1 Introduction
Consider a set of points in the plane, called sites. Each site supplies a quantity of a certain unitpriced product that needs to be collected and has an operating time window of , where , for some constant , . The distance between sites and is the euclidean distance. We are also given a fleet of vehicles , each having the same capacity . All vehicles are assumed to travel at unit speed and have unit fuel consumption per distance travelled. Only one vehicle may visit any given site and all vehicles must start and end their tour at the same given depot . The goal is to compute, for every vehicle , a route that collects a quantity of the good such that the total profit of all routes is maximized, where the profit of a route is the total quantity collected (called reward) minus the total distance travelled via (called costs). We call this problem the Maximum Profit pickup Routing Problem (MPRP).
MPRP was introduced in [1, 3], which focuses on the case with fixed quantities and only one vehicle per site. In this paper, we consider the case where multiple vehicles may visit one site. We first study the MPRP problem with Multiple Vehicles Per Site and fixed quantities (MPRPM), in which ’s are constant in time. We also consider the MPRP problem with Multiple Vehicles Per Site and TimeVariable Quantities supplied (MPRPMVS), in which ’s are 0 for and linearly increasing in time for . Compared to the single vehicle versions, this adds the availability constraint that . However, a vehicle may not visit a site more than once.
MPRP is known to be strongly NPhard [3]. By extensions, all the variants proposed above are also strongly NPhard. That is, they have no algorithm running in time polynomial in the value of the input unless P = NP.
Applications of the results presented in this paper include public transportation in various settings, such as cities [3], intercity railway transportation, domestic or international flight, etc. Another important application is in forprofit waste pickup, in which the reward is proportional to the amount of waste collected.
1.1 Related work
General graph TSP is proven to be hard to approximate within any constant factor [16]. However, TSP on special graphs, such as complete graphs with metric distances ane euclidean distances, do have approximation algorithms. For instance, general metric TSP has a time approximation algorithm by Christofides [9]. Euclidean metric TSP even has a PTAS achieving performance ratio via Arora’s time algorithm [4]. On the other hand, metric TSP was shown to be APXcomplete by Papadimitriou et. al [15] and, to date, the best known lower bound is by Karpinski et. al [14].
Various generalizations of TSP have also been considered. Bansal et. al introduced the DeadlineTSP and the Time WindowTSP problems [5]. In DeadlineTSP, every site has a deadline , while in Time WindowTSP, every site has a time window . Both of these problems associate every site with a reward and ask for a single tour that maximized the total reward collected. For DeadlineTSP, they provide an approximation algorithm, while for Time WindowTSP they describe an approximation.
There are also variants involving multiple vehicles. Fisher introduced the Vehicle Routing Problem with capacity constraint (VRP), in which a fleet of vehicles is given, and each customer has a demand of a certain product [11]. Their solution uses an iterative lagrangian relaxation of the constraints. Later, Fisher et. al introduced the Vehicle Routing Problem with Time Windows and capacity constraints (VRPTW), in which customers have a demand of the product, as well as operating time windows, and the vehicles have nonuniform capacity constraints [12]
. They solve this problem using a linear programming approach.
Golden et. al studied the Capacitated Arc Routing Problem (CARP) with uniform vehicle capacity where edges, rather than vertices, have customer demands, and the goal is to minimize the travelled distance subject to meeting all demands. They prove that CARP can be approximated within a constant factor when the triangle inequality is satisfied [13]. Later, van Bevern extended the result to general undirected graphs [6].
Wang et. al considered the MultiDepot Vehicle Routing Problem (MDVRPTW) with Timw Windows and Multitype Vehicle Number Limits [17]
. The main difference to our problem is that the goal is to minimize the number of vehicles used (if feasible) or maximize the number of visited customers (if infeasible). Although they solve the problem using a genetic algorithm approach, their algorithm is iterative and has no performance guarantee with respect to the approximation factor.
Versions of vehicle routing with multiple vehicles per customer have also been considered. In [10], Drexl does a comprehensive survey of variations of vehicle routing with multiple synchronization constraints (VRPMS), e.g. in which a customer may be visited by two vehicles (VRPTT) [7], or by 3 or more vehicles [8].
1.2 Our contributions
We give a constantfactor approximation algorithm for MPRPM and MPRPMVS, which uses clever reductions to some known problems. Specifically, we solve MPRPM in time for an approximation ratio of and MPRPMVS in time within an approximation ratio of .
2 MprpM
We come up with a reduction from MPRPM to MPRP.
Let be the distance from the depot to site .
An instance of MPRP (or MPRPM) can be described as a set , where .
A solution to is a set . Here is a routing function assigning, to every pair of sites or depot , a number , denoting the index of the vehicle operating the path from to , or 0 if no such vehicle exists. is a pickup function assigning, to every site on vehicle ’s route, a quantity to be picked up by .
A solution to is a routing function with the same properties as the function in.
Denote by (resp., ) the set of MPRP (resp., MPRPM) instances.
Starting from , let . That is, has the same nodes and distances as . We solve using the algorithm in [3] and denote by the solution.
We transform the solution to an MPRP instance , into a solution to an MPRPM instance , as follows. For every vehicle of , let be the total quantity collected by from its assigned tour . It may be possible that some sites in need to be visited multiple times in , producing a new tour . To figure out which ones need to, we look at tuples of sites in where
(1) 
The goal is to maximize the total amount picked up from and . Denote by the quanity assigned to be picked up by from , by the quanity assigned to be picked up by from , and by the quanity assigned to be picked up by from . We have
(2) 
and our goal is to maximize . To do that, we set , and we come to the following linear program.
Minimize s.t.
(3) 
After solving the linear program, we check the additional constraint . If it is satisfied, we assign to pickup from , and to pickup from and from in , where is a solution to the linear program. In order for to achieve a better profit than through this reassignment, the following need to hold. Let denote that is visited before in .
(4) 
where is the time when visits . That is, insertion of into in does not introduce time window violations
See Figure 2 for an illustration of this reassignment.
For each vehicle and site , we look at all possible tuples satisfying the abovementioned additional constraints and solve the linear program to find the optimal quantities to pickup, then select the tuples that maximize the profit gain after reassignment, and then perform the reassignment.
One arising concern is that the order in which the pairs are selected produces different results. However, while this may happen, it may not impact the optmiality of the reassignment by more than a constant factor, as we shall see below.
Lemma 1
Let be an ordering of the pairs and let be the gain obtained through reassignment when selecting pair (or 0, if reassignment is infeasible). Then , where does not depend on .
Proof
From (4), it follows that
where indicates that is before in .
Since , we get
We have
Let
.
Since , it follows that
Moreover, note that .
That is, , where does not depend on .
Note that
since otherwise one would be able to create an optimal metricspace MSTbased tour of cost , which is a contradiction of the result in [9].
Thus,
Since
for sufficiently large , which is reasonable since is constant, we get
That is, .
Thus, we first run a MPRP solver to obtain a routing and then select the pairs in the order given by each for every . For each of the pairs, we inspect all possible tuples in time per tuple, which is the time required to verify constraint 2. That is, our reduction is done in time.
As for the correctness of the reduction, suppose there exists a routing for yielding a profit increased by more than a factor of 4 compared to . In order to do that, must pickup a quantity from some . However, this implies that either is not reassigned in and thus gives an increased profit for , or is reassigned to one of and thus gives an increased profit for one of these sites. Both options lead to a contradiction. Now suppose there exists a routing for yielding an increased profit compared to . In order for that to happen, either some site that was never added or involved in a reassignment picks up a quantity , implying an increased profit for for in , or some site involved in a reassignment with picks up a quantity , implying a quantity was picked up from some site (wlog assumed to be ) in which is more than 4 times the one picked from the samee site in . In both cases, a contradiction follows due to Lemma 1. Hence, we have reduced MPRPM to MPRP within an approximation ratio of 4 in time.
By using the approximation for MPRP in [3] for time, we thus solve MPRPM in time.
We have proved the following result.
Theorem 2.1
MPRPM can be solved in time for an approximation ratio of .
3 MprpMvs
In MPRPMVS, quantities supplied at sites vary linearly as a function of time, i.e. .
We adapt the algorithm in [2] for solving MPRPVS, to work in our case, by using a similar reduction as the one from MPRPM to MPRP described in the previous section, to reduce MPRPMVS to MPRPVS.
Given an instance of MPRPMVS, we apply the reduction in [2] to obtain an instance , and then solve MPRPM using the algorithm in Section 2. Again, this is not straightforward since the approach in [2] depends on single vehicle assignment which is not the case here. Thus, we need a more insightful construction of than directly applying the algorithm in [2].
For each site , we split into intervals, for some , where is the smallest constant such that . Denote these intervals by , i.e. . After performing this split, we construct a set of sites where, for each original site , contains sites . To each newly constructed site , we assign a quantity and a time window . We then run the MPRPM Option 1 algorithm described in Section 2 on the transformed instance (note that since the sites in now supply constant quantities).
Denote by the routing obtained by running the MPRPM algorithm on . We now analyze the properties of in order to put a bound on its performance ratio.
We know from [2] that an optimal MPRP algorithm run on may collect a quantity at least as much as an optimal MPRPVS algorithm run on . Since for , an instance of MPRPM also belongs to MPRP, and an instance of MPRPMVS is also an instance of MPRPVS, we get the following.
Lemma 2
For , an optimal MPRPM algorithm when run on may collect a quantity at least as high as an optimal MPRPMVS algorithm when run on .
Lemma 3
[2] Let be an algorithm for a MPRP (resp., MPRPM) and let be the profit obtained by when run on an instance with one vehicle. Then, on an instance with vehicles, the profit obtained by is .
By running the optimal MPRPM algorithm in Section 2 on , we get the following a profit at least as high as an optimal MPRPMVS algorithm when run on .
Since travel costs are less than the rewards generated by the quantities collected, a performance ratio of in the quantities induces a performance ratio of in the profits as well. Putting this together with the lemma above, we get the following.
Lemma 4
When run on , the MPRPM algorithm in Section 2 may obtain a profit at least as high as an optimal MPRPMVS algorithm when run on .
Note that the reduction described before Lemma 2 takes time. Then, running MPRPM on which has sites, takes time. Thus, we have proved the following result.
Theorem 3.1
MPRPMVS can be solved in time within an approximation ratio of .
4 Conclusions and Future Work
We solve the Multiple Vehicles per Site versions of the MaximumProfit Routing Problem, specifically, the FixedSupply version and the TimeVariable Supply version.
We leave for future consideration probablisitc approaches for all versions of MPRP, e.g. algorithmic solutions whose output is, with high probability, within a certain bound of the optimum for the given instance. Proving negative results or lower bounds, e.g. inaproximability within a certain ratio, would also be of interest.
References
 [1] B. Armaselu, O. Daescu, Approximation Algorithms for the Maximum Profit Pickup Problem with Time Windows and Capacity Constraint, arXiv:1612.01038, December 2016
 [2] B. Armaselu, An APX for the MaximumProfit Routing Problem with Variable Supply, arXiv:2007.09282, July 2020
 [3] B. Armaselu and O. Daescu, Interactive Assisting Framework for Maximum Profit Routing in Public Transportation in Smart Cities, PETRA 2017: 1316
 [4] S. Arora, Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems, Journal of ACM 45 (5), 1998, pp. 753782
 [5] N. Bansal, A. Blum, S. Chawla, and A. Meyerson, Approximation Algorithms for DeadlineTSP and Vehicle Routing with Time Windows, STOC 2004: 166174
 [6] RR. van Bevern, S. Hartung, A. Nichterlein, and M. Sorge, Constantfactor approximations for Capacitated Arc Routing without triangle inequality, Operations Research Letters 42 (4), June 2014, pp. 290292
 [7] Bredstr¨om D, R¨onnqvist M. Combined Vehicle Routing and Scheduling with Temporal Precedence and Synchronization Constraints. European Journal of Operational Research 191: 19–29 (2008)

[8]
B¨urckert H, Fischer K, Vierke G.
Holonic Transport Scheduling with TELETRUCK.
Applied Artificial Intelligence 14: 697–725 (2000)

[9]
N. Christofides,
Worstcase analysis of a new heuristic for the traveling salesman problem,
N. Christofides. Worstcase analysis of a new heuristic for the travelling salesman problem, Management Sciences Research (388) report, 1976  [10] Drexl M. Synchronization in Vehicle Routing  A Survey of VRPs with Multiple Synchronization Constraints. Transp. Sci. 46(3): 297316 (2012)
 [11] Fisher ML. Optimal solution of Vehicle Routing Problems using Minimum KTrees. Oprations Research 42 (4): 626642 (1994)
 [12] Fisher ML, Jornstein KO, Madsen OB. Vehicle Routing with Time Windows: Two Optimization algorithms. Operations Research 45 (3), 1997. DOI: 10.1287/opre.45.3.488
 [13] B.L. Golden and R.T. Wong, Capacitated arc routing problems, Networks 11 (3), 1981, pp. 305315
 [14] Karpinski M, Lampis M, Schmied R. New Inapproximability bounds for TSP. Journal of Computer and System Sciences 81 (8): 1665–1677 (2005), arXiv:1303.6437, doi:10.1016/j.jcss.2015.06.003
 [15] Papadimitriou CH, Yannakakis M. The traveling salesman problem with distances one and two, Math. Oper. Res. 18: 1–12 (1993). doi: 10.1287/moor.18.1.1
 [16] S. Sahni and T. Gonzalez, Pcomplete approximation problems, J. ACM, 23 (3), 1976, pp. 555565
 [17] X. Wang, C. Xu, and H. Shang, Multidepot Vehicle Routing Problem with Time Windows and Multitype Vehicle Number Limits and Its Genetic Algorithm, WiCOM 2008, DOI: 10.1109/WiCom.2008.1502
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