In this paper, we consider the -median with discounts problem. Formally, in an instance of -median with discounts (MedDis), given clients , facilities , a finite metric and discounts , our goal is to choose at most facilities such that the sum of discounted distances from each client to its nearest facility in is minimized, i.e., to minimize where and . This problem recovers standard -median [byrka2017improved, charikar2012dependent] by having for each . MedDis also recovers -supplier [gonzalez1985clustering, hochbaum1985best] using the following reduction: By setting a uniform discount for all , the instance has optimum 0 if and only if where is the optimum of the -supplier problem on the same input. Since there are a polynomial number of possible values for , solving the -supplier instance is equivalent to solving MedDis for each possible .
An algorithm is said to be a bi-criteria -approximation for MedDis, if it always outputs a solution s.t. , where is the optimal solution. Using the same reduction as above, we observe that such an algorithm recovers an -approximation to -supplier and a -approximation to -median. Using the APX-hardness of -supplier [hochbaum1986unified] and -median [jain2002greedy], one has and unless . Under current theoretical complexity assumptions, this rules out pure approximations and shows bi-criteria -approximations are the strongest type of guarantee obtainable. Guha and Munagala [guha2009exceeding] consider -median with uniform discounts (Uni-MedDis), where the input is the same as MedDis but the discounts are uniform. They directly use the primal-dual algorithm by Jain and Vazirani [jain2001approximation] and give a bi-criteria -approximation algorithm. They develop a constant factor approximation for unassigned stochastic -center using Uni-MedDis. Very recently, Ganesh et al. [ganesh2021universal] give a bi-criteria -approximation algorithm for MedDis, using a similar primal-dual algorithm. They also use this algorithm as a key ingredient in their bi-criteria constant-factor approximations for universal -clustering problems.
Uni-MedDis is also closely related to -facility -centrum, where the input is the same as -median, but the objective is the sum of largest connection costs. This problem unifies -supplier and -median via . Given a bi-criteria -approximation for Uni-MedDis, one recovers a -approximation for -facility -centrum, by exhaustive search for the -th largest connection cost in the optimum and setting a uniform discount [chakrabarty2019approximation]. An improved bi-criteria -approximation for Uni-MedDis is given in [chakrabarty2019approximation] in the more general setting of multi-dimensional discounts and ordered -median.
We present an LP rounding algorithm for MedDis and achieve better approximation factors than the previously-known bi-criteria -approximation [ganesh2021universal]. We employ a modestly improved iterative rounding framework by Krishnaswamy et al. [krishnaswamy2018constant], inspired by the “quarter ball chasing” technique in [gupta2021structual].
In the original iterative rounding framework [krishnaswamy2018constant], one starts with a solution to the natural relaxation, and obtains an (almost) integral solution by iteratively modifying an auxiliary LP. Like several previous LP based algorithms for median clustering [charikar2012dependent, swamy2016improved], the main idea is to have a basic solution such that the tight constraints at this solution consists of laminar families. In [krishnaswamy2018constant], the laminar family is obtained by dynamically maintaining a “core set” of clients and creating packing constraints indexed by these clients. Gupta et al. [gupta2021structual] further refines the framework by adding an additional laminar family in the auxiliary LP and analyzing the structure of the basic solutions.
To obtain an approximation for MedDis, we adapt the framework by having a new objective, where the contribution of assigning facility to client is for and discretized cost function . Compared with the real contribution, this objective prescribes a larger discount for each client, which in turn provides an upper bound on the auxiliary objective after we discretize the given metric
. Since our formulation is free of outliers, we are able to have a simpler analysis of the approximation guarantee for using two laminar families in the auxiliary LP.
The LP based algorithm also gives rise to the first constant factor approximations for matroid median with discounts (MatMedDis) and knapsack median with discounts (KnapMedDis). In these two problems, the input is the same as MedDis, except that the cardinality constraint is absent. We need to either choose an independent set of facilities in a given matroid , or choose weighted facilities that have a total weight no more than a given threshold. The reduction from matroid center/median, knapsack center/median to MatMedDis, KnapMedDis follows similarly.
We also obtain constant-factor approximations for clustering problems with uncertain and stochastic clients. These are application results which rely on previously-existing frameworks, e.g., the -approximation algorithm for unassigned stochastic -center by Guha and Munagala [guha2009exceeding] and the bi-criteria -approximation for universal -median by Ganesh et al. [ganesh2021universal]. See Section 5 for more details.
-center and -median are two of the most fundamental clustering problems. -center is NP-hard to approximate to a factor better than [hsu1979bottleneck], and simple 2-approximations are developed [gonzalez1985clustering, hochbaum1985best]. There is a long line of research for -median, with approximations given by primal-dual methods [jain2002greedy, jain2001approximation, li2016approximating], local search [arya2004local] and LP rounding [charikar2002constant, charikar2012dependent]. Currently the best approximation ratio is due to Byrka et al. [byrka2017improved]. No performance guarantee better than is obtainable in polynomial time unless [jain2002greedy].
Clustering problems with stronger combinatorial constraints are extensively studied. Hochbaum and Shmoys [hochbaum1986unified] give a tight 3-approximation for knapsack center. Chen et al. [chen2016matroid] give a tight 3-approximation for matroid center. Several approximations are developed for median clustering [charikar2012dependent, krishnaswamy2011matroid, kumar2012constant, swamy2016improved] with the current best ratios [krishnaswamy2018constant] for matroid median and [gupta2021structual] for knapsack median. Many variants are considered in the literature, e.g., robust matroid and knapsack center [chen2016matroid, harris2019lottery] where a certain number of clients can be discarded, and the more general robust -center problem [chakrabarty2018generalized], where some clients are discarded and is an independence system containing all feasible solutions.
Cormode and McGregor [cormode2008approximation] introduce the study of stochastic -center and obtain a bi-criteria constant-factor approximation. Guha and Munagala [guha2009exceeding] improve their result and obtain -approximations for both the unassigned and assigned versions of stochastic -center. Huang and Li [huang2017stochastic] consider the unassigned version of stochastic -center and devise the first PTAS for fixed and constant-dimensional Euclidean metrics. Alipour and Jafari [alipour2018improvement] study multiple variants of assigned stochastic -center and give various constant-factor approximations, some of which are later improved by Alipour [alipour2021improvements].
2 -Median with Discounts
2.1 The Natural Relaxation
In this section, we consider MedDis. The natural relaxation is given as follows, where is the extent we assign facility to client , and is the extent we open facility .
Let be the optimum of the instance of MedDis. The optimum of is at most , since the integral solution induced by the optimal solution is feasible, and the objective is equal to . Fix an optimal solution to in what follows. We assume w.l.o.g. that is distance-optimal, i.e., whenever there exists , for each s.t. , we have . We note that though this is NOT guaranteed by the LP objective due to the discounts, we can always modify the solution such that it becomes distance-optimal while keeping the objective value unchanged.
2.2 Metric Discretization and Auxiliary LP
In the sequel, we adapt the iterative rounding framework in [krishnaswamy2018constant]. W.l.o.g., we assume for any that are not co-located. We fix and discretize the metric as follows: Let , , and for any , where and is a parameter we will determine later. For any , define . One notices that is not necessarily a metric. We obtain the following lemma.
When , the expectation is 0 and the inequality is trivial. We assume in what follows. Let where and . Since for and is distributed uniformly on , when , and when . This shows , hence
Using the obtained solution to , we employ the standard facility duplication technique to make sure that (see, e.g., [charikar2012dependent, krishnaswamy2018constant]). Let be the outer ball of and thus . Let be the smallest integer such that for each called the radius level of , and be the inner ball of . Let , and initially, and we will iteratively move all of to , and maintain a subset . We first define the auxiliary LP.
In the beginning, we have and the other two are empty, thus satisfies all the constraints by definition of . We start by letting be uniformly distributed on . Using Lemma 1 and the linearity of expectation, the expectation of under is
If we increase continuously from 0 to , is piece-wise non-decreasing and right-continuous for any by definition of . Specifically, suppose for and , then the expression is non-decreasing on and . Therefore, the objective of under , is also piece-wise non-decreasing and right-continuous, with respect to and a polynomial number of intervals. Therefore, it is easy to find that minimizes the objective, which is at most the expectation. ∎
Suppose we choose such using Lemma 2 in the sequel. We use the iterative rounding algorithm to iteratively change , maintain a feasible solution, and keep the objective value non-increasing. The rigorous procedures are in Algorithm 1, and here we set the step size .
Algorithm 1 returns an integral in polynomial time, i.e., , and , .
We first notice that, for each , it can only enter from once. In each iteration, we either move some from to , or reduce the radius level of some by 1. Since the latter happens only when , and by definition of , we must have , otherwise , contradicting . There are possible radius levels, where , thus the algorithm returns in polynomial time.
When Algorithm 1 returns, none of the constraints in corresponding to or is tight. By definition of the subroutine , for each in , we have unless (recall that ), thus the remaining tight constraints at in form two laminar families, i.e., those in with odd radius levels and those with even radius levels. Hence is an integral basic solution. ∎
After each iteration of Algorithm 1, is feasible to the new LP with a no larger objective.
We consider two cases. In the first case, there exists some . The old contribution of to is . After we move to , because , satisfies the new constraints corresponding to in and (if it is added to ), and its contribution becomes . Since is partitioned into and each must satisfy by definition, the contribution of stays the same.
In the second case, there exists for some . The old contribution of to is since . To avoid confusion, let denote the modified values of after the iteration. After we reduce the radius level of and attempt to add to , the new satisfies , hence still satisfies the constraints, if is indeed added to . The contribution of becomes , where . Since the new is partitioned into , and each must satisfy by definition, the contribution is unchanged. ∎
After Algorithm 1 returns , its objective in is at most that of in the original auxiliary relaxation. Moreover, for each , there exists such that and , .
The first assertion follows from Lemma 4, since the objective of is non-increasing within each iteration, and we find an optimal basic solution at the beginning of each iteration. For the second assertion, we first notice that is an integral solution in .
If and during Algorithm 1, according to the subroutine , only with can remove from , but this implies which is impossible. Therefore, whenever there exists with , it cannot be removed from . We first need the following claim.
If is added to when it has radius level , there exists with in the final solution, and , .
We prove it using induction on . If , the claim follows from the argument above since cannot be removed from , and each satisfies . Suppose the claim holds for radius levels up to , we consider added to with radius level . If it remains in to the end, then the claim follows similarly as for each . If it is removed by another later in the algorithm, it means that since the radius level of is non-increasing. Using the induction hypothesis, there exists with and , thus using the triangle inequality, we have . This finishes the induction. ∎
For any client with a final radius level of , when we reduce to and invoke the subroutine on it, if indeed we can add to , we directly invoke creftype 6 and get the desired result. Otherwise, cannot be added to due to there being s.t. and , in which case we use creftype 6 on the time we add to with radius level and conclude the existence of with and . ∎
Using the final output , we define the solution and remove any co-located copies of facilities, if there are any (notice this does not affect Lemma 5). We prove the following main theorem.
Let and . There exists a polynomial time bi-criteria -approximation algorithm for MedDis for any .
We consider each client and its distance to , i.e., . Since is an integral solution, and when Algorithm 1 finishes, it follows that for each by definition of the algorithm. Using Lemma 4, the objective of is at most that of before we run the algorithm, that is,
Theorem 7 provides a trade-off between the approximation factors. For instance, one can choose s.t. minimizing , or s.t. minimizing . Both improve previous bi-criteria -approximation algorithms for MedDis [ganesh2021universal, guha2009exceeding].
3 Matroid Median with Discounts
In this section, we consider MatMedDis. Formally, MatMedDis has the same input as MedDis except that we need to open facilities that constitute an independent set of an input matroid , instead of having an upper bound on the number of open facilities. MedDis is a special case of MatMedDis where the given matroid is uniform with rank , that is, . The natural relaxation is the same as , except that we replace the cardinality constraint with the constraints of a matroid polytope, that is, for as the rank function of , for each , following a classic result by Edmonds [edmonds2001submodular].
Our algorithm proceeds very similarly to MedDis. We solve the relaxation and obtain a fractional solution that is distance-optimal. We discretize the metric into and construct the same auxiliary LP as , except that we replace the cardinality constraint with matroid constraints, too. The discretized metric is constructed in a way such that its objective is at most times the optimum, akin to Lemma 1.
We use Algorithm 1 with a smaller step size . This is because, unlike MedDis with the cardinality constraint, the matroid constraints in the auxiliary LP are non-trivial and the algorithm only admits an integral solution if the sets is a single laminar family, as in [krishnaswamy2018constant]. We obtain an integral solution and define the solution in the same way. Using the same arguments as Theorem 7, we obtain the following result. When , we have , recovering the current best result for matroid median [krishnaswamy2018constant].
Let and . There exists a polynomial time bi-criteria -approximation algorithm for MatMedDis for any .
4 Knapsack Median with Discounts
In KnapMedDis, we are given a knapsack constraint instead of a cardinality constraint as in MedDis. Formally, each facility has a weight and we need to open facilities with a combined weight no more than a given threshold , that is, . MedDis is a special case of KnapMedDis where each and .
4.1 Sparsify the Instance
The natural relaxation for the standard knapsack median problem has an unbounded integrality gap [kumar2012constant]. We overcome it by adapting the pre-processing by Krishnaswamy et al. [krishnaswamy2018constant]. First, we let , and fix an unknown optimal solution to the problem. Let be the optimal objective thereof, that is, , and be the largest contribution of any single client to it, that is, . It follows that .
Fix a small constant . There are a polynomial number of possible values for , and for each , there are possible values for . Thus, we enumerate all possible pairs and assume it is as desired in what follows, i.e., is exactly the largest contribution of any client in the unknown optimum, and either when or when .
For a KnapMedDis instance , we start by creating a new “sparse” instance where some facility set is pre-selected and some clients are pre-connected to . The formal definition is given below.
We begin with the premise that is known to us and remedy this requirement later. We construct one such extended instance in the following two phases. Set and initially. It follows from the conditions in the theorem that .
First, iteratively for each that satisfies , we set . After this phase, for each , 1 is satisfied. Because is the nearest facility to in , and at most facilities are added to in this phase.
Then, iteratively for each s.t. , we set , and remove from all clients within distance from , that is, . For each such removed , using the triangle inequality, the nearest open facility in is at a distance at least , so the total contribution to the true objective of removed is at least After this phase, 2 is satisfied. Since the optimal objective of is at most , at most facilities are added to and at most that many closed balls are removed from . This finishes the construction.
Since and is obtained from by removing at most that many closed balls, we can enumerate all possible such procedures and the number of choices is at most . This eliminates the dependence on , and the existence of such an extended instance is guaranteed, thus 1 follows.
Finally, in the second phase above, for each removed by , using the triangle inequality, . We thus have