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Page 1: 2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2014, Vol. 284, Suppl. 1, pp. S87–S95.c© Pleiades Publishing, Ltd., 2014.Original Russian Text c© I.I. Eremin, E.Kh. Gimadi, A.V. Kel’manov, A.V. Pyatkin, M.Yu.Khachai, 2013,published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 2.

2-Approximation Algorithm for Finding a Clique

with Minimum Weight of Vertices and Edges

I. I. Eremin†, E. Kh. Gimadi1, A. V. Kel’manov1,A. V. Pyatkin1, and M. Yu. Khachai2

Received February 10, 2013

Abstract—The problem of finding a minimum clique (with respect to the total weight of itsvertices and edges) of fixed size in a complete undirected weighted graph is considered along withsome of its important subclasses. Approximability issues are analyzed. The inapproximabilityof the problem is proved for the general case. A 2-approximation efficient algorithm with timecomplexity O(n2) is suggested for the cases when vertex weights are nonnegative and edgeweights either satisfy the triangle inequality or are squared pairwise distances for some pointconfiguration of Euclidean space.Keywords: complete undirected graph, clique of fixed size, minimum weight of vertices andedges, subset search, approximability, polynomial time approximation algorithm, approximationguarantee, time complexity.

DOI: 10.1134/S0081543814020084

1. INTRODUCTION

We investigate discrete optimization problems induced, in particular, by important problemsof data analysis and pattern recognition. Our aim is to analyze the computational complexity ofthese problems, construct efficient algorithms, and prove approximation guarantees.

In [5], it was proved that the problem of selecting from a set of vectors in Euclidean space asubset of given size minimizing the sum of all squared pairwise distances between its elements isNP-hard in the strong sense. This statement is one of the possible formalizations of the clusteranalysis problem concerning grouping similar objects, which is very important in data analysis.

One of the possible motivations for the problem is the following. Consider a table whose rowscontain the results of measuring a fixed collection of numerical informationally significant featuresfor a set of physical objects under analysis. Some elements of this collection (their number is knownin advance) are identical and, therefore, have equal values of the measured features. The featuresof the remaining objects are not fixed and may have any admissible values. The data in the table

†Deceased.1Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akad. Koptyuga 4,Novosibirsk, 630090 Russiaemails: [email protected], [email protected], [email protected]

2Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16,Yekaterinburg, 620990 Russia; Institute of Mathematics and Computer Science, Ural Federal University,pr. Lenina 51, Yekaterinburg, 620000 Russiaemail: [email protected]

S87

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S88 EREMIN et al.

are distorted by an unbiased random independent noise with unknown distribution. The aim isfinding the subset of families corresponding to the identical objects and estimate the true valuesof their features based on the given results of measurements, using the minimum criterion for thesum of squared distances.

Sometimes, the initial data in the problem are given only by a matrix of pairwise comparisonsof objects (values from the family of numerical features are unavailable), and comparison criteriamay be varied. Of particular importance are problems where the entries of the input matrix:

(1) satisfy the triangle inequality;(2) are combinations of the weights of pairwise compared objects and Euclidean distances

between them;(3) are combinations of the weights of pairwise compared objects and squared Euclidean

distances between them;(4) are arbitrary nonnegative values.In this paper, we solve some of these cases. We show that the general case of the problem is

NP-hard and inapproximable. Nevertheless, in some rather natural cases, the problem, while stillintractable, admits 2-approximation polynomial time algorithms.

2. PROBLEM STATEMENT AND ANALYSIS OF ITS COMPLEXITY

Consider a complete simple weighted undirected graph G = (V,E, a, c) and weight functionsa : V → Q and c : E → Q, which define the weights of its vertices and edges, respectively. The sum∑

v∈V av +∑

e∈E ce is called the weight of the graph G. The problems considered in this paper aresubclasses of a general combinatorial optimization problem given below.

Weighted clique problem (WCP). Input: a complete weighted undirected graph G =(V,E, a, c), where a : V → Q and c : E → Q, and a positive integer m. Output: a completesubgraph (clique) of the graph G of order m with the smallest (largest) weight.

Without loss of generality, we identify the sets V and Nn = {1, . . . , n}, where n = |V |, and usethe short notation cij for the image ce of an edge e = {i, j} ∈ E under the weight mapping c.

By analogy with the method proposed in [3], we assign to each WCP the polynomial timeequivalent problem of finding the extremum of the linear function

n∑

i=1

aixi +∑

1≤i<j≤n

cijyij → maxxi,yij∈{0,1}

(min

xi,yij∈{0,1}

)(1)

on the feasible set

n∑

i=1

xi = m, xi + xj − 1 ≤ yij ≤ min{xi, xj} (i < j).

The computational complexity of WCP is determined by the polynomial time reducibility of theknown Clique problem, which is NP-complete in the strong sense [1], to it.

Clique problem. Input: an undirected graph G and a positive integer m. Question: is therea clique of order m in this graph?

Proposition 1. WCP is NP-hard in the strong sense and remains intractable if the optimiza-tion criterion (max /min) is fixed, even for ai ≡ 0 and cij ≥ 0.

Proof. Let us consider the maximum and minimum problems separately.

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2-APPROXIMATION ALGORITHM S89

1. For the maximum problem, we assign to the original graph G a complete weighted graphG′ = (V,E′, a, c) where a ≡ 0 and the function c is the indicator function of the subset E ⊂ E′:

ce ={

1 if e ∈ E,

0 otherwise.

Consider an instance of WCP defined by the graph G′ and a positive integer m. Evidently,OPT ≤ m(m − 1)/2, and equality is attained if and only if G contains a clique of order m.

2. Consider the case of the minimum problem. Similarly, for the original graph G, we constructa complete weighted graph G′ where the weight function c is defined by the equality

ce ={

0 if e ∈ E,

1 otherwise.

Evidently, OPT ≥ 0, and the existence of a clique of order m in the graph G is still a necessaryand sufficient condition for equality.

Proposition 1 is proved.

Everywhere below, we consider the minimum WCP with nonnegative values of the vertex weightfunction a and the edge weight function c. We will denote this problem Min-EWCP.

It is known [2] that the optimization variant of Clique is in general inapproximable. Followingthe idea of the proof of Proposition 1, it is easy to find a polynomial time reduction of Clique toMin-EWCP, which preserves the gap. Therefore, Min-EWCP does not belong to the class APX(unless P = NP).

Proposition 2. If there exists a polynomial time approximation algorithm with an arbitraryfixed approximation guarantee α > 1 for Min-EWCP, then P = NP .

Proof. Assume that, for some α > 1, Min-EWCP has a polynomial time approximationalgorithm with approximation guarantee α; i.e., for each instance I of the problem, a clique ofrequired order can be found in polynomial time with respect to |I| such that the weight of the cliqueis upper bounded by αOPT(I). Consider an instance of Clique defined by the graph G = (V,E)and positive integer m. Let K = m(m− 1)/2. Similarly to the proof of Proposition 1, consider theinstance I of Min-EWCP defined by the complete graph G′ = (V,E′), the identically zero weightfunction a, and the function

ce ={

1 if e ∈ E,

αK otherwise.

If the graph G has a clique of order m, then OPT(I) = K; otherwise, OPT(I) > α K. The assumedexistence of an α-approximation polynomial time algorithm for Min-EWCP implies the polynomialtime solvability of Clique, i.e., the equality P = NP.

Proposition 2 is proved.

Remark 1. Actually, we have shown that a polynomial time O(2n)-approximation algorithm(where n is the order of the graph G′) for Min-EWCP cannot exist unless P = NP.

Let us show that Min-EWCP on a graph G = (V,E, a, c) with weight functions a : V → Q andc : E → Q can be reduced to Min-EWCP on the graph G = (V,E, 0, w) with zero weight function a

and appropriately modified weight function w.

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S90 EREMIN et al.

Proposition 3. Min-EWCP with the modified edge weight function

wij =

⎧⎨

aim − 1 + cij + aj

m − 1 if i �= j,

0 if i = j(2)

can be written in the equivalent form

F (C) =∑

e∈E(C)

we → minC⊂V ;|C|=m

, (3)

where E(C) is the set of edges that connect vertices from the subset C (m-clique), or in the form

F (C) =12

i∈C

j∈C

wij → minC⊂V ;|C|=m

.

Proof. Indeed, putting A = A(C) =∑

i∈Cai and combining (3) and (2), we obtain

F (C) =∑

e∈E(C)

we =12

i∈C

j∈C

wij =12

i∈C

j∈C\i

(ai + aj

m − 1+ cij

)

=12

i∈C

( ∑

j∈C\i

ai

m − 1+

j∈C\i

aj

m − 1+

j∈C\icij

)=

12

i∈C

(ai +

A − ai

m − 1+

j∈C\icij

)

=∑

i∈C

(m − 2)ai + A

2(m − 1)+

12

i∈C

j∈C\icij =

i∈C

ai +∑

e∈E(C)

ce.

Proposition 3 is proved.

Similarly to linear function extremum statement (1), Min-EWCP in terms of the modified edgeweight function takes the following form:

1≤i<j≤n

wijyij → minxi,yij∈{0,1}

(4)

on the feasible set

n∑

i=1

xi = m, xi + xj − 1 ≤ yij ≤ min{xi, xj} (i < j).

Proposition 4. If entries of the matrix c = (cij) in Min-EWCP on a graph G = (V,E, a, c)satisfy the triangle inequality, i.e., if cij + cjk ≥ cik for all i, j, and k, then entries of the matrixW = (wij) obtained according to (2) also satisfy this condition.

Proof. Indeed, by (2), for any i, j, and k,

wij + wjk =( ai

m − 1+ cij +

aj

m − 1

)+

( aj

m − 1+ cjk +

ak

m − 1

)≥ ai

m − 1+ cik +

ak

m − 1= wik.

Proposition 4 is proved.

Now, let us consider the above mentioned data analysis problem whose input is an n×n matrixof pairwise comparisons of objects.

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2-APPROXIMATION ALGORITHM S91

Row’s subset of symmetric matrix problem (RSSM). Input: a symmetric n×n matrixW = (wij) with nonnegative entries and wii = 0, a positive integer m and a positive number D.Question: does the set of rows of W contain a subset C of cardinality m such that

F (C) =12

i∈C

j∈Cwij ≤ D? (5)

It is easy to show that RSSM is polynomial time equivalent to formulation (4) of Min-EWCPin the form of property verification problem.

Let us consider some important subclass of RSSM.Problem 1. Entries of the matrix W = (wij) satisfy the metric axioms; in particular, wij +

wjk ≥ wik for any i, j, and k.

Problem 2. Entries of the matrix W = (wij) are calculated by formula (2) in accordance withthe input data of Min-EWCP: the nonnegative weight function a = (ai) and the matrix c = (cij)consisiting of squared pairwise distances in some point configuration of Euclidean space.

Let us analyze the computational complexity of Problems 1 and 2.Theorem 1. Problem 1 is NP-complete in the strong sense.Proof. The NP-completeness of Problem 1 follows from the polynomial time reducibility of

the known problem [1] on an independent set (an edgeless subgraph) of a graph to it; the latterproblem is NP-complete in the strong sense [1].

Independent set problem. Input: an undirected graph G of order n and a positive number k.Question: does the graph G contain a subgraph consisting of isolated vertices (an edgeless subgraph)of order k?

Indeed, let G = (V,E). Consider the instance of Problem 1 defined by the complete graphG′ = (V,E′) and the weight matrix W = (wij):

wij =

⎧⎪⎪⎨

⎪⎪⎩

0 if i = j,

1 if i �= j and (i, j) �∈ E,

2 otherwise.

Entries of W satisfy the triangle inequality. Let m = k and D = k(k− 1)/2. Clearly, the answer inthe constructed instance of Problem 1 is affirmative (there exists a subset of rows of C of cardinality k

satisfying inequality (5)) if and only if the subset C ⊂ V is independent in G.Theorem 1 is proved.Theorem 2. Problem 2 is NP-complete in the strong sense.

Proof. To prove the NP-completeness of Problem 2, we will need the following problem, whichis NP-complete in the strong sense [5].

Vector subset 3 (VS-3) problem. Input: a set Y = {y1, . . . , yn} of vectors from Rq, apositive integer m > 1, and a positive number D′. Question: is there a subset C′ ⊂ Y such that

y∈C′

z∈C′

‖y − z‖2 ≤ D′

under the constraint |C′| = m on the cardinality of C′?Evidently, we can reduce VS-3 to Problem 2 in polynomial time by setting wij = ‖yi −yj‖2 and

D = D′/2. Thus, Problem 2 is also NP-complete in the strong sense.Theorem 2 is proved.

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S92 EREMIN et al.

3. ANALYSIS OF AN APPROXIMATION ALGORITHM FOR PROBLEMS 1 AND 2

Consider the following approximation Algorithm A.

Step 1. For each j = 1, . . . , n, find a set Bj that consists of indices of m smallest entries inthe jth row of W including the index j itself. Define S(Bj) =

i∈Bj

wij.

Step 2. Denote by k∗ the value j for which S(Bj) takes the minimum value

S∗ = S(B∗) =∑

i∈B∗

wik∗. (6)

Take C = B∗ as an approximate solution of RSSM.

3.1. On approximate solution of Problem 1.

Theorem 3. Algorithm A finds a solution of Problem 1 with approximation guarantee 2 intime O(n2). The approximation guarantee 2 of the algorithm is asymptotically achievable.

Proof. Finding each Bj at Step 1 takes O(n) operations (e.g., using the algorithm for findingthe mth smallest number in an array of n numbers [4, Sect. 3.6]). Thus, finding the solution B∗

requires time O(n2). Let us show that this solution is at most twice worse than the optimal solution.Denote by C∗ the optimal solution of Problem 1. On the one hand, we have

2F (C∗) =∑

i∈C∗

j∈C∗

wij ≥∑

i∈C∗

j∈Bi

wij =∑

i∈C∗

S(Bi) ≥∑

i∈C∗

S(B∗) = mS∗. (7)

On the other hand, with the use of the triangle inequality, we obtain

2F (B∗) =∑

i∈B∗

j∈B∗

wij ≤∑

i∈B∗

j∈B∗

(wik∗ + wk∗j) = m∑

i∈B∗

wik∗ + m∑

j∈B∗

wk∗j = 2mS∗. (8)

Using (7) and (8), we bound the approximation guarantee of Algorithm A in Problem 1:

αA =F (B∗)F (C∗)

≤ 2mS∗

mS∗ = 2.

Let us show the asymptotic achievability of the approximation guarantee. Consider an example.

Example 1. Let n = 2m, where m > 3 is even. Let the matrix W have a block structure:

W =(

W0 W2

W2 W1

)

, where each of the matrices W0, W1, and W2 is of dimension m × m. The

main diagonal of W0 is filled with 0, all other entries of its first row and first column are 1, and allthe remaining entries of the matrix are 2. The main diagonal of W1 is filled with 0, the secondarydiagonal is filled with 2, and all the remaining entries are 1. All entries of W2 are 2. Evidently,entries of the matrix W satisfy the triangle inequality. Below, we present the matrix W for m = 6:

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2-APPROXIMATION ALGORITHM S93

W =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 1 1 1 1 2 2 2 2 2 21 0 2 2 2 2 2 2 2 2 2 21 2 0 2 2 2 2 2 2 2 2 21 2 2 0 2 2 2 2 2 2 2 21 2 2 2 0 2 2 2 2 2 2 21 2 2 2 2 0 2 2 2 2 2 22 2 2 2 2 2 0 1 1 1 1 22 2 2 2 2 2 1 0 1 1 2 12 2 2 2 2 2 1 1 0 2 1 12 2 2 2 2 2 1 1 2 0 1 12 2 2 2 2 2 1 2 1 1 0 12 2 2 2 2 2 2 1 1 1 1 0

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The algorithm finds the solution B∗ = B1 = {1, 2, . . . ,m}, and the value of the objectivefunction for this solution is m2 − 2m + 1. However, the optimum of the objective function m2/2 isattained for C∗ = {m + 1,m + 2, . . . , 2m}. Thus, the ratio tends to 2 (from below) as m increases.

Theorem 3 is proved.

3.2. On approximate solution of Problem 2. In [6], a 2-approximation algorithm withtime complexity O(qn2) for solving VS-3 was suggested. Recall that VS-3 is equivalent to min-WCP on the graph G = (V,E, a, c) with zero weight function ai and matrix c = (cij) consisting ofsquared pairwise distances between points x1, . . . , xn in Euclidean space Rq. However, in Problem 2,vectors xi are not given explicitly, and the input data are the weight function ai and entries of thematrix c = (cij); moreover, it is known that these entries are squared pairwise distances for somepoint configuration of Euclidean space.

Let us show that Algorithm A solves Problem 2 with the same approximation guarantee.

Theorem 4. Problem 2 is solved in time O(n2) with achievable approximation guarantee 2.

Proof. The complexity of Algorithm A was found in Theorem 3. Let us bound the approxi-mation guarantee of the obtained solution. On the one hand, we have

2F (C∗) =∑

i∈C∗

j∈C∗

wij ≥∑

i∈C∗

j∈Bi

wij =∑

i∈C∗

S(Bi) ≥∑

i∈C∗

S∗ = mS∗. (9)

On the other hand,F (B∗) ≤ mS∗. (10)

To prove this inequality, we need the following statements.

Proposition 5.∑

j∈B∗

i∈B∗

cij ≤ 2m∑

i∈B∗

cik∗ .

Proof. There exist points x1, . . . , xn in Euclidean space that generate the matrix c = (cij) ofsquared distances between pairs of these points: cij = ||xi − xj ||2, 1 ≤ i, j ≤ n. Let x = x(B∗) =1m

i∈B∗xi. We have

i∈B∗

cij =∑

i∈B∗

||xi−xj||2 =∑

i∈B∗

||xi−x+x−xj||2 =∑

i∈B∗

||xi−x||2+m||x−xj||2+2∑

i∈B∗

⟨xi−x, x−xj

=∑

i∈B∗

||xi − x||2 + m||x − xj||2 + 2⟨ ∑

i∈B∗

xi − mx, x − xj

⟩=

i∈B∗

||xi − x||2 + m||x − xj||2, (11)

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S94 EREMIN et al.

where⟨x, y

⟩denotes the scalar product of vectors x and y.

Sum (11) over j ∈ B∗. In view of the inequality∑

i∈B∗

||xi − x||2 ≤∑

i∈B∗

||xi − xk∗ ||2,

we have∑

j∈B∗

i∈B∗

cij =∑

j∈B∗

i∈B∗

||xi − xj||2 = m

( ∑

i∈B∗

||xi − x||2 +∑

j∈B∗

||x − xj ||2)

= 2m∑

i∈B∗

||xi − x||2 ≤ 2m∑

i∈B∗

||xi − xk∗||2 = 2m∑

i∈B∗

cik∗ .

Proposition 5 is proved.

Proposition 6.∑

i∈B∗

cik∗ ≤ S∗ − A(B∗)m

.

Proof. Indeed, using formulas (2) and (6) and the obvious chain of inequalities

A(B∗ \ k∗)m − 1

+ ak∗ ≥ A(B∗)m − 1

≥ A(B∗)m

,

we obtain∑

i∈B∗

cik∗ =∑

i∈B∗\k∗

( ai

m − 1+cik∗ +

ak∗

m − 1

)−

(A(B∗ \ k∗)m − 1

+ak∗

)≤

i∈B∗

wik∗−A(B∗)m

= S∗−A(B∗)m

.

Proposition 6 is proved.

Inequality (10) is proved with the help of Proposition 3 and the inequalities proved in Propo-sitions 5 and 6:

F (B∗) =12

i∈B∗

j∈B∗

wij = A(B∗) +12

i∈B∗

j∈B∗

cij ≤ A(B∗) + m∑

i∈B∗

cik∗ ≤ mS∗.

Inequalities (9), (10) imply a bound for the approximation guarantee of Algorithm A for Problem 2:

αA =F (B∗)F (C∗)

≤ 2mS∗

mS∗ = 2.

It remains to show that the approximation guarantee of 2 is achievable by Algorithm A.Consider the following example.

Example 2. Suppose that n = 4, m = 3, and the weights of vertices are 0. Let the edgeweight matrix be generated by the set {(0, 0), (1, 0), (−1, 0), (1/2,

√3/2)} of points on the plane:

W =

⎜⎜⎜⎝

0 1 1 11 0 4 11 4 0 31 1 3 0

⎟⎟⎟⎠

.

The optimal solution here is the subset C∗ = {1, 2, 4}; the optimal value is OPT = F (C∗) = 3.Since S(B∗

1) = S(B∗2) = 6, the following two algorithmic solutions are equally possible: B∗

1 ={1, 2, 3} and B∗

2 = {1, 2, 4}. For these solutions, F (B∗1) = 6 and F (B∗

2) = 3. For the first solution,F (B∗)/F (C∗) = 2; i.e., the approximation guarantee of 2 is achievable by the algorithm.

Theorem 4 is completely proved.

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2-APPROXIMATION ALGORITHM S95

3.3. Inapproximability of Min-EWCP. Note that Min-EWCP (4) is inapproximable inthe general case (by Proposition 2 and Remark 1). For example, the approximate solution obtainedby Algorithm A can be arbitrarily distant (in terms of the functional) from the optimal solution,which is illustrated, in particular, by the following example.

Example 3. Consider the instance of Min-EWCP defined by m = 3 and the matrix

W =

⎜⎜⎜⎝

0 2 2 22 0 1 12 1 0 x

2 1 x 0

⎟⎟⎟⎠

.

It is clear that the entries of the matrix W do not satisfy the triangle inequality for x > 2 becausew34 > w32 + w24 in this case.

It is easy to verify that C∗ = {1, 2, 3} and OPT = F (C∗) = 5 for arbitrary x > 3, whereasB∗ = B2 = {2, 3, 4} and F (B∗) = x + 2 −−−→

x→∞∞.

4. CONCLUSIONS

We investigated the computational complexity of the problem of finding a minimum clique (withrespect to the total weight of vertices and edges) of fixed order in an undirected weighted graph G

and considered some of its important subclasses. We proved the inapproximability of the problemin the general case and suggested a 2-approximation algorithm with time complexity O(n2) for thecase when the weights of vertices in the graph G are nonnegative and the weights of edges eithersatisfy the triangle inequality or are squared pairwise distances for some point configuration ofEuclidean space.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project nos. 12-01-00090, 12-01-00093, 12-01-33028-mol-a-ved, 13-01-00210, 13-07-00070, and 13-07-00181) and byintegration projects of the Ural and Siberian Branches of the Russian Academy of Sciences (projectnos. 12-P1-1016, 12-S1-1017/1, and 7B). The fifth author was also supported by the Program forState Support of Leading Universities of the Russian Federation (agreement no. 02.A03.21.0006 ofAugust 27, 2013).

REFERENCES

1. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness(Freeman, San Fransisco, 1979).

2. J. H̊astad, “Clique is hard to approximate within n1−ε,” Acta Math. 182 (1), 105–142 (1999).

3. K. Park, K. Lee, and S. Park, “An extended formulation approach to the edge-weighted maximal clique problem,”Europ. J. Oper. Res. 95 (3), 671–682 (1996).

4. A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading,MA, 1974; Mir, Moscow, 1979).

5. A. V. Kel’manov and A. V. Pyatkin, “NP-completeness of some problems of choosing a vector subset,” J. Appl.Ind. Math. 5 (3), 352–357 (2011).

6. A. V. Kel’manov and S. M. Romanchenko, “An approximation algorithm for solving a problem of search for avector subset,” J. Appl. Ind. Math. 6 (4), 90–96 (2012).

Translated by Ya. Salii

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