1 Online Weighted Bipartite Matching with Capacity Constraints 59 2 60 3 61 4 Hao Wang, Zhenzhen Yan, Xiaohui Bei 62 5 Nanyang Technological University 63 6 Singapore 64 7 65 8 66 ABSTRACT Once the capacity is used up, it becomes unavailable in demand ful- 9 67 fillment. This new generation of ridesharing significantly increases 10 We investigate an online edge-weighted bipartite matching problem 68 vehicle occupancy rates and the efficiency of urban transportation 11 with capacity constraints. In this problem, the supply vertices are 69 systems, consequently reducing congestion and pollution. 12 offline with different capacities. Demand vertices arrive online and 70 In the grab hitch platform, drivers are not allowed to pick up 13 each consumes a certain amount of resources. The goal is to maxi- 71 passengers by themselves via self-arrangements, but can only take 14 mize the total weight of the matching. This framework can capture 72 ride requests assigned by the platform. Therefore, one key problem 15 several real-world applications such as the trip-vehicle assignment 73 faced by the platform is to automatically match ride requests to 16 problem in ridesharing. We model the offline optimization problem 74 available drivers in real-time so as to maximize the total profit. 17 as a deterministic linear program and provide several randomized 75 A rich literature has been devoted to the study of online matching 18 online algorithms based on the solution to the offline linear pro- 76 algorithms with different models and under different assumptions. 19 gram. We analyze the performance guarantee of each algorithm 77 Karp et al. [19] first studied the online bipartite matching problem 20 measured by its competitive ratio. Importantly, we introduce a re- 78 to maximize the number of matches under the assumption that the 21 solving heuristic that periodically re-computes the offline linear 79 arrival process is determined by the adversary. In their paper, each 22 program and uses the updated offline solution to guide the online 80 resource has a single unit of capacity and each request is assumed 23 algorithm decisions. We find that the algorithm’s competitive ratio 81 to consume only one unit of resource. Feldman et al. [14] revisited 24 can be improved when re-solving at carefully selected time steps. Fi- 82 the problem by assuming the arrival process to be stochastic, that 25 nally, we investigate the value of the demand distribution in further 83 is, the arrival of online vertices follows a known independent and 26 improving the algorithm efficiency. 84 identical distribution (i.i.d). Brubach et al. [8] extended the earlier 27 85 work to the model to maximize the vertex-weighted sum of matches 28 86 KEYWORDS and the edge-weighted sum of matches. They again assumed unit 29 87 online bipartite matching, randomized algorithm, re-solving heuris- demand size for each online request and unit capacity for each 30 88 tic, competitive ratio offline resource. However, in practice, the resources often have 31 89 general capacities, and each request may consume multiple units 32 90 of capacity in one match. The ride hitch problem described earlier 33 91 is an example. Different cars have different number of vacancies to 34 1 INTRODUCTION 92 accommodate passengers. Each ride request could involve multiple 35 In a typical online bipartite matching problem, requests arrive se- 93 passengers, hence occupy multiple vacancies. The model studied in 36 quentially following some probability distribution. Upon the arrival 94 this paper considers such a general problem settings. We assume 37 of each request, a decision has to be made to either match the re- 95 there are multiple types of resources and each type of resource 38 quest to an appropriate resource, or reject it. If the request gets 96 has a general capacity. Each online request consumes a single type 39 matched, it consumes a certain amount of resources. Resources 97 of resource but by multiple units. We also assume an i.i.d arrival 40 can be replenishable or non-replenishable. Each match made be- 98 process. Our goal is to maximize the edge-weighted sum of matches. 41 tween the request and the resource generates a profit. The goal is 99 To the best of our knowledge, this is the first paper considering the 42 to maximize the total profit generated from all matches. 100 online matching problem in such a general setting. 43 The online bipartite matching has been widely applied to var- 101 44 ious resource allocation problems. Examples include airline seat 102 45 allocation, clinic appointment slot allocation, and car allocation in Our Contribution 103 46 the online ride hitch problem. Particularly, ride hitch is a recent To solve this general online matching problem, we start by propos- 104 47 innovation of ride sharing. It refers to a mode of transportation in ing a simple randomized algorithm based on linear program round- 105 48 which private car drivers offer to share their journeys to multiple ing. The algorithm allocates an appropriate resource to each request 106 49 passengers based on coordination through a centralized dispatch. with certain probability that is based on the solution of an offline 107 50 For example, drivers may share part of their ride on the way to linear program. We analyze the performance of this simple algo- 108 51 work with other passengers who have similar itineraries, and the rithm. Following the convention in the literature, we measure the 109 52 drivers will receive remuneration to compensate the petrol and efficiency of an online algorithm by the competitive ratio, which is 110 53 labor costs. An example is the grab hitch service launched by Grab defined as the total profit generated from the algorithm, dividedby 111 54 in 2015, the leading super app in Southeast Asia. A ride request the maximum profit achievable if full information on the arrival 112 55 may involve multiple passengers. In each trip, a driver could take of demand requests is known beforehand. Next, as the main result 113 56 multiple ride requests as long as the capacity permits. Each ride of this paper, we introduce a re-solving heuristic to the random- 114 57 trip of a driver can be regarded as a non-replenishable resource. ized algorithm. The idea of re-solving is to periodically re-compute 115 58 1 116 Conference’17, July 2017, Washington, DC, USA Wang, et al.
117 the offline linear program and uses the updated offline solutionto related work is Xu et al. [27].In their paper, there are multiple types 175 118 guide the online algorithm. We show that re-solving at a right time of resources and each request could consume at most one unit of 176 119 could help significantly improve the performance of the algorithm. each type. The objective is to maximize the vertex-weighted sum 177 120 1 178 We also investigate the value of demand distribution of the online of matches. They designed an algorithm which achieves 4∆ com- 121 request to further improve the algorithm’s efficiency. petitive ratio, where ∆ denotes the maximal number of resources 179 122 Finally, we conduct extensive experimental studies to test the requested by arrivals. Although their paper shares a similar setting 180 123 efficiency of our proposed algorithms. On average, our online algo- to ours, our paper distinguishes from theirs in the following aspects: 181 124 rithm achieves 70% − 80% of the optimal profit on both synthetic First, the profit in our paper is defined on edges instead of vertices. 182 125 and real-world datasets. In particular, the proposed randomized The edge-weighted matching is known to be much more nebulous 183 126 algorithm with the re-solving heuristic significantly outperforms than the vertex-weighted case [13]. Second, we assume each arrival 184 127 all the other algorithms. We observe that by re-solving the linear only requests one type of resource but could consume multiple 185 128 program at our proposed time, the profit obtained is increased by units of resources. 186 129 almost 20% compared with the standard randomized online algo- Another relevant problem to online bipartite matching is the 187 130 rithm. We also show that the advantage of our proposed algorithms online generalized assignment problem. In the online generalized 188 131 becomes more salient when the demand in the market increases. assignment problem, there are m (static) bins each with a capacity 189 132 We summarize our main contributions as follows: limit. Items arrive online and consume some capacity of the assigned 190 133 (1) We solve a general online bipartite matching problem where bins. Alaei et al. [4] provided an algorithm for this problem with 191 134 − 1 192 the offline resources are equipped with multiple capacities and 1 k competitive ratio, assuming that no item consumes more than 135 each online request consumes multiple units of capacity once 1 193 k fraction of any bin’s capacity. Kesselheim et al. [20] and Naori et 136 matched. The arrival process is assumed to be i.i.d and the goal al. [26] considered the online generalized assignment problem in 194 137 is to maximize the edge-weighted sum of matches. the random arrival model and provided the best-known competitive 195 138 (2) We propose a randomization algorithm based on the solution to 1 196 ratio of 6.99 . 139 an offline linear program and establish its competitive ratioas Other extensions include generalizing the graph to a general 197 140 a function of the maximal demand among requests. In a special network structure and allowing a matching delay, i.e. the request 198 1 141 case that the maximal demand is 2, the competitive ratio is 4 , is allowed to wait for some time before being matched (c.f. Chen 199 142 which is comparable to the existing results on similar problem et al. [9], Adamczyk et al. [1], Adamczyk et al. [2], Baveja et al. 200 143 settings. [7], Mehta et al. [24], Ashlagi et al. [5], Dickerson et al. [12] and 201 144 (3) We further introduce a re-solving heuristic to the randomiza- Lowalekar et al. [21]). 202 145 tion algorithm and show that re-solving at the right time could 203 146 significantly improve the performance of the algorithm. 3 MODEL 204 147 (4) We investigate the value of demand distribution of the online 205 We define our online bipartite matching problem in a ride hitch 148 request to further improve the algorithm’s efficiency. 206 G = (U ,V , E) U 149 context. Consider a bipartite graph , where denotes 207 V 150 the set of drivers’ offers and denotes the set of possible riders’ 208 u ∈ V c 151 2 RELATED WORK requests. Each offer is associated with a capacity u , and 209 each request v has a demand d . An edge (u,v) exists for u ∈ U 152 The problem of online bipartite matching has been intensively stud- v 210 and v ∈ V if the offer u can can be matched to the request v. The 153 ied, and the literature is too vast to survey here. We provide an 211 edge is also associated with a weight (i.e. revenue) w . 154 overview of the work most directly relevant to ours. The first algo- uv 212 In our problem definition, we will treat drivers’ offers as offline 155 rithm for the single-capacity unweighted online bipartite match- 213 resources and riders’ requests as online demands. This is because 156 ing problem was given by Karp et al. [19]. They introduced the 214 1 generally, the driver has a much longer time tolerance to be matched. 157 RANKING algorithm and proved a tight 1 − competitive ratio 215 e He/She might plan to pick up some friends in the evening but put 158 with adversary online vertex arrival order. The analysis was later 216 up an offer in system in the morning. In contrast, a rider’s request 159 simplified by Devanur et al.11 [ ]. Aggarwal et al. [3] generalized 217 usually needs to be matched in seconds. If no suitable drivers can 160 the problem by considering weigthed offline vertices. Mehta et al. 218 be found within one minute, the request will be rejected. 161 [25] investigated the multi-demand case known as the AdWords 219 1 We assume an i.i.d. distribution model of request arrival over 162 problem and presented a 1 − competitive algorithm. 220 e an online time horizon of T rounds. That is, the set U is made 163 Another line of works considers the random arriving model, 221 available offline beforehand. In each round, a rider request v is 164 in which the online vertices arrives in a uniformly random order. 222 sampled with replacement from a known distribution {p } over V . 165 Karande et al. [18] and Mahdian et al. [22] independently showed v 223 The distribution is independent and identical in every round. Upon 166 that the RANKING algorithm can achieve a competitive ratio better 224 1 the arrival of request v, a decision has to be made to either reject v, 167 than 1 − in the random arrival model. Huang et al. [16] further 225 e or to match it to some neighbor offer u ∈ U that still has enough 168 generalized the analysis to the vertex-weighted setting. Devanur 226 remaining capacity. If a pair (u,v) is matched, a revenue of w is 169 and Hayes [10] presented a 1 − ϵ competitive algorithm for the uv 227 generated and the capacity of u is decreased by d . 170 AdWord problem in the random arrival model. v 228 Below is the list that summarizes the notations. 171 Finally, a third line of works, which also includes this work, as- 229 172 sumes the arrival of online vertices follows a known independent • T : total numbers of online requests. 230 173 and identical distribution [6, 14, 15, 17, 23]. In this model, a closely • pv : the probability of type v vertex in each arrival. 231 174 2 232 Online Weighted Bipartite Matching with Capacity Constraints Conference’17, July 2017, Washington, DC, USA
233 • dv : the demand of online request v. Algorithm 1: Samp(α) Algorithm 291 234 • D: the maximal demand of all online requests. Result: Online matchings M 292 235 • cu : the capacity of offer u. Solve the LP (3) and get the optimal solution y∗; 293 236 • C: the maximal capacity of all offers. 294 Time t = 1; 237 • w : the weight (i.e., revenue) associated with edge (u,v) 295 uv Matchings M = ϕ; 238 296 while t ≤ T do 239 MILP formulation. Given a realized arrival sequence s of requests, 297 Online vertex v arrives; 240 we can solve a mixed integer linear program (MILP) to optimally 298 ∗ 241 match them to the offers. Randomly choose u with probability αyuv ; 299 242 Õ if cu ≥ dv then 300 max wuv Xuv (s) Match u and v: c = c − d , M = M + (u,v); 243 (u,v)∈E u u v 301 244 Õ else 302 s.t. d X (s) ≤ c , u ∈ U 245 v uv u Reject v; 303 v:(u,v)∈E ∀ 246 (1) end 304 Õ 247 Xuv (s) ≤ Nv (s), v ∈ V t = t + 1; 305 ∀ 248 u:(u,v)∈E end 306 249 Xuv (s) ∈ {0, 1}, (u,v) ∈ E 307 250 ∀ We omit the proof due to space constraints. 308 251 Here Nv (s) denotes the number of type v vertex appeared in In the subsequent sections, we will use linear program (3) to aid 309 252 sequence s. The first set of constraint restricts the total consumption our design of the randomized online algorithms and analyze their 310 253 of each resourceu below its capacity and the second set of constraint competitive ratios. 311 254 specifies the matched request cannot exceed the total arrivals. 312 ( ) 255 We denote the optimal solution to (1) as H s and the optimal 4 A RANDOMIZED ALGORITHM — SAMP (α) 313 objective value as OFF(s). Then the expected revenue generated by 256 Our first online algorithm takes a common linear-program-rounding 314 optimally solving each possible arrival realization can be formulated 257 approach. First solve the optimal solution y∗ from (3). Then for 315 as E[OFF] = Í P(s)OFF(s), where P(s) denotes the probability of uv 258 s each arrival request v, an offer u is randomly chosen to match v 316 259 sequence s among all possible sequences. ∗ 317 with probability αyuv . Here α is a parameter that controls how 260 318 Competitive Ratio Analysis. Note that one usually cannot achieve aggressively the online algorithm makes the matches. 261 319 E[OFF] via an online algorithm due to an unforeseen circum- Let Cut denote the amount of capacity of offer u consumed by 262 320 stance in the future. In this paper, we aim to design online al- the requests before time t. We have 263 321 gorithms to achieve as a larger expected revenue as possible. To 264 t−1 t−1 322 evaluate the performance of an online algorithm L, we adopt a Õ Õ ∗ Õ Õ ∗ 265 E[Cut ] = pv′αyuv′dv′ = αyuv′dv′pv′ 323 commonly used performance criterion— competitive ratio (CR). For t ′=1 v′ ∈V t ′=1 v′ ∈V 266 (4) 324 a given sequence s, we denote the outcome achieved by an algo- t−1 267 Õ αc αc (t − 1) 325 rithm L as ALGL(s). Then the expected outcome by the algorithm is ≤ u = u 268 Í 326 E[ALGL] = s P(s)ALGL(s). The competitive ratio of an algorithm ′ T T 269 t =1 327 L is defined as 270 where the last inequality is from offline vertex’s capacity constraint. 328 E[ALGL] 271 CRL = (2) By Markov’s inequality, we can write the probability that v is 329 E[OFF] 272 matched to u at time step t as 330 273 Noticed that OFF(s) of (1) is a concave function in Nv (s) for each 331 αc 274 sequence s. By taking the expectation over all possible arriving ( , , ) ≥ ∗ − ( − ) u 332 xuv Prα u v t αpvyuv 1 t 1 sequences, replacing E[N (s)] by p T , and defining y = , T (cu − dv + 1) 275 v v uv pv T 333 E[ ] 276 we get upper bound OFF by the following linear program. This allows us to bound the expected number of times that edge 334 277 Õ (u,v) is matched by the algorithm during the first t ′ steps. 335 278 max Tpvwuvyuv 336 ( )∈ ′ 279 u,v E t 337 ′ Õ Õ cu Nα (u,v, t ) ≥ Prα (u,v, t) 280 s.t. pvdvyuv ≤ , u ∈ U 338 T t=1 281 ( )∈ ∀ 339 v: u,v E (3) ′ 282 t 340 Õ Õ ∗ αcu yuv ≤ 1, v ∈ V = αpvy 1 − (t − 1) 283 ∀ uv T (c − d + 1) 341 u:(u,v)∈E t=1 u v 284 (5) 342 y ≥ 0, (u,v) ∈ E " ′ 2 2 ′ # 285 uv t α cu t ∗ 343 ∀ ≥ − + α Tpvyuv 286 We use OPT to denote the optimal value of (3). T 2(cu − dv + 1) T 344 287 345 " ′ 2 2 ′ # 288 Proposition 3.1. OPT ≥ E[OFF], i.e., E[OFF] is upper bounded t α D t ∗ 346 ≥ − + α Tpvyuv 289 by the optimal value of (3). T 2 T 347 290 3 348 Conference’17, July 2017, Washington, DC, USA Wang, et al.
349 The last inequality holds as 1 ≤ cu ≤ C and 1 ≤ dv ≤ D and sequence has less than two v0. According to the i.i.d assumption, 407 350 ≥ 1 1 408 cu dv , which implies that for every u and v, we have q0 = 16 and q1 = 4 . Therefore we have: 351 409 cu dv − 1 D − 1 3 352 = 1 + ≤ 1 + = D. E[OFF] = E[ALG] + 2q + q = E[ALG] + 410 cu − dv + 1 cu − dv + 1 1 0 1 353 8 411 Therefore, we can further bound the expected performance of the 354 Next we study the effect of re-solving heuristic. Suppose we re-solve 412 Samp(α) algorithm as follows. 355 the linear program after the first arrival. Consider such a realized 413 Õ 356 ALGα = [wuv Nα (u,v,T )] sequence s0 = (v1,v1,v0,v0). The re-solved LP will suggest to 414 1 357 (u,v)∈E accept v0 with probability 1 and accept v1 with probability 3 from 415 358 2 2 the second arrival on. Hence the expected profit generated under 416 α D Õ ∗ α D 2 1 359 ≥ − + α Tpvwuvy ≥ − + α OPT this sequence is ×2w + (1+w) by the re-solving method, which is 417 2 uv 2 3 3 360 (u,v)∈E equivalent to say the expected loss compared to the offline optimal 418 (6) w−1 361 solution is 3 . Noticed that the probability of having a sequence 419 362 1 420 Proposition 4.1. The Samp(α) algorithm has competitive ratio s = s0 is 16 . Then we can upper bound the expected outcome of 363 ≥ 1 this re-solving method as follows 421 CR 2D . 364 422 The proof of the proposition is a straightforward result from the ′ w − 1 Õ 365 E[ALG ] ≤ P(s0) OFF(s0) − + P(s)OFF(s) 423 analysis above and is omitted here. 3 366 s,s0 424 367 w − 1 425 4.1 Re-solving Heuristic ≤ E[OFF] − 368 48 426 369 Note that in algorithm Samp(α), the linear program (3) is solved 427 By setting w > 19, we have E[ALG′] < E[ALG]. In other words, 370 once at the beginning of the algorithm. Then the solution will be 428 to re-solve LP could generate a worse performance. 371 used to guide the online algorithm matching probability throughout 429 372 the whole time span. However, in the middle of the process, due 4.1.2 Re-solving at the right time helps. In this section we focus on 430 373 to the randomness of the request arriving sequence, it is probably the problem of when to re-solve. Consider the following question: if 431 374 that the capacities of the orders are consumed disproportionately. we are only allowed to re-solve the LP once during the whole time 432 375 In such cases, the original LP can no longer capture the correct span, when should we re-solve the LP to maximize the expected 433 376 resource configuration. As such, we need to re-solve the linear performance of the algorithm? Suppose we re-solve at time t ′ = 434 377 program with the updated capacity information, and update the LP (1 − β)T = γT . Let T ′ denote the remaining time period. Then 435 378 ′ ′ 436 solution to guide the subsequent allocation. We call this refinement T = βT . Let Xu denote the remaining capacity of u at time t . Then 379 step the re-solving heuristic. the new LP becomes 437 380 438 In this section, we will analyze the re-solving heuristic. Our goal Õ ′ 381 is to decide whether and when this heuristic will improvement the max T pvwuvyuv , 439 382 algorithm performance. (u,v)∈E 440 383 Õ Xu 441 4.1.1 Re-solving does not always help. First we try to answer the s.t. pvdvyuv ≤ ′ , u ∈ U , 384 T ∀ 442 question of whether the re-solving heuristic, regardless of when it v:(u,v)∈E 385 (7) 443 is applied, always helps the algorithm to generate a better solution. Õ 386 yuv ≤ 1, v ∈ V , 444 Intuitively this may seem true. However, in the following we show ∀ 387 u:(u,v)∈E 445 via a counterexample that the re-solving heuristic may make things 388 y ≥ 0, (u,v) ∈ E 446 worse sometimes. uv 389 ∀ 447 yuv = 0, (u,v) ∈ E, Xu < dv . 390 • 1 offline vertex: c = 2 ∀ 448 391 • 2 online vertices: v0 and v1 We add the last constraint because when running the algorithm 449 • 1 392 v0: p0 = 2 , d0 = 1, w0 = w > 1 for several rounds, some offline vertex might have less remaining 450 1 393 • v1: p1 = 2 , d1 = 1, w1 = 1 capacity than the demand. In this case, the edge between them still 451 394 • T = 4 exists but the matching is not possible. Denote the optimal solution 452 1 ′∗ 395 In this case, D = 1 hence α = = 1. By solving the offline of the re-solved linear program as y and the optimal value as 453 D ′ 396 LP (3), we know that algorithm Samp will always accept v0 as long OPT . The optimal solution and optimal value of the new linear 454 397 as the capacity permits and reject v1. To understand the perfor- program depend on the remaining capacity, which is a random 455 ′∗ ′ 398 mance of algorithm Samp, let s denote the realized sequence. If variable. Hence the optimal solution y and optimal value OPT 456 399 s = (v1,v1,v1,v1), then the offline optimal profit is 2 while Samp are also random variables. To simplify the analysis, we condition on 457 400 gives 0. If there is exactly one v0 in s, the offline optimal profit is a particular realization of the remaining capacity in the subsequent 458 ′ ′∗ 401 1 + w while Samp gives w. If there are more than one v0 in s, both presentation, i.e. Xu = cu , in which case, y and optimal value 459 ′ 402 offline optimal profit and the expected profitby Samp are 2w. Let q0 OPT are deterministic. 460 ′ 403 denote the probability that s has no v0 and q1 denote the probability Let Nα (u,v,T ) be the expected number of (u,v) that is matched by 461 ∗ 404 that s has one v0. In summary, the expected profits generated by re-solved algorithm, which is to match the resource based on αy 462 ′ ′∗ ′ 405 the offline linear program and Samp are different only if the realized before t and on α1y after t . Following a similar analysis above, 463 406 4 464 Online Weighted Bipartite Matching with Capacity Constraints Conference’17, July 2017, Washington, DC, USA
′ 465 we have from the re-solve point t , the expected match between a 4.1.3 Re-solving Many Times At the Right Time Further Helps to 523 466 resource-request pair (u,v) can be analyzed as follows: Improve the Bound. Consider such a randomization algorithm. Ran- 524 467 " ′ 2 ′ ′ # domly allocate the resource based on Samp(α) until γT . After ith 525 T α cu T ′ 468 ′ ′ 2 1 ∗ re-solve, the allocation is based on Samp(αi ) and the new re-solve 526 Nα (u,v,T ) ≥ −( ′ ) ′ + ′ α1 Tpvyuv 469 T 2(cu − dv + 1) T time point is at γ proportion of the remaining time period. We call 527 (8) 470 " 2 # such an algorithm as a log-resolving algorithm with parameter γ 528 α1D ′∗ 471 ≥ − + α Tpvy to specify the re-solving time point. Denote the remaining time 529 2 1 uv 472 (i) (0) (i) 530 period at ith re-solve as T . T = T . Let Xu denote the remain- ′ 473 1 ′ ( ′) ≥ 1 ∗ (i) 531 By choosing α1 = D , we have Nα u,v,T 2D Tpvyuv . The ing capacity right before ith re-solve. Notice that Xu is a random 474 ′∗ ′ 532 second inequality holds as yuv = 0 if cu < dv , which implies variable, we first study the bound of the expected ratio between 475 ′ ′ 533 ∗ ′ ≥ cu ≤ dv −1 ≤ the optimal values of two consecutive linear programs conditioned 476 for any yuv > 0, cu dv , hence c′ −d +1 1 + c′ −d +1 dv . ( ) 534 u v u v on a realization of X i . 477 Therefore, u 535 478 2 (i) (i) 536 ′ α D 2 ∗ 1 ′∗ Proposition 4.4. Conditioned on Xu = cu , u, under the con- 479 Nα (u,v,T ) ≥ − γ + αγ Tpvyuv + Tpvyuv (9) R ∀ 537 2 2D dition that (i) << 1, we have 480 T 538 The expected outcome of re-solve algorithm is 481 OPT + 539 E i 1 c(i) ≥ (1 − γ )e−αγ R (13) 482 2 540 Õ ′ α D 2 1 ′ OPTi 483 wuv Nα (u,v,T ) ≥ − γ + αγ OPT + OPT 541 uv 2 2D 484 We omit the proof here as it is similar to Proposition 4.2. With 542 485 Then we can bound the competitive ratio from below, i.e., this proposition, we are now ready to establish our first main result 543 in the paper. 486 h 2 i ′ 2 2 ′ 544 CR′ ≥ − α D γ 2 + αγ + 1 OPT = −γ α D + γα + 1 OPT . 487 2 2D OPT 2 2D OPT 1 545 Theorem 4.5. The log-resolving algorithm with γ = D gives a 488 (10) 546 ′ 1 1−C K+1 1 − R OPT competitive ratio at least , where C = 1 − e D . This 489 Now we want to show the lower bound (LB) of OPT and check if 2D 1−C D 547 490 we need to re-solve according to this LB. We construct a solution value increases with the number of re-solving time K and converges 548 ′ 1 491 y for LP’ based on the optimal solution of LP. We denote U+ as the to D( −C) when K approaches infinity. 549 ′ 2 1 492 set of offline vertices whose capacity without change, i.e., cu = cu 550 Proof: To start the proof, we first define some notations: denote the 493 and U− as the rest of the offline vertices. Consider such a solution 551 time for ith re-solve as ti , the ratio between the maximal remaining 494 ( ∗ 552 ′ yuv u ∈ U+ capacity and the minimum demand as Ri and the optimal value of 495 y = (11) 553 uv u ∈ U the ith re-solved LP as OPT . Samp(α) is applied at the beginning 496 0 − i 554 and after ith re-solve, Samp(αi ) is applied. ALG(s,T ) denotes the ex- 497 It is easy to check that y′ is feasible for the re-solved linear program 555 pected outcome from time s to T using the designed randomization 498 (7). Therefore, 556 algorithm, which is to match the resource according to the solution 499 ′ ′ ′ ′ ′ 557 OPT T (w (U+) + w (U−)) w (U+) from the corresponding re-solved deterministic linear program in 500 ≥ = β (12) 558 OPT T (w(U ) + w(U )) w(U ) + w(U ) each time interval. We can derive a lower bound if the expected 501 + − + − 559 Í ∗ Í ′ performance of our re-solving heuristic in the following way: from 502 where wu = v ∈Ad j(u) pvwuvyuv , w(S) = u ∈S wu and w (S) = 560 Í Í ′ the beginning of the time period to the first re-solve point t1, the 503 u ∈S v ∈Ad j(u) pvwuvyuv . Bearing in mind that the remaining 561 ′ lower bound is derived from (5); from t1 onward the algorithm is 504 capacity Xu is a random variable, OPT is also random. From the 562 based on the designed randomization algorithm. Therefore, 505 analysis above, we have 563 2 2 506 ′ γ α D 564 maxu cu h OPT i −αγ R Proposition 4.2. Define R = , E ≥ (1−γ )e E[ALG(0,T )] ≥ − + γα OPT + E[ALG(t1,T )] 507 minv dv OPT 2 565 508 566 We omit the proof of the proposition due to space constraints. Hence we have 509 1 567 Based on the result in Proposition 4.2. Let α = 1 and γ = . 2 2 510 D E[ALG(0,T )] γ α D E[ALG(t1,T )] 568 We can further establish the following proposition to demonstrate OPT ≥ − 2 + γα + OPT 511 2 2 h ( ) i 569 that re-solving at γT helps to achieve a better lower bound for the = −γ α D + γα + E ALG t1,T OPT1 512 2 OPT1 OPT 570 competitive ratio. 513 571 Notice that the remaining capacity at time t1 is a random variable, 514 T 572 Proposition 4.3. Re-solving at D helps to generate a better com- hence the re-solved LP is also a random variable determined by 515 ′ − R 573 ≥ 1 1 ( − 1 ) D the remaining capacity. Hence we can reformulate the last term as petitive ratio which is CR 2D + 2D 1 D e 516 h ALG(t ,T ) OPT i h ALG(t ,T ) ( )i h OPT i 574 E 1 1 = E 1 c 1 · E 1 . From propo- 517 The first term in the established competitive ratio is exactly the OPT1 OPT OPT1 OPT 575 h i 518 one we have built for algorithm Samp. The second term is non- OPT1 −αγ R 576 sition 4.4, we have E OPT ≥ (1 − γ )e . Hence we have 519 negative as long as D ≥ 1, which indicates that to re-solve at our 577 2 2 520 proposed time will generate at least the same competitive ratio as E[ALG(0,T )] γ α D ALG(t1,T ) ( ) − 578 ≥ − +γα + E c 1 (1 −γ )e αγ R 521 Samp(α). OPT 2 OPT1 579 522 5 580 Conference’17, July 2017, Washington, DC, USA Wang, et al.
(1) h ALG(t1,T ) (1)i pql 581 Conditioning onc ,OPT is deterministic, hence E c group q, we define the demand distribution as p | = , where 639 1 OPT1 l q pq 582 Í Í 640 represents the expected performance of the randomization algo- pql = pv and pq = pql . 583 v ∈V:q(v)=q,d =l l ∈L 641 rithm. Starting from t1, we can follow a similar analysis to get the v q 584 following inequality From the definition of groups, it is easy to get 642 585 643 h ALG(t1,T ) (1)i Lemma 4.6. There exists an optimal solution to (3), such thatyuv = 586 E c ′ 644 OPT1 ′ ( ) ( ) ′ yuv is q v = q v and dv = dv . 587 γ 2α 2D h ( ) i 645 ≥ E − 1 + γα c(1) + E ALG t2,T OPT2 c(1) 588 2 1 OPT2 OPT1 Then we can revise the linear program (3) accordingly as below 646 2 2 589 γ α D ( ) h ALG(t ,T ) ( ) ( )i − max Í Tw Íp y 647 ≥ E − 1 + γα c 1 + E 2 c 1 ,c 2 (1 − γ )e α1γ R uq ql uql 590 2 1 OPT2 (u,(q,l))∈E l 648 c γ 2α 2D h i s.t. Í Í p ly ≤ u , u ∈ U 591 1 ALG(t2,T ) (1) (2) −α1γ R ql uql T 649 ≥ − + γα1 + E c ,c (1 − γ )e q ∈Q l ∈L ∀ 592 2 OPT2 q (15) 650 (14) Í y ≤ , (q,l) ∈ V 593 uql 1 651 With the same analysis we have for each i, u ∈Ad j(q,l) ∀ 594 y ≥ 0, (u, (q,l)) ∈ E 652 h ( ) i γ 2α 2D uql 595 E ALG ti,T c(1),...,c(i) ≥ − i + γα + ∀ 653 OPTi 2 i 596 We prove in the following lemma that the revised LP is equivalent 654 h ALG(t + ,T ) i E i 1 c(1),...,c(i+1) (1 − γ )e−αi γ R to (3). 597 OPTi+1 655 598 In the last period, we have Lemma 4.7. (15) is equivalent to (3). Specifically, For all v that 656 599 ( ) = , = ∗ 657 2 q v q dv l, we have an optimal solution y to (15) such that 600 ALG(tK ,T ) ( ) ( ) α D ∗ ∗ ∗ 658 1 K ≥ − K y = yuv for every adjacent vertex u, where yuv is an optimal E c ,...,c + αK uql 601 OPTK 2 solution to (3). 659 602 ALG(ti,T ) (1) (i) 660 Denote E[ | c ,...,c ] as li . Hence we have 603 OPTi In other words, we can regard the vertices in the same group and 661 γ 2α 2D capacity as a super vertex with arrival probability p and solve 604 i −αi γ R ql 662 li ≥ − 2 + γαi + li+1(1 − γ )e , l = 0,..., K − 1 605 ∀ the revised linear program. For each arrival, a resource is matched 663 α 2 D according to the revised linear program. The randomization algo- 606 and l ≥ − K + α . Set α = 1 , α = 1, i = 0,..., K − 1 and 664 K 2 K K D i rithm based on the revised linear program will generate the same 607 1 1 1 ∀ − R 665 γ = . We have l ≥ and l ≥ + l + (1 − γ )e D , i = 608 D K 2D i 2D i 1 results as the original linear program. In the following analysis, 666 0,..., K − 1. Aggregate all the inequalities, we have ∀ 609 we will focus on the revised linear program. Denote the optimal 667 K+1 ∗ R , , 610 − solution to (15) as yuql for each u q l. 668 K i 1− (1−γ )e D 1 − R 1 ′ 611 l ≥ Í (1 − γ )e D = Let Nα (u,q,l, t ) be the expected number of matched edge (u,q,l) 669 0 2D 2D − R ′ 612 i=0 1−(1−γ )e D from t = 1 to t = t . Following a similar analysis to (5), we have 670 → 1 1 = 1 1 613 2D − R 2D − R ′ 2 ′ 671 1−(1−γ )e D 1−(1− 1 )e D ′ t α cu t ∗ D ( ) ≥ −( )2 614 Nα u,q,l, t + α Tpqlyuql 672 T 2(cu − l + 1) T 615 (16) 673 t ′ α2l t ′ 616 4.1.4 Time Complexity. For each arrival, the matching decision is ≥ −( )2 ∗ 674 + α Tpqlyuql 617 done by flipping a coin based on the optimal solution to a linear pro- T 2 T 675 618 cu l−1 676 gram and checking the capacity availability. Hence the computation as ≤ 1 + ≤ l since l ≤ cu . cu −l+1 cu −l+1 619 efficiency is mainly determined by the computation time of alinear ′ 2 ′ 677 Let A (t ′) = −( t )2 α l + t α and A (t ′) is decreasing in l for 620 program. For Samp(1), we need to solve O(1) linear programs, and l T 2 T l 678 any 0 < t ′ ≤ T . 621 it requires O(log |V |) linear programs for the re-solving heuristic. 679 622 Then write down the expected outcome of this algorithm: 680 Õ 623 4.2 With Information of Demand Distribution ALGα = [wuq Nα (u,q,l,T )] 681 624 Previous section has established the competitive ratio as a func- u,q,l 682 625 Õ 683 tion of the maximal demand. But we have not made use of the ≥ Tp w A (T )y∗ 626 ql uq l uql 684 demand distribution. To see the value of the information, we first u,q,l 627 685 aggregate the type of each online arrival as follows: For online ver- Õ Õ Õ ∗ (17) 628 ≥ Tp A (T ) w y 686 tices with the same adjacent offline vertices and the same weight ql l uq uql 629 ∈ u 687 for each corresponding incident edges, we aggregate them into a q Q l ∈Lq 630 ′ ′ Õ Õ 688 group. In other words, the set defined by Q(v) = {v | Adj(v ) = = Tp A (T )W 631 ql l ql 689 ′ Adj(v),wuv = wuv , u ∈ Adj(v)} contains all the online vertices q ∈Q l ∈Lq 632 in the same group as∀v. Then we can distinguish online arrival 690 For notation simplicity, we omitT and use A to denote A (T ). To 633 vertices by its affiliated group and demand size. We denote the l l 691 further analyze the lower bound, we first establish some properties 634 whole set of groups as Q and the demand set in group q as L . For 692 q of vertices in the same group. We define W = Í w y∗ for 635 any vertex v ∈ V, its group is defined by q(v) ∈ Q. The probability ql u uq uql 693 636 Í a fixed group q to represent the expected profit generated by the 694 of getting a vertex in group q ∈ Q is pv . For vertex in 637 v ∈V:q(v)=q vertices in the group. 695 638 6 696 Online Weighted Bipartite Matching with Capacity Constraints Conference’17, July 2017, Washington, DC, USA
′ 697 Lemma 4.8. If q(v) = q(v ) = q and dv = l < m = dv′ , then If the number of re-solving time K increases the ratio can be better. 755 698 Wql ≥ Wqm. When K is large, this ratio converges to 756 699 757 Lemma 4.8 indicates that within the same group, the vertices 1 1 700 758 with a smaller demand generates a higher expected profit. Based 2M 1 − R 701 e 1 − (1 − )e Me 759 on such an observation, we can build the following lemma. Me 702 760 Í 703 Lemma 4.9. For each type q ∈ Q, given a fixed TpqlWql , we 761 ∈ 704 l Lq 5 EXPERIMENT 762 Í ≥ (− 1 E[ ] 2 ) E[ ] 705 have l ∈Lq Tpql AlWql 2 Dq α + α Wq , where Dq is the 763 Í We test the online algorithms proposed in Section 4 over several 706 expected demand of type q arrivals and Wq represents TpqlWql . 764 ∈ synthetic data sets and a New York city taxi data set. Each data 707 l Lq 765 set specifies a bipartite graph, which is represented by G(U ,V , E), 708 766 Denote Me = maxq ∈Q E[Dq ]. Then we are ready to present where U ,V represents the offline and online request set respectively 709 | | 767 a new lower bound of the Samp(α) when demand distribution is = V 710 and E denotes the arc set. Let r |U | denote the ratio between the 768 available. 711 number of online and offline vertices. We compare our algorithms 769 712 Theorem 4.10. With the distribution of demand, Samp(α) gener- to the greedy algorithm, which is a widely used benchmark in the 770 1 713 ates a competitive ratio , where M = max ∈ E[D ]. literature (c.f. Xu et al. [27], Dickerson et al. [12] and Lowalekar et 771 2Me e q Q q 714 al.[21]). Specifically, we analyze the following algorithms in this 772 Proof: From Lemma 4.9, we have for all possible type q, 715 section. 773 716 Õ 1 2 1 2 • Greedy: Assign an arriving request v to the resource u with 774 Tpql AlWql ≥ (−E[Dq ] α + α)Wq ≥ (−Me α + α)Wq 717 2 2 the largest weight on the edge (u,v) among all the available 775 l ∈Lq 718 resources; if no available resource found, reject v. 776 Í Í ( ) ≥ (− 1 2 ) Í • ( ) ( ) 719 Therefore, q ∈Q l ∈Lq Tpql Al T Wql Me 2 α +α q ∈Q Wq = Samp 1 : Refer to the Samp α algorithm and choose α = 1. 777 1 2 • RES-γ : Under the Samp(1) framework, update the offline linear 720 (−Me 2 α + α)W . In other words, the competitive ratio is lower 778 1 2 1 program at t = T (1 − (1 − γ )i ) for 1 ≤ i ≤ K, where K is the 721 bounded by −Me α +α. By setting α = , we can get a constant 779 2 Me maximal re-solve times. Set K = 10. We test over different γ 722 a competitive ratio 1 . 780 2Me 1 1 723 including γ = D = 3 as D = 3 in the data. 781 724 4.2.1 Re-solve with Demand Distribution. With information of de- The comparison is based on the empirical competitive ratio (ECR). 782 725 mand distribution, we can again apply the re-solving heuristic to see The empirical competitive ratio (ECR) is defined as the ratio between 783 726 whether we can further improve the competitive ratio. According the total profit generated from an algorithm and the total offline 784 727 to Lemma 4.7, the deterministic linear program (3) can be reformu- optimal profit for a sample of request sequences. 785 728 lated as (15). When revolving the problem, the deterministic linear Í 786 s ∈S PL(s) 729 program can be revised accordingly as follows: ECRL = Í , 787 s ∈S OFF(s) 730 Í Í 788 max Twuq pqlyuql 731 (u,(q,l))∈E l where S denotes the set of sampled request sequences and PL(s) 789 Í Í cu 732 s.t. pqllyuql ≤ T , u ∈ U denotes the profit by algorithm L for the sequence s. Noticed that 790 q ∈Q l ∈L ∀ 733 q different algorithms share the same offline optimal profit, thecom- 791 Í y ≤ 1, (q,l) ∈ V (18) 734 uql parison of empirical competitive ratio is equivalent to a comparison 792 u ∈Ad j(q,l) ∀ 735 of profit. For a given bipartite graph G and a planing horizon T , 793 yuql ≥ 0, (u, (q,l)) ∈ E 736 ∀ we generate M request sequences by uniformly sampling each re- 794 yuql = 0, (u, (q,l)) ∈ E,cu < l 737 ∀ quest from the demand pool. We run tests for T = k|V | where 795 738 Follow a similar analysis in the previous section, we have k = 1, 2, 3, 4, 5. 796 739 " ′ 2 2 ′ # The test is organized in the following manner: We first test 797 t α Me t 1 ′ 740 E[ALG(0,T )] ≥ − + α OPT + E[OPT ] different algorithms in New-York city taxi data, with the request 798 T 2 T 2Me 741 weights ranging from 1 to 5. We examine the performance of each 799 (19) 742 algorithm in different markets and investigate the effect of different 800 743 Following a similar analysis to Proposition 4.3, we can easily derive re-solving time points. Lastly, we extend the test to some synthetic 801 1 1 1 − R 744 the new competitive ratio + (1 − )e Me if re-solving data generated and test different request weight ranges. 802 2Me 2Me Me 745 T The first column of Figure 1 presents the performance of differ- 803 once at M . The same analysis applies to the case with multiple 746 e k r 804 times of re-solve. ent algorithms in different markets. The parameter and indicate 747 the unbalance between the supply and demand in the market. The 805 748 Theorem 4.11. When demand distribution is available, the log- larger they are, the more overwhelming the demand is. From the 806 749 resolving algorithm with γ gives a competitive ratio after Kth re-solve figure we can see that in general, the re-solving method outper- 807 750 is: forms both greedy method and Samp(1) algorithm. On average, 808 K+1 1 − R 751 1 − (1 − )e Me incorporating re-solving heuristic in the randomization algorithm 809 1 Me 752 improves the profit by 20%. Greedy method performs well when 810 2M 1 − R 753 e 1 − (1 − )e Me demand is not high. But its performance drops significantly when 811 Me 754 7 812 Conference’17, July 2017, Washington, DC, USA Wang, et al.
813 871 814 872 815 873 816 874 817 875 818 876 819 877 820 878 821 879 822 880 823 881
824 (a) r = 1 (b) r = 1 (c) Synthetic data: 1 ≤ wuv ≤ 5 882 825 883 826 884 827 885 828 886 829 887 830 888 831 889 832 890 833 891 834 892 835 893 836 894 (d) r = 2 (e) r = 2 (f) NYC data, 1 ≤ wuv ≤ 2 837 895 838 896 839 897 840 898 841 899 842 900 843 901 844 902 845 903 846 904 847 905 848 906
849 (g) r = 3 (h) r = 3 (i) Synthetic data: 1 ≤ wuv ≤ 2 907 850 908 Figure 1: Result Summary 851 909 852 demand increases. In contrast, the performance of the proposed ran- data with edge weights in a smaller range. It is observed that when 910 853 domization algorithms is more robust across different markets. One the weight range becomes smaller, online algorithms’ performance 911 854 possible reason is that when demand is high, strategically skipping increases. This observation is more significant when it applies to 912 855 some inferior demand can better utilize the resource capacity. the greedy method. It implies that the randomization algorithm is 913 856 To further understand how important it is to re-solve at a right more robust across different data sets. 914 857 time. We compare algorithms with different re-solving time points. 915 858 The second column in Figure 1 compares the performance of al- 6 CONCLUSIONS 916 859 gorithms with different γ when applying RES-γ algorithm. From We study the online weighted bipartite matching problem with 917 860 the figure we can see that resolving at our proposed time which capacity constraints in this paper. We propose a randomized algo- 918 861 1 1 919 is to set γ = D = 3 outperforms the other time points. It could rithm based on the solution to an offline linear program and analyze 862 achieve up to 12% improvement compared to some other resolving its competitive ratio. We further introduce a re-solving heuristic 920 863 time. Consistent with the observation in the column in Figure 1, the to the randomized algorithm and demonstrate that re-solving at 921 864 randomization algorithms gets better performance when γ becomes the right times could significantly improve the performance of the 922 865 larger, for all the tested γ s. algorithm. Finally, we investigate the value of demand distribution 923 866 We extend the test to the synthetic data generated and plot the of the online request to further improve the algorithm’s efficiency. 924 867 results in Figure (a) in the last column of Figure 1. The observations Several experiments are conducted to test the performance of the 925 868 are consistent with those in the first column . We further test the proposed algorithms based on an application in ride hitch. 926 869 927 870 8 928 Online Weighted Bipartite Matching with Capacity Constraints Conference’17, July 2017, Washington, DC, USA
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