1 Introduction
In this paper, we consider the MaximumProfit Routing Problem with Variable Supply (MPRPVS), stated as follows. Consider sites in the euclidean plane, each site being specified by its coordinates , the production rate of a certain good, and an interval (in hours) called time window. The values are assumed to be integers in the interval , where is a given constant, and supply loading is assumed to be instantaneous (that is, no service time needed). For every site , the quantity of the good available at time at site , is assumed to be 0 outside the time window, and piecewise linearly increasing at the rate within the time window. 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).
MPRPVS is a generalization of MaximumProfit Pickup Problem with Time Windows and Capacity Constraint (MPRP), introduced in [2] (see also [1]. MPRP is similar to the problem considered in this paper, the difference being that the quantity of the good supplied at each site is constant in time. In [2], two polynomialtime approximation algorithms with constant approximation factors are given for MPRP. They both run in time, assuming , for some constant . The best performance ratio, , among the two algorithms, is achieved by an approach involving wellseparated pair decompositions.
MPRPVS is a variant of the Euclidean Travelling Salesman Problem (TSP), which is known to be strongly NPhard [8, 10]. Hence, MPRPVS is also a strongly NPhard problem, which means there is no algorithm with a running 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 [2], 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
Various cases and generalizations of TSP have been studied, and various approximation algorithms and inapproximability results have been given. It is known that general TSP cannot be approximated within any constant factor [14]. However, constant factor approximations have been proposed for some restricted versions. For metric TSP, the bestknown result is Christofides’s algorithm [7], which runs in time for an instance of sites, and has an approximation factor of . For the euclidean case, Arora et. al [3] presented an time randomized algorithm for the planar case, with a approximation ratio. They also show how to extend their algorithm for the dimensional case, in which case their running time increases to .
Fisher et. al studied the Vehicle Routing Problem with Capacity Constraint (VRP) [11], a generalization of TSP in which there are vehicles of equal capacity , there is only one depot for all vehicles, and each customer has a demand of a specified good. They gave a nearoptimal iterative algorithm for the problem, using an iterative lagrangian relaxation of the constraints. Later, study the VRP with Time Windows (VRPTW) [12], which is a generalization of VRP with time windows and different capacity constraints for the vehicles. They adapt their algorithm in [11]
to this version, as well as give an additional linear programming approach.
Golden et. al introduced the Capacitated Arc Routing Problem (CARP) [9], in which every edge has a demand and a cost, every vehicle has the same capacity, and the goal is to minimize total cost while meeting all demands. They prove that CARP can be approximated within a constant factor when the triangle inequality is satisfied. Later, van Bevern extended the result to general undirected graphs [5].
Wang et. al [15] considered a somewhat similar problem, namely the MultiDepot Vehicle Routing Problem with Time Windows and Multitype Vehicle Number Limits [15]
. 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.
The DeadlineTSP problem was studied by Bansal et. al [4]. In DeadlineTSP, we are given sites, where each site has a deadline and a reward , and we want to find a tour that maximizes the total reward of sites that can be visited. They propose an algorithm with an approximation factor of . They also show how to extend it to the more general TSP with Time Windows (TSPTW), in which the sites also have release times . For TSPTW, they give an algorithm with an approximation. They also give an approximate algorithm for the relaxed version of the problem, in which we are allowed to extend the deadlines by a factor of .
1.2 Challenges
The problem we study involves major challenges.
First, the reward maximization part of our solution relies on a bicriteria solver for the Travelling Salesman Problem with time windows (TSPTW), which is known to be hard to approximate within a constant factor for every in the general case [6]. In the case where time windows are integer, Bansal et. al [4] show how to solve TSPTW within an approximation factor, where is the latest deadline. However, even for constant , the constants involved in their approximation factor are still very large.
Second, it is not sufficient to maximize rewards, since the highest reward tour may have high travelling costs, which may render the optimalreward tour not profitoptimal. Moreover, if a vehicle takes too long to complete its tour, it may miss time windows of sites it may otherwise be able to visit under the rewardoptimal tour.
Third, a rewardoptimal tour that takes into account only the maximum or average quantity supplied at each site may not even be rewardoptimal for the variablesupply version, as a site visited later gives a better reward, but visited too late gives no reward.
1.3 Our contributions
We give a constantfactor approximation algorithm for MPRPVS, which uses clever reductions to some known problems. Specifically, our algorithm runs in time and has an approximation ratio of . In addition, our algorithm can be adapted to the MPRP problem to improve upon the algorithm in [2], under certain assumptions about the quantities supplied.
The rest of the paper is structured as follows. In Section 2, we formulate the problem in formal, mathematical ways, make some preliminary observations, and list some reasonable assumptions about the given data. Then, in Section 3, we solve the problem on a restricted onevehicle case, show how to reduce to the problem in [1], and analyze the algorithm. After that, in section 4, we generalize to multiple vehicles and provide the APX for the general case. Finally, in Section 5, we conclude and list some future directions.
2 Preliminaries
MPRPVS can be mathematically formulated as:
Maximize (1)
subject to
(2),
(3),
(4),
(5),
(6), and
(7),
where is the timestamp when vehicle visits site .
Here (1) is the profit maximization objective function, (2) is the first and last site visitation constraint, (3) is the capacity constraint, (4) is the time window constraint, (5) is the travelling constraint, (6) is the production constraint, and (7) is the persite vehicle uniqueness constraint. Note that the depot is treated as the 0th site.
We call the above Formulation 1.
Alternatively, one can formulate the problem as follows.
Max. (1)
subject to
(2),
(3),
(4),
(5),
(6),
(7.1),
(7.2),
where if vehicle arrives from at at time .
Here (1) is the profit maximization objective function, (2) is the capacity constraint, (3) is the time window constraint, (4) is the production constraint, (5) is the persite vehicle uniqueness constraint, (6) is the tour connectivity constraint, and (7.1), (7.2) are the constraints enforcing starting and ending at the depot.
We call this alternative Formulation 2.
We assume that all sites start ”empty”, that is, . Further, we suppose the discrepancies in the quantities supplied are small, specifically , for some constant , and .
Since the quantities supplied increase linearly in time, we have .
Note that, if , then the capacity constraint is always satisfied. Thus, from now on, we also assume .
3 Onevehicle case
We first consider a restricted version of the MPRPVS problem, say MPRPVS1, in which only one vehicle is available and show how to reduce it to the MPRP problem with one vehicle (MPRP1).
Lemma 1. MPRPVS1 problem can be reduced to MPRP1 within time and a ratio of , for any . That is, the vehicle assigned for an instance of MPRPVS1 collects a reward , where is the quantity collected by the vehicle assigned for and is the corresponding instance of MPRP1.
Proof. Let be an instance of the restricted MPRPVS1 problem. We construct an instance of MPRP1. Let . For every site on instance , we consider the intervals , with
.
Here is the smallest integer such that , that is, (see Figure 2 for an illustration). Note that, the length of the interval containing timestamp is a nondecreasing function of . Indeed, , and , which is increasing in . For each interval , we add a site labeled to , with the same coordinates as site and with fixed quantity and time window . For every two nodes in , the pointtopoint distance is defined as follows.
After constructing , we approximately solve the optimization SubsetSum problem, in which the goal is to find a set of numbers that add up to as close as possible to without exceeding . The input set of numbers for SubsetSum will be . Denote by the set of numbers output by the SubsetSum solver.
Using the rewardmaximization algorithm in [4], we compute a routing on . Then, for every , we take the latest interval for which , and remove all nodes such that ., to ensure that every site is visited only once. The resulting route is .
By construction, the list of input nodes in is the same as the one in . Moreover, for each site , the time interval in which visits is the same as the time window . Also, we solve the SubsetSum problem optimally in , using the method in [13], where ignores a polylogarithmic factor. Since , the running time becomes time. The quantity collected by is at least , with , where and are the maximum (respectively, minimum) quantity collectible in interval of site . It is easy to check that by construction. Hence, the result follows. ∎
4 General case
Now we show how to extend the reduction to solve the general MPRPVS problem (for multiple vehicles).
Given an instance of MPRPVS, we do the following.
1. Decompose into a set of subsets for each vehicle . For each subset , create an instance of MPRPVS1 with the set of sites, the same capacity constraint and the same time windows.
2. For each instance , construct an instance of MPRP1 from , using the reduction in Section 3.
3. Solve each instance and denote by the resulting route.
4. For each site , if two or more vehicles visit , retain only the vehicle that would collect the largest .
5. For each site and vehicle , reassign to if and only if the profit would increase.
6. Set
7. Repeat steps 16 until either there are no sites left unassigned or no more sites are assigned during an iteration.
Now we describe Steps 1 and 3 in detail.
4.1 Decomposing into subsets
We first construct a wellseparated pair decomposition (WSPD) of the sites, using the separation factor . Note that
is a binary treelike hierarchical clustering of the sites such that, for any pair
of sites, exactly one pair of subsets in separates them. That is, either or viceversa. There are subsets in , and each subset or is further decomposed into pairs of subsets . Since this is a hierarchical pair decomposition, there are levels in the hierarchy, and the union of the subsets on any level in the whole input set . Assume wlog that the pairs of subsets are ordered by the total number of sites contained in them.After that, for every vehicle , we construct an instance of MPRPVS1 where the set of sites is assigned to as follows.
For vehicles , assign subsets . For every , for vehicles , assign subsets , and remove all sites in from .
It is easy to check that no subsets are overlapping.
Moreover, since the subgraph induced by the subsets is an spanner, it follows that any minimumlength routing algorithm on the induced subgraph outputs a total length of at most , where is the cost output by the algorithm when run on the input set of sites.
This gives us the following result.
Lemma 2. The total length of a minimumlength routing for each subset is at most , where is the total length of a minimumlength routing on .
4.2 Solving an instance
First, we compute a route that approximately maximizes the quantity collected from , using the algorithm in [4]. After that, we traverse and, for every two consecutive arcs , replace them by a single arc if and only if it meets and time windows and increases the profit. Denote the resulting route by .
Lemma 3. has a profit , where is the maximum profit obtainable by a tour visiting sites in with capacity constraint .
Proof. With the algorithm in [4], picks up a total quantity of at least , where is the total quantity collected by an optimal algorithm.
After arc replacements on , the total profit of the resulting route becomes
,
where is the total length of the shortest tour visiting all sites in with capacity constraint .
4.3 Combining routes and reassigning sites
We analyze Let be at the beginning of the th iteration.
Lemma 4. During some iteration , after Step 5 of the algorithm, the total quantity collected from is at least , where is the optimal quantity collectible from .
Proof. Suppose on the contrary that, in any iteration , the total quantity collected after Step 5 is less than . Then there exists a site from which a quantity less than is collected, where is the quantity collected from by the optimal MPRP1 algorithm. By Lemma 1, it follows that the optimal MPRPVS1 algorithm collects a quantity less than . Since the travel costs on are at most times the optimal value, it follows that the algorithm misses a site satisfying due to missing its time window. However, by Step 7, if a site is missed during an iteration, it will eventually be visited during a further iteration . ∎
4.4 Analysis
We now analyze the performance of the algorithm.
By Lemma 2, the total distance travelled by all on all is at most times higher than the optimal distance. By Lemma 1, for each , the quantity collected from all sites in is at least of the quanity collected by an MPRP1 solver on . Since is the output of an MPRP1 solver on and, by Lemma 3, the profit collected by from has a profit at least the optimal profit, it follows that the profit for each is at least times the optimal for an MPRPVS1 solver on . Now the profit of the algorithm on the original problem instance would be times the optimal in absence of time winodws. The extra tour length factor of may cause more than as much revenue to be lost due to missing time windows. However, by Lemma 4, we collect a total quantity at least while increasing travel costs at most times, where is the optimal total quantity collectible.
That is, the total profit for the algorithm becomes at least
,
where is the optimal profit.
Letting , we get
.
Additionally, note that Step 2 produces a routing on a total of sites. Since Step 3 takes time when run on a et of sites [2], it runs in time in our case.
Since , we get a ruinning time of .
Reassigning as , we can state the following result.
Theorem 5. For any parameter , MPRPVS can be solved within time and has an approximation ratio of
.
Notice that the algorithm presented in this paper can also be used to solve MPRP. When the quantities supplied at each site are constant in time, step 2 of the algorithm can be skipped, and step 3 can be run on instead of . Since the extra factor in the approximation ratio and the extra is due to the reduction used in step 2, we obtain the following.
Corollary 6. MPRP can be solved in time and has an approximation factor of , assuming , for some constant , and .
Note that this is an improvement over the algorithm in [2] when the aforementioned assumption holds.
5 Conclusions and Future Work
We presented an APX for the MaximumProfit Routing Problem with Variable Supply (MPRPVS). We also improve upon the results in [2] for the MaximumProfit Routing Problem (MPRP) under certain conditions.
We leave for future work proving lower bounds on the approximation ratio (assuming polynomial running time) for the problem, as well as coming up with randomized algorithms that are either faster or have better approximation factor, with high probability.
We also leave for future consideration finding approximation algorithms for the other versions of MPRP, listed below.
MPRPM. More than one vehicle is allowed per site, and the quantity is fixed.
MPRPMVS. More than one vehicle is allowed per site, and the quantity varies in time.
Acknowledgement
The author would like to thank Dr. Ovidiu Daescu for helpful discussions.
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