CC BY
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Труды ИСП РАН, том 27, вып. 5, 2015 г..
Approximating chromatic sum coloring of bipartite graphs in expected polynomial time
1 A.S. Asratian <[email protected]>
2 N.N. Kuzyurin <[email protected]>
1 Linkopings Universitet, Department of Mathematics, Sweden, 581 83,
2 ISP RAS, Russia, 109004, Moscow, Solzhenitsyna, 25
Abstract. It is known that if P^NP the sum coloring problem cannot be approximated within 1 + £ for some constant £. We propose for arbitrary small £>0 an approximation scheme for this problem that works in expected polynomial time.
Keywords: sum coloring problem, bipartite graphs, expected polynomial time
1. Introduction
Let G = (Vt,V2, E) be a bipartite graph with n + m vertices such that | Vx \—m, \V2\=n, m < n . By a coloring we mean a mapping: ciV^V-z —> {1,2 ,...,n+m}.
A coloring is proper if c(v) f c(u) whenever (u, v) <s E .
Let S(G,C) = y^ r c(v). By a chromatic sum we mean S(G) = mmcS(G ,c)
where minimum is taken over all proper colorings of G . The problem of finding S(G) is called the SUM COLORING PROBLEM.
The notion of chromatic sum was first introduced in [6] where it was shown that the SUM COLORING PROBLEM is NP-complete on arbitrary graphs. A few b -approximation algorithms which find a coloring C with S(G, c) < b ■ S(G) were
presented. In [7] a 10/9-approximation polynomial algorithm for die SUM COLORING PROBLEM on any bipartite graph was described. This result was improved in [8] where an 27/26-approximation algorithm for the same problem was constructed. On the other side, in [7] the authors have shown that there exists £ > 0, such that there is no (1 + s) -approximation polynomial algorithm for the SUM COLORING PROBLEM on bipartite graphs, unless P = NP.
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In this paper we present for any positive S an (1 + s) -approximation algorithm for this problem with expected polynomial time. The probabilistic distribution is uniform over all bipartite graphs with N vertices, N = n + m, m<n . Note that the first example of approximation algorithm with expected polynomial time guaranteeing approximation ratio better than inapproximability threshold in the worst case was presented in [9]. Probabilistic analysis of algorithms for random graphs is the focus of much research now [1-5, 9].
2. Approximation scheme with expected polynomial time
Let N = n + m . We consider now a straightforward approach testing all possible colorings of G and choosing the one with the best possible color sum.
Algorithm 1. Test all possible vertex colorings of a bipartite graph and choose a proper coloring with minimum color sum.
Lemma 1. The time complexity of Algorithm 1 is 0(NN ) = 0((2n)ln ).
Let 8 be a positive number, 0 < S < 1 and
V;={v^Vx-.{\-8)^<degv<(\ + 5)^},
У’г =(vgV2 :(l-8)j<degv<(l + 8)^j,
v\ = v,\K
v'2 = v2\v.i
2.1 Algorithm VERTEX-COLOR.
Input: A bipartite graph G = (l\, V2, E) such that | Vx |= m, \ V2\ = П, m<n , and a parameter e > 0.
Output: A proper coloring c of G such that S(G) < S(G,c) < (1 + s)S(G) .
1. If s < тах{40и~0 5,и~° 2,50и~0 3} then goto 7.
2. If m < no s then goto 7.
3. Set J = min{—, — -nT03}.
50 50
4. Count the number tx =| V\ \, and t2 =| V'2 \.
5. If tx > -J~n or t2 > n0A then goto 7.
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6. Color V2 by color 1 and color l \ by color 2 and STOP.
7. Run Algorithm 1 and STOP.
Theorem 1. For any fixed £ > 0 Algorithm VERTEX-COLOR finds a proper coloring within 1 + £ of the optimum color sum in expected polynomial time.
Proof. Note that at step 2 and step 5 of the algorithm we get S(G,c) = n + 2m using very simple coloring strategy. The main idea of the proof is to extract sufficiently large almost regular bipartite subgraph (j = (l\\ V2,E') of G such that for any vgE/ (1 — S')r < deg v < (1 + S')r, and for any vgV2 (1 — S')k < deg v < (1 + S')k . Such an almost regular subgraph can guarantee a tight lower bound on S(G) close to the upper bound S(G) <n + 2m . The main difficulty is to estimate the probability that the size of such subgraph is large enough.
We use m' and n' for denoting | V/1 and | V2 | respectively.
Lemma 2. For any 0 < 5' < — and an induced subgraph G' = (l\', V2, E') as above
ri + 2m' -10S'm' < S(G') <n' + 2m'.
Proof of Lemma 2. The upper bound is evident (we color V[ by color 2 and color V2 by color 1). To prove the lower bound we use the folowing inequalities
(1 + 8')г^ф) + (\ + 5')к^с{у)> J] (ф) + ф))>
vepy e=(«,v)Gfi'
>Ъ\Е'\>Ъг(\-д')т'.
This implies the inequality
Yf(y) + - 2C(V) -3m> \—T- - 3m'(1 - 2S')-
к
Adding to both parts of the inequality (1-rc(v) and taking into account
that c(v) > 1 for any V we obtain that for any proper coloring c of G'
S(G',c) = ^c(v)+^c(v)>3m'~6S,m, + (l—)^c(v)>
veV~ Г vsKi
veV:
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> 2m' + m' -6S'm' + (!- — )ri = 2m' + ri + m' - 6S'm' - —ri >
r r
2m’ + ri + m'~ 6 S'rri - m’ - 4S'm' = ri + 2m' -1 OS'm'.
Here we used the inequality m'r{\ + S') > n'k(l - S') which for any
0 < S' < — implies 2
—ri < m'^ + ^ = m'( 1 + —^ ) < m'(l + 4S'). r 1 -S' 1 -S'
The proof of Lemma 2 is complete.
Now we estimate the size of G'.
Lemma 3. There is c > 0 depending on S such that
Pr{ | V'2\>yfc} < exp{Jn logn-cn3 2}.
Pr{| V\ |> w04} < exp{w04 logn-cri 2}.
Proof. We need the following lemma.
Lemma ([5]). Let x1,...,xn be independent random variables such that X takes two values: 0 and 1, and Pr{xi = 1} = p, Рг{х{ = 0} = 1 - p.
Let X = ^ xf and EX = np . Then the following inequalities hold: for any 8 > 0
Pr{X-EX < -SEX) < exp{-(£2/2)£T}, for any 0<£<1
Pr{X -EX> SEX} < exp{-(^2 /3)EX).
Using this Lemma we have for V e V(:
Pr{d{v) < n(\-5)2) < ехр{-(^2/2)и/2},
Pr{d{v) > n(\ + S)2) < ехр{-(с^2/3)и/2}.
We give the proof for Vi. The proof for Vi is similar.
To do this we estimate the following probability:
Pr{| V'i |>k}<n (Pr{fixed kxverticesv in V'2 have d(v) < (1 - 5)n!2} ■
к
Pr {fixed k2verticesv in V' 2 have d (v) > {1 + S)n!2}),
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where к — k1+k2. Using the Lemma and taking into account independence of the corresponding events we have
Pr{fixed kxverticesv in V2 have d(v) < (1 - 5)n/2\ < exp{-(S2/3)ktm/2 J < exp{-cmkfi,
Pr {fixed k2verticesv in V'2 have d(v) > (1 + S)m/2} < exp{-(S2/3)k2m/2} < exp{-cmk2}, where c depends on S .
Letting in the last inequalities к = n° 4 we obtain
Pr{I V'21>k}<n exp{-cm(kx +k2)}<
к
exp{& log n - cmk} < exp{«°4 log n-cn12}.
To finish the proof of Theorem 1 it is necessary to estimate the approximation ratio of the algorithm VERTEX-COLOR and its expected running time.
2.2 Approximation ratio
If the algorithm terminates at step 2 then we use the inequality
n + m< S(G) < n + 2m.
This gives that for the proper coloring c obtained at step 2
S(G, c) = n + 2m< S(G) ■ = ^(G)(l + <
n+m n+m
<S(G)(\ + n02)<S(G)(\ + s),
because s > n 0 2 (in the opposite case the algorithm always finds an optimal solution at step 7).
Because at step 7 we always find an optimal solution it is sufficient to estimate approximation ratio for step 6. To do this we use Lemma 2. If the algorithm
terminates at step 6 then tx < *Jn and t2 < n0 4. Thus we have n' = n — tx >n — yfn . m' = m - i2>m- -Jn . Because the degree of a vertex in G' can decrease by at most -\fn we can estimate S' as follows:
d,gv>(l^)^-^ = (l-^)f,
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which implies 8' — 8 +
2 4n
m
By Lemma 2
n + 2m -1-tx-t2< S(G') < S(G) <n + 2m.
This implies the inequality
n + 2m -108m -23~Jn < S(G) <n + 2m, and then the inequality
(n + 2m)(\-\08-Щ=) < S{G) <n + 2m.
4n
Thus, for the coloring c that the algorithm outputs at step 6 the following inequality holds
S(G, c) < S(G)( 1 -108- 4=У ■
ЫП
Now we use the following technical lemma.
1 s
Lemma. Let 0 < 8 < mini—,—}, s > AQn~0 5. Then
50 50
(1-10 8-^jLy1 <l + £.
у/ll
Proof. We have
25
(1-10£—7=)-(l + £)>l yjtt
This is equivalent to
25
e-108(l + s)—j=(\ + s)
V и
e-(\ + s)(108+^)>0.
This implies
8 25
---->10£ + -=.
\ + e
Taking into account the inequality 8 < el50 we have
1200
n>——
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This inequality follows from the condition of the Lemma: s > AOn 0 5.
2.3 Expected running time
Step 4 is performed in quadratic (in n) time. By Lemmas 1 and 3 the expected time of step 7 is at most
0((2n)ln) exp{yfn logn -cn12} <
cexp{2n\og2n + y[n\ogn-cn12} —» 0 as n tends to infinity.
References
[1] . B. Bollobas, Random graphs, second ed., Cambridge Univ. Press, Cambridge, 2001.
[2] . B. Bollobas, The chromatic number of random graphs, Combinatorica, 1988, v. 8, c. 49-
55.
[3] . A. Frieze,C. McDiarmid, Algorithmic theory of random graphs, Random Structures and
Algorithms, Jan.-March 1997, v. 10, n. 1-2, c. 5^42.
[4] . L. Kucera, The greedy coloring is a bad probabilistic algorithm, J. of Algorithms, Dec.
1991, v. 12, n. 4, c. 674-684.
[5] . R. Motwani and P. Raghavan, Randomized algorithms, Cambridge Univ. Press, 1995.
[6] . E. Kubicka, A.J. Schwenk, An introduction to chromatic sums, Proc. of ACM Computer
Science Conference, 1989, с. 39A5.
[7] . A. Bar-Noy, G. Kortsarz, Minimum color sum of bipartite graphs, J. of Algorithms,
1998, v. 28, c. 339-365.
[8] . K. Giaro, R. Janczewski, M. Kubale, M. Malafiejski, A 27/26-approximation algorithm
for the chromatic sum coloring of bipartite graphs, Proc. APPROX 2002, LNCS v. 2462, c. 135-145.
[9] . M. Krivelevich , V.H. Vu, Approximating the independence number and the chromatic
number in expected polynomial time, J. Combin. Optimization, 2002, v. 6, c. 143-155.
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Trudy ISP RAN [The Proceedings of ISP RAS], vol. 27, issue 5, 2015.
Приближенный алгоритм для хроматической раскраски двудольных графов за полиномиальное в среднем
время
гА.С. Асратян <[email protected]>
2Н.Н. Кузюрин <[email protected]>
1 Linkopings Universitet, Department of Mathematics, Швеция, 581 83 2 ИСП РАН, Россия, 109004, Москва, ул. Солженицына, 25
Аннотация. Известно что если P^NP то задача аппроксимации суммарной раскраски двудольных графов не может быть осуществлена в полиномиальное время с точностью 1 + £ для некоторой константы £. Мы предлагаем для сколь угодно малого £>() приближенный алгоритм для данной проблемы который работает за полиномиальное в среднем время.
Ключевые слова, проблема хроматической раскраски, двудольные графы, полиномиальное в среднем время
Литература
[1] . В. Bollobas, Random graphs, second ed., Cambridge Univ. Press, Cambridge, 2001.
[2] . B. Bollobas, The chromatic number of random graphs, Combinatorica, 1988, v. 8, c. 49-
55.
[3] . [1] A. Frieze,C. McDiarmid, Algorithmic theory of random graphs, Random Structures
and Algorithms, Jan.-March 1997, v. 10, n. 1-2, c. 5-42.
[4] . L. Kucera, The greedy coloring is a bad probabilistic algorithm, J. of Algorithms, Dec.
1991, v. 12, n. 4, c. 674-684.
[5] . R. Motwani andP. Raghavan, Randomized algorithms, Cambridge Univ. Press, 1995.
[6] . E. Kubicka, A. J. Schwenk, An introduction to chromatic sums, Proc. of ACM Computer
Science Conference, 1989, c. 39U5.
[7] . A. Bar-Noy, G. Kortsarz, Minimum color sum of bipartite graphs, J. of Algorithms,
1998, v. 28, c. 339-365.
[8] . K. Giaro, R. Janczewski, M. Kubale, M. Malafiejski, A 27/26-approximation algorithm
for the chromatic sum coloring of bipartite graphs, Proc. APPROX 2002, LNCS v. 2462, c. 135-145.
[9] . M. Krivelevich , V.H. Vu, Approximating the independence number and the chromatic
number in expected polynomial time, J. Combin. Optimization, 2002, v. 6, c. 143-155.
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