approximation algorithms for graph approximation problems

ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2011, Vol. 5, No. 4, pp. 569–581. c© Pleiades Publishing, Ltd., 2011.Original Russian Text c© V.P. Il’ev, S.D. Il’eva, A.A. Navrotskaya, 2011, published in Diskretnyi Analiz i Issledovanie Operatsii, 2011, Vol. 18, No. 1, pp. 41–60.

Approximation Algorithms for Graph Approximation Problems

V. P. Il’ev1*, S. D. Il’eva2**, and A. A. Navrotskaya3***

1Omsk State University, pr. Mira 55a, Omsk, 644077 Russia2Omsktelecom, ul. 1-ya Zavodskaya 23, Omsk, 644040 Russia

3Omsk State University, ul. Mira 55a, Omsk, 644077 RussiaReceived July 20, 2010; in final form, November 29, 2010

Abstract—Several versions of the graph approximation problem are under study. Approximationalgorithms for these problems are proposed, and performance guarantees of the algorithms areobtained. In particular, it is shown that the problem of approximation by graphs with a boundednumber of connected components belongs to the class APX.

DOI: 10.1134/S1990478911040120

Keywords: graph approximation problem, approximation algorithm, performance guarantee

INTRODUCTION

The graph approximation problems along with the minimal cut problems are the most adequatemathematical models for the problems of classification of interconnected objects. However, in contrastto the minimal cut problem, in the graph approximation problem not only the number of “superfluous”connections between the classes is minimized, but also the number of “missing” connections inside theclasses. For statements and various interpretations of this problem see [5, 6, 8, 11, 12].

We consider only the simple graphs, i.e., the graphs without loops and multiple edges. Refer toa graph as an M-graph if each connected component of it is a complete graph. Denote by M(V ) theclass of all M-graphs on a vertex set V ; by Mk(V ), the class of all M-graphs on a vertex set V withexactly k nonempty connected components; and by M1

k(V ), the class of all M-graphs on a vertex set Vwith at most k connected components, where 2 � k � n.

Introduce the concept of distance in the class of all graphs with labelled vertices. The distancebetween two labelled graphs G1 = (V,E1) and G2 = (V,E2) is defined as ρ(G1, G2) = |E1�E2|, where

E1�E2 = (E1 \ E2) ∪ (E2 \ E1).

The three versions of the graph approximation problem are considered in the literature:

Problem A. Given an arbitrary graph G = (V,E), find some graph M∗ ∈ M(V ) satisfying

ρ(G,M∗) = minM∈M(V )

ρ(G,M).

Problem Ak. Given a graph G = (V,E) and an integer k with 2 � k � n, find some graphM∗ ∈ Mk(V ) satisfying

ρ(G,M∗) = minM∈Mk(V )

ρ(G,M).

*E-mail: [email protected]**E-mail: [email protected]

***E-mail: [email protected]

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570 IL’EV et al.

Problem A1k. Given a graph G = (V,E) and an integer k with 2 � k � n, find some graph

M∗ ∈ M1k(V ) satisfying

ρ(G,M∗) = minM∈M1

k(V )ρ(G,M).

The weighted and directed versions of these problems were also considered.The computational complexity of the graph approximation problems remained unknown for a long

time. In 2004, V. P. Il′ev and A. S. Talevnin proved [7] that the weighted Problem Ak is NP-hard forevery fixed k � 2. In 2006, A. V. Kononov showed that Problem A2 is NP-hard on cubic graphs. Thiswas used in [1] to prove that all versions of the graph approximation problem are NP-hard, which impliesthat they are NP-hard for the directed graphs as well.

Until recently any algorithms for solving graph approximation problems were essentially unavailable.It is shown in [10] that the problem of approximation of a graph without triangles reduces to constructinga maximum matching in it. In 1971, G. A. Veiner proposed [2] an algorithm for the approximationproblem for the graphs containing no 4-vertex subgraphs with exactly five edges, but did not prove thatthe algorithm indeed produces some M-graph optimally approximating the given graph. The Veineralgorithm was justified by G. Sh. Fridman [9].

The goal of this article is to develop and analyze some approximation algorithms for the graphapproximation problems. Recall that an algorithm for solving a combinatorial optimization problemis called an α-approximation if, for every input I, it finds in polynomial time an feasible solution tothe problem I whose weight differs from that of an optimal solution by a factor of at most α. Saythat a combinatorial optimization problem belongs to the class APX whenever there exists an α-approximation algorithm for it with some constant α. A family Aε of algorithms is called a polynomialtime approximation scheme if, for all ε > 0, each algorithm in Aε is an (1 + ε)-approximation.

Consider an arbitrary family Gn of graphs on n vertices. Refer to the graphs in Gn as nondenseif |E| � αnβ for every graph G = (V,E) in Gn, where α and β are some constants with α > 0 and0 < β < 2.

Consider some class K of minimization problems. Given I ∈ K, let OPT(I) denote the optimal valueof the objective function, and let A(I) stand for the value found by an approximation algorithm A. Referto A as a guaranteed asymptotically exact algorithm whenever

A(I) � (1 + εn)OPT(I)

on the set Kn ⊂ K of dimension n problems, where εn → 0 as n → ∞ (about the algorithms withperformance bounds, see [3]).

The existence of a polynomial time approximation scheme for Problem A12 is proved in [1], while

[4] proposed simple constant-approximation algorithms for Problems A12 and A2. We now propose

some approximation algorithms for solving various versions of the graph approximation problem. Wedevelop a guaranteed asymptotically exact algorithm and a polynomial time approximation scheme forProblem A1

k on the nondense graphs, as well as a 3-approximation algorithm for Problem A2. We provethat Problem A1

k belongs to the class APX for every fixed k � 2. We also propose an approximationalgorithm for Problem A and obtain some performance guarantees for this algorithm.

1. A POLYNOMIAL TIME APPROXIMATION SCHEME FOR PROBLEM A1k

ON NONDENSE GRAPHS

Consider the following equivalent statement of Problem A1k:

Given a graph G = (V,E), find a partition P = (V1, V2, . . . , Vk) of V on which the function

f(P ) = |{uv /∈ E | u, v ∈ Vi, i ∈ {1, . . . , k}}|+ |{uv ∈ E | u ∈ Vi, v ∈ Vj , i ∈ {1, . . . , k − 1}, j ∈ {i + 1, . . . , k}}|

attains its minimum. Some of the sets V1, . . . , Vk can be empty.

JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS Vol. 5 No. 4 2011

APPROXIMATION ALGORITHMS FOR GRAPH APPROXIMATION PROBLEMS 571

It is obvious that f(P ) = ρ(G,M), where M ∈ M1k(V ) is the M-graph with the connected compo-

nents induced by V1, V2, . . . , Vk.

Put

NG(v) = {u ∈ V | uv ∈ E}, NG(v) = V \ (NG(v) ∪ {v}).

To every vertex v ∈ V of G and a partition P = (V1, . . . , Vk) associate the quantity

bi(v, P ) = |NG(v) ∩ Vi| + |NG(v) ∩ (V \ Vi)|, i ∈ {1, . . . , k}.Observe that bi(v, P ) is the contribution of a vertex v to the objective function when v is placed into Vi.Refer to the quantity bi(v, P ) as the penalty of v ∈ Vi.

Lemma 1. Every feasible solution P = (V1, . . . , Vk) to Problem A1k on an n-vertex graph G =

(V,E) satisfies

k∑

i=1

∑

v∈Vi

bi(v, P ) = 2f(P ).

Proof. Summing |NG(v) ∩ Vi| over all v ∈ Vi, we count twice every missing edge uv in the subgraphof G induced by Vi, i ∈ {1, . . . , k}. The sum |NG(v) ∩ (V \ Vi)| over all v ∈ Vi determines the numberof edges of the cut between Vi and V \ Vi. As we sum over all i ∈ {1, . . . , k}, every edge of the cut iscounted twice. Thus,

k∑

i=1

∑

v∈Vi

bi(v, P ) = 2|{uv /∈ E | u, v ∈ Vi, i ∈ {1, . . . , k}}|

+ 2|{uv ∈ E | u ∈ Vi, v ∈ Vj, i ∈ {1, . . . , k − 1}, j ∈ {i + 1, . . . , k}}|.

Hence, the definition of f(P ) yields the required equality. The proof of Lemma 1 is complete.

Lemma 2. Consider an arbitrary partition P = (V1, . . . , Vk) of the vertex set V of a graphG = (V,E). Then, for every v ∈ V , we have

k∑

i=1

bi(v, P ) = n − 1 + |NG(v)|(k − 2).

Proof. Take an arbitrary vertex v ∈ V . Considering that

|NG(v) ∩ (V \ Vi)| =∑

j �=i

|NG(v) ∩ Vj |,

we obtain

k∑

i=1

bi(v, P ) =k∑

i=1

(|NG(v) ∩ V1| + . . . + |NG(v) ∩ Vi−1| + |NG(v) ∩ Vi|

+ |NG(v) ∩ Vi+1| + . . . + |NG(v) ∩ Vk|).

Collect the terms into the two sums: one consisting of |NG(v) ∩ Vi|, and the other consisting of|NG(v) ∩ Vi|. Then

k∑

i=1

bi(v, P ) =k∑

i=1

|NG(v) ∩ Vi| + (k − 1)k∑

i=1

|NG(v) ∩ Vi|.


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It is not difficult to observe that the first sum equals the number of vertices not adjacent to v, or|NG(v)| = n − 1 − |NG(v)|; while the second, the degree of v, i.e., |NG(v)|. Thus,

k∑

i=1

bi(v, P ) = |NG(v)| + (k − 1)|NG(v)| = n − 1 + (k − 2)|NG(v)|.

The proof of Lemma 2 is complete.

Lemma 3. Consider an feasible solution P = (V1, . . . , Vk) to Problem A1k on a graph G = (V,E)

and v ∈ Vp, where p ∈ {1, . . . , k}. If

bp(v, P ) − bq(v, P ) > 0

for some q ∈ {1, . . . , k} then, by moving v into the component Vq, we decrease the value of f(P ).

Proof. It is clear that p = q. Take a partition P ′ = (V ′1 , . . . , V ′

k) such that V ′i = Vi for i /∈ {p, q}, V ′

p =Vp \ {v}, and V ′

q = Vq ∪ {v}. Calculate the difference f(P ) − f(P ′). Lemma 1 yields

2f(P ) − 2f(P ′) =k∑

i=1

∑

u∈Vi

bi(u, P ) −k∑

i=1

∑

u∈V ′i

bi(u, P ′).

Since in P ′ all vertices except v remain in the same part of V as in the partition P , it follows that u ∈ V ′i

for all u = v and i ∈ {1, . . . , k} if and only if u ∈ Vi. Thus,

2f(P ) − 2f(P ′) =k∑

i=1

∑

u∈Vi,u �=v

(bi(u, P ) − bi(u, P ′)) + bp(v, P ) − bq(v, P ′).

Moreover, it is not difficult to observe that bi(u, P ) − bi(u, P ′) = 0 for all i /∈ {p, q} and u ∈ Vi, whilebq(v, P ) = bq(v, P ′). Consequently,

2f(P ) − 2f(P ′) =∑

i∈{p, q}

∑

u∈Vi,u �=v

(bi(u, P ) − bi(u, P ′)) + bp(v, P ) − bq(v, P ).

Consider bi(u, P ) − bi(u, P ′) for a arbitrary vertex u ∈ Vi with u = v, where i ∈ {p, q}:

bi(u, P ) − bi(u, P ′) =

⎧⎨

⎩1, if u ∈ Vp ∩ NG(v) or u ∈ Vq ∩ NG(v),

−1, if u ∈ Vp ∩ NG(v) or u ∈ Vq ∩ NG(v).

Hence, we obtain

2f(P )− 2f(P ′) = |Vp ∩NG(v)|+ |Vq ∩NG(v)| − |Vp ∩NG(v)| − |Vq ∩NG(v)|+ bp(v, P )− bq(v, P )

= |Vp ∩ NG(v)| + |Vq ∩ NG(v)| + |V \ (Vp ∪ Vq) ∩ NG(v)|− |V \ (Vp ∪ Vq) ∩ NG(v)| − |Vp ∩ NG(v)| − |Vq ∩ NG(v)| + bp(v, P ) − bq(v, P )

= |NG(v) ∩ Vp| + |NG(v) ∩ (V \ Vp)| − |NG(v) ∩ Vq| − |NG(v) ∩ (V \ Vq)|+ bp(v, P ) − bq(v, P ) = bp(v, P ) − bq(v, P ) + bp(v, P ) − bq(v, P ).

Consequently, 2f(P ) − 2f(P ′) = 2(bp(v, P ) − bq(v, P )) > 0 and f(P ) > f(P ′). The proof of Lemma 3is complete.



To solve Problem A1k approximately we propose some local improvement algorithm. The algorithm is

based on a step-by-step decrease of the value of the objective function. At each step of the algorithm, wechoose a vertex that satisfies Lemma 3 and move it into the corresponding component.

Local improvement algorithm LA.

Step 0. Put V1 = V, V2 = V3 = . . . = Vk = ∅ and define the partition P = P (V1, V2, . . . , Vk).

Step s, s � 1. Choose two numbers p, q and a vertex v ∈ Vp satisfying

bp(v, P ) − bq(v, P ) = maxi,j∈{1,...,k},

u∈Vi

(bi(u, P ) − bj(u, P )).

If bp(v, P ) − bq(v, P ) > 0 then put Vp = Vp \ {v} and Vq = Vq ∪ {v}, then go to Step s + 1, andotherwise halt.

End.

Lemma 4. The solution P to Problem A1k on an n-vertex graph G = (V,E) obtained by

Algorithm LA satisfies

f(P ) � n(n − 1) + 2|E|(k − 2)2k

. (1)

Proof. Verify that, once the algorithm halts, the estimate

bp(v, P ) � n − 1 + |NG(v)|(k − 2)k

(2)

holds for all p ∈ {1, . . . , k} and v ∈ Vp. Lemma 2 yields

k∑

i=1

bi(v, P ) = n − 1 + |NG(v)|(k − 2)

for all v ∈ V . Then, for every v ∈ V , there is a number q ∈ {1, . . . , k} such that

bq(v, P ) � (n − 1 + |NG(v)|(k − 2))/k.

Once Algorithm LA halts, for all p ∈ {1, . . . , k} and v ∈ Vp, we have bp(v, P ) − bq(v, P ) � 0. Therefore,

bp(v, P ) � bq(v, P ) � (n − 1 + |NG(v)|(k − 2))/k,

and so (2) is valid.Summing (2) over all v ∈ V , we obtain

k∑

i=1

∑

v∈Vi

bi(v, P ) �k∑

i=1

∑

v∈Vi

n − 1 + |NG(v)|(k − 2)k

=n(n − 1) + 2|E|(k − 2)

k.

Hence, Lemma 1 yields

f(P ) � n(n − 1) + 2|E|(k − 2)2k

.


Theorem 1. The solution P by Algorithm LA to Problem A1k for an arbitrary nondense graph

G ∈ Gn satisfies

f(P ) �(

1 +(n + 4αnβ)(k − 1)n2 − kn − 2αnβ

)f(P ∗),

where P ∗ is an optimal solution to Problem A1k.


574 IL’EV et al.

Proof. Every solution P to Problem A1k satisfies

f(P ) =n(n − 1)

2− |E| + 2C −

k−1∑

i=1

k∑

j=i+1

pipj,

where C is the number of edges uv with u ∈ Vi and v ∈ Vj for i, j ∈ {1, . . . , k} and i = j, while pi = |Vi|.It is easy that

k−1∑

i=1

k∑

j=i+1

pipj � n2(k − 1)2k

.

Considering that, for nondense graphs, |E| � αnβ and C � 0, we obtain

f(P ) � n(n − 1)2

− αnβ − n2(k − 1)2k

� n2 − 2kαnβ − kn

2k.

Observe that this inequality holds for every partition P including the optimal P ∗. Applying (1),estimate the value of the objective function for the partition obtained by Algorithm LA:

f(P ) � n(n − 1) + 2αnβ(k − 2)2k

=(

1 +(4αnβ + n)(k − 1)n2 − 2kαnβ − kn

)(n2 − 2kαnβ − kn

2k

)

�(

1 +(4αnβ + n)(k − 1)n2 − 2kαnβ − kn

)f(P ∗).

The proof of Theorem 1 is complete.

It is obvious that

(4αnβ + n)(k − 1)/(n2 − 2kαnβ − kn) → 0

as n → ∞ for k fixed.

Corollary 1. The local improvement algorithm is guaranteed asymptotically exact on non-dense graphs.

Indeed, it suffices to put

εn =(4αnβ + n)(k − 1)n2 − 2kαnβ − kn

.

We can split the construction of a polynomial time approximation scheme for Problem A1k on

nondense graphs into the two stages. At the first stage, given ε define n = nε from

ε = (4αnβ + n)(k − 1)/(n2 − 2kαnβ − kn).

At the second stage, depending on the dimension of the concrete problem, formulate some AlgorithmAε for solving Problem A1

k. Denote the dimension of this problem by n.If n � nε then we solve the problem by Algorithm LA and, by Theorem 1, obtain

f(P ) �(

1 +(4αnβ + n)(k − 1)n2 − 2kαnβ − kn

)f(P ∗) � (1 + ε)f(P ∗).

For n < nε, we find the exact solution using the branch-and-bound method or exhaustive search, whichguarantees the required estimate. For the search, we need on the order of kn−1n2 operations at worst.However, for every fixed nε, there exists a number γ such that nγ � kn−1n2 for n < nε; i.e., for given ε,there exists a polynomial time algorithm Aε finding a solution to Problem A1

k on which the value of theobjective function differs from the optimal by at most a factor of 1 + ε. Therefore, we have constructeda polynomial time approximation scheme.



2. APPROXIMATION ALGORITHMS FOR PROBLEMS A12 AND A2

Assume that n � 3. Take two labelled graphs G1 = (V,E1) and G2 = (V,E2) with n = |V |. LetD(G1, G2) denote the graph on the vertex set V with the edge set E1ΔE2. It is not difficult to observethat ρ(G1, G2) equals the number of edges in D(G1, G2), which implies

Lemma 5. The minimum degree dmin of vertices in D(G1, G2) satisfies

ρ(G1, G2) � ndmin/2.

Given V1, V2, . . . , Vs ⊆ V satisfying Vi ∩ Vj = ∅ for all i, j ∈ {1, . . . , s} and V1 ∪ V2 ∪ · · · ∪ Vs = V ,let M(V1, V2, . . . , Vs) denote the M-graph of class M1

s(V ) with the connected components induced byV1, . . . , Vs. Some Vi can be empty.

Consider an optimal approximating M-graph M∗ = M(V ∗1 , V ∗

2 ) ∈ M12(V ) for G = (V,E) and put

D = D(G,M∗). Let dD(v) be the degree of a vertex v in D.

Lemma 6. Take a vertex v ∈ V of minimum degree in D,

dD(v) = minu∈V

dD(u) = dmin,

and the M-graph Mv = M(V1, V2), where V1 = {v} ∪ NG(v) and V2 = V \ V1 = NG(v). Then, forn � 3, we have ρ(G,Mv) � (3 − 6/n)ρ(G,M∗).

Proof. Verify that the M-graph Mv can be obtained from M∗ by moving dmin vertices into the othercomponent. Without loss of generality, assume that v ∈ V1 ∩ V ∗

1 .By definition, those and only those vertices are adjacent in D that are either adjacent in G and lie in

distinct components of M∗ or are not adjacent in G and lie in the same component of M∗. Consequently,u ∈ V ∗

1 for every vertex u ∈ V \ {v} if and only if either u ∈ NG(v) \ ND(v) or u ∈ NG(v) ∩ ND(v)(hence, u ∈ V ∗

2 if and only if either u ∈ NG(v) ∩ ND(v) or u ∈ NG(v) \ ND(v)).

Put U1 = NG(v) ∩ ND(v) and U2 = NG(v) ∩ ND(v). It is not difficult to see that

U1 ⊆ V ∗1 , U2 ⊆ V ∗

2 , V1 = (V ∗1 \ U1) ∪ U2, V2 = (V ∗

2 \ U2) ∪ U1.

Since

|U1| + |U2| = |NG(v) ∩ ND(v)| + |NG(v) ∩ ND(v)| = |ND(v)| = dmin,

the graph Mv can be obtained from M∗ by moving dmin vertices into the other component.For dmin = 0, the graphs Mv and M∗ coincide; therefore,

ρ(G,Mv) = ρ(G,M∗) � (3 − 6/n)ρ(G,M∗)

for n � 3, and the claim of the lemma holds.Suppose that dmin � 1. Verify that movement of one vertex cannot increase the value of the objective

function by more than n − 3. Take u ∈ ND(v). Let M ′ denote the M-graph obtained from M∗ bymoving u vertices into the other component. It is obvious that D and D′ = D(G,M ′) differ only byan edge of the form uw, w ∈ V , while their other edges coincide. Consequently,

ρ(G,M ′) − ρ(G,M∗) = |ND′(u)| − |ND(u)|.Since v ∈ ND(u) and ND′(u) ⊆ V \ {u, v}; therefore,

ρ(G,M ′) − ρ(G,M∗) � n − 2 − 1 = n − 3.

Thus, as we move one vertex, the objective function increases by at most n − 3. Since Mv is obtainedfrom M∗ by moving dmin vertices, it follows that

ρ(G,Mv) � ρ(G,M∗) + dmin(n − 3).

Estimate ρ(G,Mv) using Lemma 5:

ρ(G,Mv) � ρ(G,M∗) + dmin(n − 3) � ρ(G,M∗) + 2ρ(G,M∗)(1 − 3/n) = (3 − 6/n)ρ(G,M∗).



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Consider the following

Algorithm N12 .

Step 1. For every vertex u ∈ V , define the M-graph Mu ∈ M12(V ) as follows: u and all adjacent

vertices of G are in one connected component Mu, while all vertices not adjacent to u are in the othercomponent.

Step 2. Among all Mu choose some graph MN satisfying

ρ(G,MN ) = minu∈V

ρ(G,Mu).

End.

It is obvious that among all vertices we will choose v ∈ V with dD(v) = dmin. Therefore,

ρ(G,MN ) � ρ(G,Mv).

Hence, by Lemma 6, we obtain a performance bound for N12 .

Theorem 2. Suppose that n � 3. Then, for every graph G = (V,E) on n vertices,

ρ(G,MN ) � (3 − 6/n)ρ(G,M∗),

where M∗ ∈ M12(V ) is an optimal solution to Problem A1

2 on G.

Slightly modifying Algorithm N12 , we can obtain a similar bound for Problem A2.

Henceforth, assume that G = Kn since Problem A2 can be solved analytically for Kn.

Algorithm N2.

Step 1. For every vertex u ∈ V , define an M-graph Mu ∈ M12(V ) as follows: u and all adjacent

vertices of G lie in one connected component of Mu, while all vertices not adjacent to u lie in the othercomponent.

Step 2. Among all graphs Mu choose M ′ ∈ M2(V ) satisfying

ρ(G,M ′) = minu∈V, Mu∈M2(V )

ρ(G,Mu).

Step 3. Take a vertex w ∈ V of a minimum degree in G. Define an M-graph M ′′ ∈ M2(V ) as follows:w lies in one connected component of M ′′ and the other vertices lie in the other. If ρ(G,M ′) � ρ(G,M ′′)then M ′

N = M ′, and otherwise M ′N = M ′′.

End.

Theorem 3. Suppose that n � 3. Then, for every graph G = (V,E) on n vertices,

ρ(G,M ′N ) � (3 − 6/n)ρ(G,M∗),

where M∗ ∈ M2(V ) is an optimal solution to Problem A2 on G.

Proof. Put D = D(G,M∗). Take a vertex v ∈ V of a minimum degree in D,

dD(v) = minu∈V

dD(u) = dmin,

and a vertex w ∈ V of a minimum degree in G.It is not difficult to verify that if dmin = 0 then the M-graph Mv is an optimal solution to Problem A2.

Consequently,

ρ(G,M ′) = ρ(G,Mv) = ρ(G,M∗).

Thus,

ρ(G,M ′N ) = min(ρ(G,M ′), ρ(G,M ′′)) = ρ(G,M ′) = ρ(G,M∗).



Example of the graph Gn for n = 10

Therefore, for n � 3, we have

ρ(G,M ′N ) = ρ(G,M∗) � (3 − 6/n)ρ(G,M∗).

Suppose that dmin � 1. The two cases are possible:

Case 1: Mv /∈ M2(V ). Then v is adjacent to all vertices of G.If w ∈ ND(v) then, in order to obtain M ′′ from M∗, we have to move dmin − 1 vertices. To this end,

move all vertices of ND(v) but w into the component containing v. As in the proof of Lemma 6, it is notdifficult to show that movement of one vertex cannot increase the value of the objective function by morethan n − 3. Estimate ρ(G,M ′′) using Lemma 5:

ρ(G,M ′′) � ρ(G,M∗) + (dmin − 1)(n − 3)� ρ(G,M∗) + 2ρ(G,M∗)(1 − 3/n) � (3 − 6/n)ρ(G,M∗).

Consequently,

ρ(G,M ′N ) � ρ(G,M ′′) � (3 − 6/n)ρ(G,M∗).

If w /∈ ND(v) then every vertex w′ ∈ ND(v) satisfies

ρ(G,M ′′) � ρ(G,M),

where M = M({w′}, V \ {w′}). Hence,

ρ(G,M ′′) � ρ(G,M) � ρ(G,M∗) + 2ρ(G,M∗)(1 − 3/n).

Therefore,

ρ(G,M ′N ) � ρ(G,M ′′) � (3 − 6/n)ρ(G,M∗).

Case 2: Mv ∈ M2(V ). Then

ρ(G,M ′) � ρ(G,Mv) � (3 − 6/n)ρ(G,M∗),

which can be proved in the same fashion as in Lemma 6. Therefore,

ρ(G,M ′N ) � ρ(G,M ′) � (3 − 6/n)ρ(G,M∗).


Remark 1. There exists an infinite family of graphs {Gn} (where n is the order of Gn, with n = 4r + 2for r = 2, 3, . . .) satisfying

ρ(Gn,MN ) = (3 − 6/n)ρ(G,M∗), ρ(Gn,M ′N ) = (3 − 6/n)ρ(G,M∗);

thus, the performance guarantees for Algorithms N12 and N2 are attained.

For every n = 4r + 2, the structure of the graph Gn is as follows: a vertex v1 of degree 2r + 1 isadjacent to a vertex v2 of degree 2r + 1; 2r vertices induce the subgraph obtained from the completegraph K2r by removing a maximum matching of cardinality r; the other 2r vertices induce an isomorphicsubgraph. The vertex v1 is adjacent to all vertices of the first subgraph, while v2 is adjacent to all verticesof the second subgraph. The figure shows an example of the graph Gn for n = 10.

It is not difficult to verify that the optimal solution to the problems A12 and A2 of approximation of Gn

is a graph M∗ with two components each of which is the clique K2r+1. The vertices of one clique are v1


578 IL’EV et al.

and all adjacent vertices of Gn except v2, while the vertices of the other are all remaining vertices of Gn.Consequently, ρ(Gn,M∗) = 2r + 1.

Observe that ρ(Gn,Mu) = 6r for every vertex u of Gn. Thus, MN = Mu, where u is an arbitraryvertex of Gn. Since ρ(Gn,M ′) = ρ(Gn,Mu) < ρ(Gn,M ′′), it follows that M ′

N = MN . Therefore,

ρ(Gn,MN )ρ(Gn,M∗)

=ρ(Gn,M ′

N )ρ(Gn,M∗)

=6r

2r + 1= 3 − 3

2r + 1= 3 − 6

n.

3. AN APPROXIMATION ALGORITHM FOR PROBLEM A1k

Consider a procedure P (G, i) whose input is a graph G = (V,E) and an integer i � 2, and whoseoutput is an M-graph M ∈ M1

i (V ).

Procedure P (G, i):

Step 1. If |V | � 2 then put M = G and halt. Otherwise go to Step 2.

Step 2. If i = 2 then solve Problem A12 on G using Algorithm N1

2 . Put M = MN , where MN is thesolution constructed by N1

2 , and halt. If i = 2 then go to Step 3.

Step 3. For every vertex u ∈ V , do:Step 3.1. Put V1 = {u} ∪ NG(u). If V1 = V then let Mu be the complete graph on the vertex set V .

Otherwise go to Step 3.2.Step 3.2. Let G1 denote the subgraph of G induced by the vertex set V \ V1. Carry out the procedure

P (G1, i − 1) and denote the resulting graph by M1. Suppose that M1 = M(V2, . . . , Vj) for some j � i.Put Mu = M(V1, V2, . . . , Vj).

Step 4. Among the constructed graphs Mu choose an M-graph M nearest to G:

ρ(G,M) = minu∈V

ρ(G,Mu).

End.

In order to solve approximately Problem A1k on G, we apply the following algorithm:

Algorithm N1k .

Step 1. Carry out the procedure P (G, k). Denote the resulting graph by M .

End.

We have the following performance guarantee for Algorithm N1k .

Theorem 4. For all k � 2, every graph G = (V,E) on n � 3 vertices satisfies

ρ(G,M) � (3 − 6/n)k−1ρ(G,M∗) < 3k−1ρ(G,M∗),

where M ∈ M1k(V ) is the M-graph constructed by Algorithm N1

k, while M∗ ∈ M1k(V ) is an

optimal solution to Problem A1k on G.

Proof. Induct on k. The base for k = 2 follows from Theorem 2.On assuming that the claim holds for k � i − 1, where i � 3, verify that it holds for k = i: for every

graph G = (V,E),

ρ(G,M) � (3 − 6/n)i−1ρ(G,M∗), (3)

where M ∈ M1i (V ) is the M-graph constructed by the procedure P (G, i), while M∗ ∈ M1

i (V ) is anoptimal solution to Problem A1

i on G.Take a vertex v of minimal degree in the graph D = D(G,M∗) and put dmin = dD(v). Consider

the M-graph M ∈ M1i (V ) obtained from M∗ by moving dmin vertices to other components of M∗:



move the vertices belonging to NG(v) ∩ ND(v) to the component containing v, and move all vertices ofNG(v) ∩ ND(v) to arbitrary components not containing v.

If dmin > 0 then in the same fashion as in Lemma 6 we can show that as we move vertices the valueof the objective function increases by at most dmin(n − 3). Hence, Lemma 5 yields

ρ(G,M ) � ρ(G,M∗) + dmin(n − 3) � ρ(G,M∗) + 2ρ(G,M∗)(1 − 3/n) = (3 − 6/n)ρ(G,M∗).

If dmin = 0 then the M-graph M coincides with the M-graph M∗, and so,

ρ(G,M ) = ρ(G,M∗) � ρ(G,M∗)(3 − 6/n)since n � 3.

Thus,

ρ(G,M ) � (3 − 6/n)ρ(G,M∗). (4)

Put V1 = {v} ∪ NG(v). The two cases are possible:

Case 1: V1 = V . Put Mv = Kn. It is not difficult to verify that then Mv = M . By (4), this impliesthat

ρ(G,Mv) � (3 − 6/n)ρ(G,M∗) � (3 − 6/n)i−1ρ(G,M∗).

Case 2: V1 = V . Denote by G1 the subgraph of G induced by the vertex set V \ V1. Denote byM∗

1 an optimal solution to Problem A1i−1 on G1. Consider the M-graph M ∈ M1

i (V ) whose connectedcomponent containing v coincides with a component of the M-graph M , while the other componentscoincide with the components of an optimal solution to Problem A1

i−1 on G1; i.e., M = M(V1) ∪ M∗1

(here M(V1) is the complete graph on the vertex set V1). It is obvious that ρ(G, M ) � ρ(G,M ), whence,by (4), we obtain

ρ(G, M ) � (3 − 6/n)ρ(G,M∗). (5)

Denote by M1 ∈ M1i−1(V \ V1) the M-graph constructed by the procedure P (G1, i − 1). Suppose

that M1 = M(V2, . . . , Vj) for some j � i. Put Mv = M(V1, V2, . . . , Vj).Let s = s(v,G) denote the sum of the number of missing edges in the subgraph of G induced by the

vertex set V1 = {v} ∪ NG(v) and the size of the cut (V1, V \ V1) in G. It is obvious that

ρ(G,Mv) = s + ρ(G1,M1), ρ(G, M ) = s + ρ(G1,M∗1 ).

If |V \ V1| � 2 then M1 = G1 and ρ(G1,M1) = 0 = ρ(G1,M∗1 ). Hence,

ρ(G,Mv) = s + ρ(G1,M1) = s + ρ(G1,M∗1 ) = ρ(G, M ).

By (5), this implies

ρ(G,Mv) � (3 − 6/n)ρ(G,M∗) � (3 − 6/n)i−1ρ(G,M∗).

Suppose that |V \ V1| � 3. Then the inductive assumption yields

ρ(G1,M1) � (3 − 6/n)i−2ρ(G1,M∗1 ).

Consequently,

ρ(G,Mv) = s + ρ(G1,M1) � s + (3 − 6/n)i−2ρ(G1,M∗1 )

≤ (3 − 6/n)i−2s + (3 − 6/n)i−2ρ(G1,M∗1 ) = (3 − 6/n)i−2ρ(G, M ).

By (5) we obtain

ρ(G,Mv) � (3 − 6/n)i−2ρ(G, M ) � (3 − 6/n)i−2(3 − 6/n)ρ(G,M∗) � (3 − 6/n)i−1ρ(G,M∗).

Thus, in both cases, we have

ρ(G,Mv) � (3 − 6/n)i−1ρ(G,M∗).

Since at Step 4 of the procedure P (G, i) the graph Mv appears among the graphs Mu, this implies (3).The proof of Theorem 4 is complete.


580 IL’EV et al.

Corollary 2. The graph approximation problem A1k belongs to the class APX for every fixed

k � 2.

4. AN APPROXIMATION ALGORITHM FOR PROBLEM A

As above, denote by s(v,G) the sum of the number of missing edges in the subgraph of G induced bythe vertex set {v} ∪ NG(v) and by the size of the cut

({v} ∪ NG(v), V \ ({v} ∪ NG(v))

)in G.

To solve Problem A approximately, we propose

Algorithm N .

Step 1. Put G1 = G. Choose a vertex v1 ∈ V satisfying s(v1, G) = min s(v,G), where the minimumis over all v ∈ V . Put V1 = {v1} ∪ NG1(v1). If V \ V1 = ∅ then halt, otherwise go to step 2.

Step i for i � 2. Denote by Gi the subgraph of Gi−1 induced by the set

V ∗ = V \ (V1 ∪ V2 ∪ · · · ∪ Vi−1).

Take a vertex vi ∈ V ∗ satisfying s(vi, Gi) = min s(v,Gi), where the minimum is over all v ∈ V ∗. PutVi = {vi} ∪ NGi(vi). If

V \ (V1 ∪ · · · ∪ Vi) = ∅,

then halt, otherwise go to step i+1.

End.

Let l denote the number of sets Vi constructed by Algorithm N (1 � l � n). Consider the M-graphMN = M(V1, V2, . . . , Vl) ∈ M(V ), where M(V1, V2, . . . , Vl) is the M-graph in which the set Vi inducesa complete subgraph, i ∈ {1, . . . , l}.

Remark 2. Take an optimal approximating M-graph M∗ for G and an optimal approximating M-graph M∗

i ∈ M(V \ (V1 ∪ V2 ∪ · · · ∪ Vi−1)) for Gi, i ∈ {1, . . . , l}. Then

ρ(G,M∗) = ρ(G1,M∗1 ) � ρ(Gi,M

∗i )

for every i ∈ {1, . . . , l}.

We have the following performance bound for Algorithm N :

Theorem 5. For an arbitrary graph G = (V,E) on n � 3 vertices, Algorithm N finds an M-graph MN satisfying

ρ(G,MN ) � (3 − 6/n)l ρ(G,M∗),

where M∗ ∈ M(V ) is an optimal approximating M-graph for G, and l is the number of connectedcomponents of MN .

Proof. Express ρ(G,MN ) as

ρ(G,MN ) = s(v1, G1) + s(v2, G2) + · · · + s(vl, Gl). (6)

Verify that

s(vi, Gi) � (3 − 6/n)ρ(Gi,M∗i )

for every i ∈ {1, . . . , l}.Consider an arbitrary index i ∈ {1, . . . , l} and denote the number of vertices of the graph Gi by ni.

If ni � 2 then Gi is an M-graph. Hence,

s(vi, Gi) = 0 � (3 − 6/n)ρ(Gi,M∗i ).

Suppose now that ni � 3. At each step, the algorithm chooses the minimal s(vi, Gi). Consequently,s(vi, Gi) � s(v∗i , Gi), where v∗i is a vertex of minimum degree in Di = D(Gi,M

∗i ). Construct a graph

Mi by modifying the optimal solution M∗i as follows. Move all vertices belonging to NGi(v

∗i ) ∩ NDi(v

∗i )



to the component containing v∗i and all vertices of NGi(v∗i ) ∩ NDi(v

∗i ) out of the component containing

v∗i to other components of the M-graph. The number of movements equals di, i.e., the degree of v∗i inDi. If di = 0 then the graphs Mi and M∗

i coincide. Thus,

s(vi, Gi) � s(v∗i , Gi) � ρ(Gi, Mi) = ρ(Gi,M∗i ) � (3 − 6/ni)ρ(Gi,M

∗i )

(the last inequality follows since ni � 3).If di > 0 then each movement cannot increase the value of the objective function by more than ni − 3.

The resulting solution is worse than the optimal by at most di(ni − 3). Thus,

s(vi, Gi) � s(v∗i , Gi) � ρ(Gi, Mi) � ρ(Gi,M∗i ) + di(ni − 3).

Lemma 5 yields

(vi, Gi) � ρ(Gi,M∗i ) + di(ni − 3) � ρ(Gi,M

∗i ) + 2ρ(Gi,M

∗i )(1 − 3/ni)

= (3 − 6/ni)ρ(Gi,M∗i ) � (3 − 6/n)ρ(Gi,M

∗i ).

Thus, s(vi, Gi) � (3 − 6/n)ρ(Gi,M∗i ) for every i ∈ {1, . . . , l}. By Remark 2, this implies that

s(vi, Gi) � (3 − 6/n)ρ(Gi,M∗) for every i ∈ {1, . . . , l}.

Inserting the last inequality into (6), we obtain

ρ(G,MN ) � (3 − 6/n)l ρ(G,M∗).


Corollary 3. For an arbitrary graph G = (V,E) on n � 3 vertices, Algorithm N finds an M-graph MN ∈ M(V ) satisfying

ρ(G,MN ) � (3n − 6)ρ(G,M∗),

where M∗ is an optimal approximating M-graph for G.

REFERENCES1. A. A. Ageev, V. P. Il’ev, A. V. Kononov, and A. S. Talevnin, “Computational Complexity of a Graph

Approximation Problem,” Diskret. Anal. Issled. Oper. Ser. 1, 13 (1), 3–15 (2006) [J. Appl. Indust. Math.1 (1), 1–8 (2007)].

2. G. A. Veiner, “On Approximation of a Symmetric Reflexive Binary Relation with the Equivalence Relation,”Trudy Tallinsk. Politekh. Inst., No. 313, 45–49 (1971).

3. E. Kh. Gimadi, N. I. Glebov, and V. A. Perepelitsa, “Algorithms with Estimates for Discrete OptimizationProblems,” Problemy Kibernet. No. 31, 35–42 (1975).

4. V. P. Il’ev and S. D. Il’eva, “Approximation Algorithms for Approximation with Graphs with a BoundedNumber of Components,” Trudy Inst. Math. Natsion. Akad. Nauk Belarusi 18 (1), 47–52 (2010).

5. V. P. Il’ev and G. Sh. Fridman, “To a Problem of Approximation with Graphs with a Fixed Number of theComponents,” Dokl. Akad. Nauk SSSR 264 (3), 533–538 (1982) [Soviet Math. Dokl. 25 (3), 666–670(1982)].

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7. A. S. Talevnin, “On Complexity of a Graph Approximation Problem,” Vestnik Omsk. Univ. No. 4, 22–24(2004).

8. G. Sh. Fridman, “A Problem of Graph Approximation,” Upravlyaemye Sistemy No. 8, 72–25 (1971).9. G. Sh. Fridman, “Some Results in the Problem of Approximation of Graphs,” in Problems of Discrete

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