Random assignment with integer costs

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1 Radom assigmet with iteger costs Robert Parviaie Departmet of Mathematics, Uppsala Uiversity P.O. Box 480, SE-7506 Uppsala, Swede Jue 4, 200 Abstract The radom assigmet problem is to miimize the cost of a assigmet i a matrix of radom costs. I this paper we study this problem for some iteger valued cost distributios. We cosider both uiform distributios o, 2,..., m, for m = or 2, ad radom permutatios of, 2,..., for each row, or of, 2,..., 2 for the whole matrix. We fid the limit of the expected cost for the 2 cases, ad prove bouds for the cases. This is doe by simple couplig argumets together with Aldous recet results for the cotiuous case. We also preset a simulatio study of these cases. Itroductio I the assigmet problem we are to choose elemets from a matrix C of costs, oe elemet from each row ad each colum, i such a way that the total cost is miimized. I other words, we are lookig for a permutatio π, that miimizes Z = C iπi. If we let the elemets of C be radom variables, we have the radom assigmet problem. Traditioally, the radom costs have bee idepedet, idetically distributed, with the expoetial or the uiform distributio. Whe the costs are i.i.d. expoetial mea there are strog cojectures for the more geeral case of k-assigmet from a m cost matrix. Let Z k, m, deote the miimal cost. Mézard ad Parisi [8], [9], cojectured that lim EZ,, = π 2 /6. This was prove by Aldous []. Parisi [] has also cojectured that EZ,, = i 2, which was improved by Coppersmith ad Sorki [3] to EZ k, m, = m i j. i+j<k

2 The last cojecture was prove by Alm ad Sorki [2] for k 4, k = m = 5, ad k = m = = 6. Liusso ad Wästlud [7] exteded this to k 6, ad k = m = = 7.. Discrete variats We will study four discrete variats of the radom assigmet problem. Case I Each row i C is a idepedet radom permutatio of {, 2,..., }, chose uiformly from the set of all permutatios. Case II Each elemet i C is a idepedet radom umber, chose uiformly from {, 2,..., }. Case III C is a radom permutatio of {, 2,..., 2 } chose uiformly. Case IV Each elemet i C is a idepedet radom umber, chose uiformly from {, 2,..., 2 }. I the first two cases we ormalize by, ad i cases III ad IV by 2, thus cosiderig the problem of miimizig Z = C iπi or Z = 2 C iπi. i The radom miimal costs will be deoted by Zi, for the four discrete cases, ad by Zc i the case of cotiuous costs. I [], Aldous proves the followig theorems, valid for ay o-egative cotiuous distributio, such that the desity of the idepedet costs have value at 0. Let π deote the permutatio givig a optimal assigmet. Theorem.. lim EZ c = π2 6. Theorem.2. C iπi coverges i distributio. The limit distributio has desity hx = e x e x + x e x 2, 0 x <. Theorem.3. lim P C iπi is the kth smallest elemet of the ith row i C = 2 k. Remark. I a simulatio study i [0], Oli oted that, eve for as small dimesios as = 50, the row rak distributio is surprisigly close to the above. i 2 Couplig argumets I this sectio we will prove the followig theorem. 2

3 Theorem 2.. Let EZi = lim EZi. The π 2 6 EZ 2, π EZ 2 π , EZ 3 = π2 6, EZ 4 = π2 6. The idea is to compare a discrete case of the problem with the case of cotiuous uiform costs. We wat to geerate matrices for both cases simultaeously, such that a optimal assigmet for oe matrix is close to optimal for the other. Whe we say that π is optimal for the matrix C, we mea that π is a permutatio givig a optimal assigmet for the radom assigmet problem, with cost matrix C. 2. Case I Let U be a matrix of i.i.d. U0, uiform o 0, radom variables. It will be coveiet to deote the rows of U by U i. We wat to use U to get a idepedet radom permutatio for each row. To achieve this, we ca use the row raks of the matrix U. If we let P i j = rak U i j, each P i will be a idepedet radom permutatio, chose uiformly from the set of all permutatios. By Theorem.3 we have, i lim P rak U j = k = 2 k. This gives, if π is the optimal assigmet for U, lim EZ lim E i P i πi = lim Erak U j = 2. For a lower boud, assume that, for i, P i is a radom permutatio of {, 2,..., }, ad that V is a matrix with i.i.d. U0, radom variables as elemets. We will ow use the permutatios P i to rearrage the rows of V. This will give us aother matrix, U, also with i.i.d. U0, elemets, such that U ij is close to P i j. To be precise, let U i j = V i P ij = the P ijth smallest elemet i row i of V, ad ote that Ek V i i k = k k/ + = k/ +, sice V k / is Beta + k +, k. We therefore have, for all permutatios π, EP i πi U i πi > 0. Now assume that π is a optimal assigmet for the discrete problem. The cost ca the be bouded below by the cost of the problem with cost matrix U: EZ = E πi P i πi = E U i πi + P iπi U > 3

4 2.2 Case II > E U i πi EZ c π2 6. Let U ij be i.i.d. U0,. To get i.i.d. radom variables from the discrete uiform distributio o {, 2,..., }, we ca simply take the iteger part of U ij ad add. Let Y ij = [U ij ] +, where [x] deotes the iteger part of x. The Y ij are i.i.d. with the desired distributio, ad the differeces Y ij U ij are uiform o 0,. Assume that π is a optimal assigmet for Y. The we still have Y iπi U iπi U0, ad EY iπi U iπi = /2, ad for the lower boud of EZ2, EZ2 = E Y iπi = E U iπi + 2 EZ c + 2. Now for the other directio. Assume that π is the optimal assigmet for U. Svate Jaso [5] has calculated the expectatio of the fractioal part of oe elemet i the optimal assigmet, {U iπi } = U iπi [U iπi ], with respect to the limit distributio, give by Theorem.2. lim EU iπi [U iπi ] = 0 {x}hxdx = k= π 2 sih 2 2π 2 k = 24 + c, where c Let Z2 π be the cost of Y give by the assigmet π. lim EZ c Z2 π = lim E U iπi Y iπi = lim EU π [U π ] = c. Sice Z2 Zπ 2, we get the upper boud 2.3 Case III EZ 2 π This is similar to the first case, but for ease of otatio we cosider a vector of 2 elemets istead of a matrix. Give a radom permutatio P of {, 2,..., 2 }, ad a vector V of 2 i.i.d. U0, 2 radom variables, let U i be the P ith smallest elemet of V, that is, U i = V P i. Coversely, give radom variables U i, i 2, i.i.d. U0, 2, defie the radom permutatio by P k = rak U k. 4

5 This gives our desired relatios betwee U ad P. By otig that, sice V k / 2 is Beta 2 + k +, k distributed, Ek V k = k 2 +, we also have for i i the optimal assigmet for either case 2 + EP i U i Now, if π is optimal for P, EZ3 = E 2 P i Ui + Ui i π i π EZ c + 2 +, ad if π is optimal for U, EZc = E 2 Ui P i + P i EZ Ad by lettig ted to ifiity, we get the limit 2.4 Case IV EZ3 = lim EZ π2 3 = 6. As i the secod case, give the i.i.d. uiform 0, 2 variables U ij, defie X ij ad Y ij by X ij = [U ij ], Y ij = X ij +. If π is optimal for Y, Z 4 = 2 Y π + + Y π 2 U π + + U π Z c. If π is optimal for U, Z c = 2 U π + + U π 2 X π + + X π Z 4. Combiig this, we get by lettig ted to ifiity 3 Simulatio EZ 4 = lim EZ 4 = π2 6. The primary purpose of the simulatio study is of course to estimate the expected miimal cost. Besides that, we look at the variace of the expected miimal cost, as well as the row rak distributio. To solve the realizatios, we used a algorithm by Joker ad Volgeat [6]. I a recet survey [4], it came out as oe of the fastest available algorithms for 5

6 cost dimesio Figure : Simulatio results, case I problems like ours. Source code writte by Joker is available o the Iteret, ad a C++ versio was used for these simulatios. The algorithm has time complexity O 3. Beside the dimesio, the time also depeds o the size of the matrix elemets, which makes the simulatios of cases III ad IV more time cosumig. cost dimesio Figure 2: Simulatio results, case II As a idicatio of how fast the implemeted algorithm really is, we ote that i the permutatio cases, the geeratio of the matrices takes about the same time as solvig the assigmet problem. I the idepedet cases the proportio of the time, spet geeratig the matrices, is about , depedig o the dimesio. A istace of dimesio 000 is solved i less tha a secod for all cases. For cases I ad II it takes about secods to solve the problem with dimesio 0000, ad 30 secods to geerate the matrix. Almost 400 MB of RAM is eeded for this dimesio. The high dimesio cases was ru o a computer with two 000 MHz Petium III processors ad 2 GB of RAM. 6

7 cost case III case IV dimesio Figure 3: Simulatio results, cases III ad IV 3. Results 3.. Mea The results are summarized i Tables 4 ad Figures 3. Note that i the tables is the umber of realizatios. For case I ad case II we simulated problems with dimesios up to The umber of realizatios varies betwee ad We see that the estimated meas stabilize quite fast. The differece betwee dimesios 2000 ad 0000 is of order 0 4, the same order as the stadard error. The 2 cases III ad IV behaves as expected. The mea icreases icely towards π 2 /6, with case IV slightly ahead. Sice these cases are more time cosumig, ad the limit is kow to be π 2 /6, we was cotet with simulatios up to dimesio dimesio Variace case I case II case III case IV dimesio Figure 4: Estimated variace 7

8 3..2 Variace Alm ad Sorki [2] cojectures that the variace i the expoetial case is 2/ + Olog / 2. It is atural to suspect the same behavior i all our four cases. Figure 4 shows times the estimated variace plotted agaist. It is iterestig to ote is that the variace i the permutatio cases is about half of that i the idepedet cases. 0 p, logscale case I case II case III, case IV, ad 0.5 k k Figure 5: Estimated rak distributio, log-scale 3..3 The rak distributio I the cotiuous cases, the limitig rak distributio is geometric, with parameter /2. For compariso, we geerated 000 matrices of dimesio 2000 for each discrete case, ad determied the rak of every elemet i the optimal assigmet give by the program. Optimal assigmets are ot ecessarily uique. I the case of ties, we gave the elemet the lowest rak. As suspected, cases III ad IV seems to have the same limitig distributio as i the cotiuous case. Also i case II a geometric distributio, but with extra weight o, fit the data very well. For case I the picture looks a bit differet. Whe plotted o a logarithmic scale, Figure 5 we o loger get a straight lie, but a slightly cocave curve. I this scale, a polyomial i k of degree 2 fit the data well. Ackowledgemet I would like to thak Svate Jaso for suggestig both the problem ad the couplig approach. Refereces [] D. J. Aldous. The ζ2 limit i the radom assigmet problem. To appear i Radom Structures ad Algorithms. 8

9 [2] S. E. Alm ad G. B. Sorki. Exact expectatios ad distributios i the radom assigmet problem. To appear i Combiatorics, Probability ad Computig. [3] D. Coppersmith ad G. B. Sorki. Costructive bouds ad exact expectatios for the radom assigmet problem. Radom Structures ad Algorithms, 52:3 44, 999. [4] M. Dell Amico ad P. Toth. Algorithms ad codes for dese assigmet problems: the state of the art. Discrete Appl. Math., 00-2:7 48, [5] S. Jaso, 200. Persoal commuicatio. [6] R. Joker ad A. Volgeat. A shortest augmetig path algorithm for dese ad sparse liear assigmet problems. Computig, 384: , 987. [7] S. Liusso ad J. Wästlud. A geeralizatio of the radom assigmet problem. Preprit, [8] M. Mézard ad G. Parisi. Replicas ad optimizatio. Joural de Physiques Lettres, 46:77 778, 985. [9] M. Mézard ad G. Parisi. O the solutio of the radom lik matchig problem. Joural de Physiques Lettres, 48:45 459, 987. [0] B. Oli. Asymptotic properties of radom assigmet problems. PhD thesis, Kugliga Tekiska Högskola, Stockholm, Swede, 992. [] G. Parisi. A cojecture o radom bipartite matchig. Preprit, 998. Table : Simulatio results, case I. dimesio mea std. dev. s.e. mea

10 Table 2: Simulatio results, case II. dimesio mea std. dev. s.e. mea Table 3: Simulatio results, case III. dimesio mea std. dev. s.e. mea Table 4: Simulatio results, case IV. dimesio mea std. dev. s.e. mea

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio