{
v
} be an integer defined for each *v* ∈ *V*, *Z* be the set of non-negative integers, and *l*(*e*) and *u*(*e*) be two non-negative integers assigned to each edge *e* ∈ *E* called a *lower bound* and an *upper bound*, respectively. Unless otherwise specified, in this study *l*(*e*) = 0 and *u*(*e*) = ∞.

**Definition 5** *A* bidirected flow (biflow) *[19*, *20] is a mapping f* : *E* → *Z such that*

l\left(e\right)\le f\left(e\right)\le u\left(e\right)\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}each\phantom{\rule{2.77695pt}{0ex}}e\in E,

(4)

{\displaystyle \sum _{e\in E}}{a}_{v,e}f\left(e\right)={b}_{v}\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}each\phantom{\rule{2.77695pt}{0ex}}v\in V.

(5)

*The cost of f is defined as W*(*f*) = ∑_{e∈E}*w*(*f*, *e*), *where w*(*f*, *e*) *is a cost of f on e* ∈ *E. A* circulation *is a biflow such that b*_{
v
} = 0 *for any v* ∈ *V*.

#### Circular chromosome graph

**Definition 6** (Circular chromosome graph) *Let G* = (*V,E*) *be a chromosome graph*, *and let v*_{
N
} *and v*_{
T
} *be new vertices. In addition*, *let E*_{
N
} *be a set of the following edges: for* 1 ≤ *i* ≤ *N*_{
C
},

\begin{array}{c}{e}_{t}\left({v}_{i,0}^{-}\right)=\u27e8-{v}_{N},+{v}_{i,0}^{-},0,0\u27e9,\\ {e}_{t}\left({v}_{i,{n}_{i}}^{+}\right)=\u27e8-{v}_{N},-{v}_{i,{n}_{i}}^{+},\phantom{\rule{2.77695pt}{0ex}}0,0\u27e9,\\ {e}_{t}\left({v}_{i,j}^{+}\right)=\u27e8-{v}_{T},-{v}_{i,j}^{+},\phantom{\rule{2.77695pt}{0ex}}0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le j\le {n}_{i}\right),\\ {e}_{t}\left({v}_{i,j}^{-}\right)=\u27e8-{v}_{T},+{v}_{i,j}^{-},\phantom{\rule{2.77695pt}{0ex}}0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le j\le {n}_{i}\right),\end{array}

*and*

\begin{array}{c}{e}_{T}=\u27e8-{v}_{N},+{v}_{T},\phantom{\rule{2.77695pt}{0ex}}{n}_{T},\phantom{\rule{2.77695pt}{0ex}}{Q}_{T}\u27e9,\\ {e}_{N}=\u27e8+{v}_{N},+{v}_{N},\phantom{\rule{2.77695pt}{0ex}}{n}_{N},\phantom{\rule{2.77695pt}{0ex}}{Q}_{N}\u27e9.\end{array}

*Also*, *let E*_{
D
} *be a set of the following edges for e* ∈ *E*_{
S
} ∪ {*e*_{
N
}, *e*_{
T
}}*:*

\overline{e}=\u27e8-d\left(e,\phantom{\rule{2.77695pt}{0ex}}{v}_{{i}_{1},{j}_{1}}\right){v}_{{i}_{1},{j}_{1}},-d\left(e,\phantom{\rule{2.77695pt}{0ex}}{v}_{{i}_{2},{j}_{2}}\right){v}_{{i}_{2},{j}_{2}},0,\phantom{\rule{2.77695pt}{0ex}}\left|e\right|\u27e9,

*where* {v}_{{i}_{1},{j}_{1}} *and* {v}_{{i}_{2},{j}_{2}}*are vertices at the ends of e. The graph*\stackrel{\u0303}{G}=\left(V\cup \left\{{v}_{N},{v}_{T}\right\},E\cup {E}_{N}\cup {E}_{D}\right)*is called a* circular chromosome graph.

See Figure 8 for an example. Let *n*(*e*_{
N
}) = *n*_{
N
} and *n*(*e*_{
T
}) = *n*_{
T
}. For *e* ∈ *E*_{
S
} ∪{*e*_{
N
}, *e*_{
T
}}, we set *l*(*e*) = *n*(*e*), l\left(\u0113\right)=0, and u\left(\u0113\right)=n\left(e\right). For *e* ∈ *E*_{
L
} ∪ *E*_{
R
}, we set *l*(*e*) = 1. We also set *l*(*e*_{
t
}(*v*)) to 1 for *v* ∈ *V*_{
W
} because these edges have to be visited in the solution.

**Lemma 4** *Let w*(*f*, *e*) = |*e*|*f*(*e*) *and W*_{0} = *Q*_{
N
}*n*_{
N
} + *Q*_{
T
} *n*_{
T
} + ∑_{e∈E}|*e*|*n*(*e*)*. For any multi-set C of chromosomes on G*, *there is a circulation f on* \stackrel{\u0303}{G} *such that*

W\left(f\right)=W\left(C\right)+{W}_{0}.

(6)

*Conversely*, *for any circulation f on*\stackrel{\u0303}{G} *that minimizes W* *(* *f* *)*, *there is a multi-set C of chromosomes on G that satisfies (6). In addition*, *C can be calculated in* O\left({\sum}_{e\in E\cup {E}_{N}\cup {E}_{D}}f\left(e\right)\right)*time*.

Let *E*_{+} = {*e* ∈ *E* ∪ *E*_{
N
} ∪ *E*_{
D
}|*l*(*e*) ≥ 1 or *n*(*e*) ≥ 1}. Note that \mathsf{\text{CC}}\left(\stackrel{\u0303}{G},E+\right) has only one weakly connected component because of WCC.

*Proof* First, we show that for any multi-set *C* of chromosomes on *G*, there exists a circulation *f* on \stackrel{\u0303}{G} that satisfies (6). Let End(*v*) be the number of chromosomes that begin or end with *v*. Consider the following *f*:

\begin{array}{llll}\phantom{\rule{0.5em}{0ex}}f(e)\hfill & =\hfill & \mathrm{max}\{n(e),\phantom{\rule{0.5em}{0ex}}m(C,\phantom{\rule{0.5em}{0ex}}e)\}\hfill & \phantom{\rule{0.5em}{0ex}}(e\in {E}_{S}),\hfill \\ \phantom{\rule{0.5em}{0ex}}f(\overline{e})\hfill & =\hfill & \mathrm{max}\{0,\phantom{\rule{0.5em}{0ex}}n(e)-m(C,\phantom{\rule{0.5em}{0ex}}e)\}\hfill & \phantom{\rule{0.5em}{0ex}}(e\in {E}_{S}),\hfill \\ \phantom{\rule{0.5em}{0ex}}f(e)\hfill & =\hfill & m(C,\phantom{\rule{0.5em}{0ex}}e)\hfill & (e\in {E}_{L}\cup {E}_{R}),\hfill \\ f({e}_{t}(v))\hfill & =\hfill & \text{End}(v)\hfill & (v\in V),\hfill \\ \phantom{\rule{0.5em}{0ex}}f({e}_{N})\hfill & =\hfill & \mathrm{max}\{{n}_{N},\phantom{\rule{0.5em}{0ex}}|C\left|\right\},\hfill \\ \phantom{\rule{0.5em}{0ex}}f({\overline{e}}_{N})\hfill & =\hfill & \mathrm{max}\{0,\phantom{\rule{0.5em}{0ex}}{n}_{N}-|C\left|\right\},\hfill \\ \phantom{\rule{0.5em}{0ex}}f({e}_{T})\hfill & =\hfill & \mathrm{max}\{{n}_{T},\text{Tr}(C)\},\hfill \\ \phantom{\rule{0.5em}{0ex}}f({\overline{e}}_{T})\hfill & =\hfill & \mathrm{max}\{0,\phantom{\rule{0.5em}{0ex}}{n}_{T}-\text{\hspace{0.17em}Tr}(C)\}.\hfill \end{array}

Then, *f* is a circulation on \stackrel{\u0303}{G} because *f* satisfies (4) and (5). Thus, we observe that

w\left(e,\phantom{\rule{2.77695pt}{0ex}}m\left(C,\phantom{\rule{2.77695pt}{0ex}}e\right)\right)=\left|e\right|f\left(e\right)+\left|e\right|f\left(\u0113\right)-\left|e\right|n\left(e\right),

for *e* ∈ *E*_{
S
}, and

\begin{array}{c}{w}_{N}\left(C\right)=\left|{e}_{N}\right|f\left({e}_{N}\right)+\left|{e}_{N}\right|f\left({\u0113}_{N}\right)-{Q}_{N}{n}_{N},\\ {w}_{T}\left(C\right)=\left|{e}_{T}\right|f\left({e}_{T}\right)+\left|{e}_{T}\right|f\left({\u0113}_{T}\right)-{Q}_{T}{n}_{T}.\end{array}

Therefore, because |*e*| = 0 for *e* ∈ *E*_{
L
}∪ *E*_{
R
}∪{*e*_{
t
}(*v*)|*v* ∈ *V*} and *w*(*f*, *e*) = |*e*|*f*(*e*), *f* satisfies (6).

Conversely, let *f* be a circulation on \stackrel{\u0303}{G} that minimizes *W*(*f*). We show how to construct a multi-set *C* of chromosomes on *G* that satisfies (6).

First, for *e* ∈ *E*_{
S
} ∪ {*e*_{
N
}, *e*_{
T
}}, we subtract f\left(\u0113\right) from *f*(*e*), and also set f\left(\u0113\right) to 0.

Second, we construct a set *R* of cycles such that *m*(*R*, *e*) = *f*(*e*) for any edge *e* in \stackrel{\u0303}{G}. For directed graphs, the *flow decomposition theorem* [35] ensures that such *R* can be obtained in O\left({\sum}_{e\in E\cup {E}_{N}\cup {E}_{D}}f\left(e\right)\right) time. This is also true for bidirected graphs.

Third, we merge cycles in *R*. Whenever a vertex is shared by two cycles in *R*, they are merged into a single cycle. Because of WCC, \mathsf{\text{CC}}\left(\stackrel{\u0303}{G},E+\right) consists of only one weakly connected component. This implies that all cycles that contain edges in *E*_{+} can be merged into a single cycle. Note that any *r* ∈ *R* contains at least one edge in *E*_{+}, because otherwise *r* can be removed to decrease *W*(*f*). Therefore, all cycles in *R* can be merged into a single cycle \stackrel{\u0303}{r}.

Finally, let *C* be a multi-set of paths generated by removal of *v*_{
N
}, *v*_{
T
}, and edges in *E*_{
N
} from \stackrel{\u0303}{r}. Because *c* ∈ *C* is connected to edges in *E*_{
N
} in \stackrel{\u0303}{r}, the first and last edge of *c* is in *E*_{
S
} due to the directions of these edges. Accordingly, *c* is a chromosome. Therefore, *C* is a multi-set of chromosomes on *G*.

All of these steps can be completed in O\left({\sum}_{e\in E\cup {E}_{N}\cup {E}_{D}}f\left(e\right)\right) time. In addition, we observe that the following equations hold:

\begin{array}{c}\left|C\right|\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}f\left({e}_{N}\right)-f\left({\overline{e}}_{N}\right),\\ \mathsf{\text{Tr}}\left(C\right)=f\left({e}_{T}\right)-f\left({\overline{e}}_{T}\right),\\ m\left(C,\phantom{\rule{2.77695pt}{0ex}}e\right)=f\left(e\right)+f\left(\overline{e}\right)\left(e\in {E}_{S}\right).\end{array}

Accordingly, w\left(e,m\left(C,e\right)\right)=w\left(f,e\right)+w\left(f,\u0113\right)+\left|e\right|n\left(e\right) for *e* ∈ *E*_{
S
}, and

\begin{array}{c}{w}_{N}\left(C\right)\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}w\left(f,\phantom{\rule{2.77695pt}{0ex}}{e}_{N}\right)+w\left(f,{\overline{e}}_{N}\right)+{Q}_{N}{n}_{N},\\ {w}_{T}\left(C\right)\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}w\left(f,\phantom{\rule{2.77695pt}{0ex}}{e}_{T}\right)+w\left(f,{\overline{e}}_{T}\right)+{Q}_{T}{n}_{T},\\ w\left(e,\phantom{\rule{2.77695pt}{0ex}}m\left(C,\phantom{\rule{2.77695pt}{0ex}}e\right)\right)=0\left(e\in {E}_{L}\cup {E}_{R}\right).\end{array}

Therefore, *C* satisfies (6).

By Lemma 4, the solution of ChrW can be obtained by calculating a circulation *f* on \stackrel{\u0303}{G} that minimizes *W*(*f*). By Lemma 1, setting *u*(*e*) = *U*(4|*V*| + 1)(|*E*| + 1) does not affect the solution. In addition, |*E*_{
N
}| = *O*(|*E*|) and |*E*_{
D
}| = *O*(|*E*|). Accordingly, the circulation *f* can be calculated in *O*(|*E*|_{2} log |*V*| log |*E*|) time by using Gabow's algorithm [20]. Therefore, the optimal solution can be calculated in *O*(|*E*|_{2} log |*V*| log |*E*|) time.

### Proof of Theorem 3

ChrL is in NP because of Lemma 1.

Here, we show that the well-known PARTITION problem [34] can be reduced to ChrL. Let *n* be a positive integer and *S* = {*i* ∈ *Z*|1 ≤ *i* ≤ *n*}. Also, let *s*(*i*) be an integer function defined for *i* ∈ *S* such that Yasuda and Miyano Page 9 of 11 *s*(*i*) > 0, and *S*_{Σ} = ∑_{i∈S}*s*(*i*). The problem of finding a subset *S'* ⊂ *S* such that

{\displaystyle \sum _{i\in {S}^{\prime}}}s\left(i\right)={\displaystyle \sum _{i\in S-{S}^{\prime}}}s\left(i\right)={S}_{\Sigma}/2

is called the *partition problem* (hereafter referred to as *PARTITION*) [34]. It is well known that PARTITION is NP-complete. We reduce PARTITION to ChrL by constructing a chromosome graph whose solution for ChrL contains two chromosomes that correspond to two subsets of a solution of PARTITION.

Let *G* = (*V*, *E*) be a chromosome graph, where

V={\displaystyle \bigcup _{1\le i\le n+1}}\left\{{v}_{i,0}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,1}^{+},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,1}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,2}^{+},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,2}^{-},\phantom{\rule{2.77695pt}{0ex}}{v}_{i,3}^{+}\right\}

is a set of vertices, and *E* = *E*_{
S
} ∪ *E*_{
L
} ∪ *E*_{
R
} be a set of edges. Here, *E*_{
S
} consists of

\begin{array}{c}{e}_{i,0}=\u27e8-{v}_{i,0}^{-},+{v}_{i,1}^{+},1,9{S}_{\Sigma}\u27e9\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le i\le n\right),\\ {e}_{i,1}=\u27e8-{v}_{i,1}^{-},+{v}_{i,2}^{+},2,\phantom{\rule{2.77695pt}{0ex}}s\left(i\right)\u27e9\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le i\le n\right)\\ {e}_{i,2}=\u27e8-{v}_{i,2}^{-},+{v}_{i,3}^{+},1,\phantom{\rule{2.77695pt}{0ex}}{S}_{\Sigma}-s\left(i\right)\u27e9\phantom{\rule{1em}{0ex}}\left(1\le i\le n\right)\\ {e}_{n+1,0}=\u27e8-{v}_{n+1,0}^{-},+{v}_{n+1,1}^{+},2,9{S}_{\Sigma}/2\u27e9,\phantom{\rule{2.77695pt}{0ex}}\\ {e}_{n+1,1}=\u27e8-{v}_{n+1,1}^{-},+{v}_{n+1,2}^{+},\phantom{\rule{2.77695pt}{0ex}}n+2,0\u27e9,\\ {e}_{n+1,2}=\u27e8-{v}_{n+1,2}^{-},+{v}_{n+1,3}^{+},2,5{S}_{\Sigma}\u27e9.\end{array}

In addition, *E*_{
R
} consists of

\begin{array}{c}{\widehat{e}}_{i,1}=\u27e8-{v}_{i,1}^{+},+{v}_{i,1}^{-},0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\le i\le n+1\right),\\ {\widehat{e}}_{i,2}=\u27e8-{v}_{i,2}^{+},+{v}_{i,2}^{-},0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\le i\le n+1\right),\end{array}

and *E*_{
L
} consists of

\begin{array}{c}{e}_{Li}=\u27e8+{v}_{i,1}^{-},-{v}_{n+1,2}^{+},0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\le i\le n\right),\\ {e}_{Li}=\u27e8-{v}_{i,2}^{+},+{v}_{n+1,1}^{-},0,0\u27e9\phantom{\rule{1em}{0ex}}\left(1\le i\le n\right).\end{array}

We set *λ*_{
i
} = 10*S*_{Σ} for any *i* ≥ 1, *Q*_{
N
} = *Q*_{
T
} = 100*S*_{Σ}, *n*_{
N
} = *n*+2, and *n*_{
T
} = 0. See Figure 9 for an example. In addition, we set *V*_{
W
} to *V*_{5} ∪ *V*_{3}, and *E*_{
W
} to *E* by making all edges in *E*_{
L
} ∪ *E*_{
R
} required so that *G* satisfies WCC.

We show that PARTITION for *S* has a solution *S'* ⊂ *S* if, and only if, there exists a solution *C* of ChrL such that *W*(*C*) = 0. First, suppose that PARTITION has a solution *S'*. Let *r*_{
S'
}be a cycle generated by merging cycles {e}_{n+1,1}{e}_{Li}{e}_{i,1}{e}_{Li}^{\prime} for *i* ∈ *S'*. We define *r*_{S-S'}in the same way. Consider a multi-set *C* = {*c*_{1},..., *c*_{n+2}}, where *c*_{
i
} ∈ *C* is a chromosome on *G* such that

\begin{array}{c}{c}_{i}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}{e}_{i,0}{\widehat{e}}_{i,1}{e}_{i,1}{\widehat{e}}_{i,2}{e}_{i,2}\phantom{\rule{1em}{0ex}}\left(1\le i\le n\right),\hfill \\ {c}_{n+1}={e}_{n+1,0}{\widehat{e}}_{n+1,1}{r}_{{S}^{\prime}}{e}_{n+1,1}{\widehat{e}}_{n+1,2}{e}_{n+1,2},\hfill \\ {c}_{n+2}={e}_{n+1,0}{\widehat{e}}_{n+1,1}{r}_{S-{S}^{\prime}}{e}_{n+1,1}{\widehat{e}}_{n+1,2}{e}_{n+1,2}.\hfill \end{array}

Then, *W*(*C*) = 0 because |*C*| = *n* + 2, Tr(*C*) = 0, and *m*(*C*, *e*) = *n*(*e*) for *e* ∈ *E*_{
S
}. In addition, *C* visits all required edges. Furthermore, |*c*_{
i
}| = 10Σ ≤ *λ*_{
i
} for 1 ≤ *i* ≤ *n* + 2.

Conversely, suppose that ChrL for *G* has an optimal solution *C* that satisfies *W*(*C*) = 0. Because *W*(*C*) = 0, we obtain |*C*| = *n* + 2, Tr(*C*) = 0, and *m*(*C*, *e*) = *n*(*e*) for *e* ∈ *E*. Because ∑_{e∈E}|*e*|*n*(*e*) = 10(*n* + 2)*S*_{Σ}, |*c*| = 10Σ for each *c* ∈ *C*. Let *c*_{
i
}be a chromosome that begins with *e*_{i,0 }for 1 ≤ *i* ≤ *n*. The other two chromosomes are denoted by *c*_{n+1 }and *c*_{n+2}. Then, *c*_{1} begins with {e}_{1,0}{\widehat{e}}_{1,1}{e}_{1,1}. Suppose that *c*_{1} does not visit {\widehat{e}}_{1,2}{e}_{1,2}. Then, there is a chromosome *c*_{
i
} that visits {\widehat{e}}_{1,2}{e}_{1,2}, whose previous edge has to be *e*_{1,1} in *c*_{
i
}. Therefore, for some paths *p*_{1} and *p*_{2},

\begin{array}{c}{c}_{1}={e}_{1,0}{\widehat{e}}_{1,1}{e}_{1,1}{p}_{1}\hfill \\ {c}_{i}={p}_{2}{e}_{1,1}{\widehat{e}}_{1,2}{e}_{1,2}.\hfill \end{array}

(7)

Because of (7), \left|{c}_{1}\right|=\left|{e}_{1,0}\right|+\left|{\widehat{e}}_{1,1}\right|+\left|{e}_{1,1}\right|+\left|{p}_{1}\right|=10{S}_{\Sigma}=\left|{e}_{1,0}\right|+\left|{\widehat{e}}_{1,1}\right|+\left|{e}_{1,1}\right|+\left|{\widehat{e}}_{1,2}\right|+\left|{e}_{1,2}\right|. Therefore, \left|{p}_{1}\right|=\left|{\widehat{e}}_{1,2}\right|+\left|{e}_{1,2}\right|. We modify *C* so that

\begin{array}{c}{c}_{1}={e}_{1,0}{\widehat{e}}_{1,1}{\widehat{e}}_{1,2}{e}_{1,2,}\hfill \\ {c}_{i}={p}_{2}{e}_{1,1}{p}_{1}.\hfill \end{array}

The modified *C* still satisfies the required conditions. After this modification is repeated for 2 ≤ *i* ≤ *n* until no more modifications can be applied, *C* satisfies {c}_{i}={e}_{i,0}{\widehat{e}}_{1,1}{e}_{i,1}{\widehat{e}}_{i,2}{e}_{i,2} for 1 ≤ *i* ≤ *n*. Another chromosome exists that visits *e*_{i,1 }for each 1 ≤ *i* ≤ *n*, which is one of *c*_{n+1 }and *c*_{n+2}. Let *S'* = {*i*|*m*(*c*_{n+1}, *e*_{i,1}) *>*0}. Then, ∑_{i∈S'}*s*(*i*) = 10*S*_{Σ} − (9*/*2+5)*S*_{Σ} = 1*/*2*S*_{Σ}. Therefore, *S'* is a solution of PARTITION.

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