Online Set Cover with Happiness Costs

Christine Markarian

Department of Engineering and Information Technology, University of Dubai, U.A.E.

Keywords:

Combinatorial Optimization, Online Algorithms, Competitive Analysis, Online Set Cover, Happiness Cost.

Abstract:

The Online Set Cover problem (OSC) and its variations are one of the most well-studied optimization problems

in operations research and computer science. In OSC, we are given a universe of elements and a collection

of subsets of the universe. Each subset is associated with a cost. As elements arrive over time, the algorithm

purchases subsets to cover these elements. In each step, an element arrives, and the algorithm needs to ensure

that at the end of the step, there is at least one purchased subset that contains the element. The goal is to

minimize the total cost of purchased subsets. In this paper, we study a generalization of OSC, in which a

request consisting of a number of elements arrives in each step. Each request is associated with a happiness

cost. A request is served by either a single subset containing all of its elements or by a number of subsets jointly

containing all of its elements. In the latter case, the algorithm needs to pay the happiness cost associated with

the request. The goal is to serve all requests upon their arrival while minimizing the total cost of purchased

subsets and happiness costs paid. This problem is motivated by intrinsic service-providing scenarios in which

clients need not only be served but are to be satisﬁed with the service. Keeping clients happy by serving them

with one service provider rather than many, is represented by happiness costs. We refer to this problem as

Online Set Cover With Happiness Costs (OSC-HC) and design the ﬁrst online algorithm, which is optimal

under the competitive analysis framework. The latter is a worst-case analysis framework and the standard

to measure online algorithms. It compares, for all instances of the problem, the performance of the online

algorithm to that of the optimal ofﬂine algorithm that is given all the input sequence at once and is optimal.

1 INTRODUCTION

The Set Cover problem (SC) is one of the most well-

known optimization problems, extensively studied in

operations research, computer science, and combina-

torics (Feige, 1998; Slavık, 1997; Feige et al., 2004;

Caprara et al., 2000). It has been shown to be NP-

complete in 1972 as one of Karp’s 21 NP-complete

problems (Karp, 1972). Given a universe of elements

and a collection of subsets of the universe, each as-

sociated with a cost, SC asks to purchase subsets, of

minimum costs, such that each element belongs to at

least one of these subsets. SC appears in many real-

world optimization scenarios, including client-server

applications, in which subsets represent servers and

elements represent clients that need to be served at

minimum possible costs. (Vemuganti, 1998) presents

a survey of applications in various areas as capi-

tal budgeting, cutting stock, scheduling, and vehicle

routing.

SC and its variations have been studied in many

contexts including complexity theory, approximation

algorithms, and online algorithms (Feige, 1998; Alon

et al., 2003; Clarkson and Varadarajan, 2007; Duh

and F

urer, 1997; Shuai and Hu, 2006; Gupta et al.,

2017a; Markarian and Kassar, 2020; Abshoff et al.,

2016). In this paper, we continue the study of SC in

the context of online algorithms, in which the input

sequence is not given all at once but arrives in portions

over time and the so-called online algorithm reacts to

each portion as soon as it arrives while minimizing

the total incurred costs. The performance of online

algorithms is measured using the competitive analysis

framework (Borodin and El-Yaniv, 2005). An online

algorithm is said to be r-competitive or has competi-

tive ratio r if the cost incurred by the algorithm, for

all instances of the problem, does not exceed r times

the cost of the optimal ofﬂine algorithm, that is in-

formed about all the input sequence in advance and is

optimal.

Many works have addressed SC in the online set-

ting (Alon et al., 2003; Abshoff et al., 2016; Gupta

et al., 2017a; Alon et al., 2005). (Alon et al., 2003)

introduced the Online Set Cover problem (OSC), de-

ﬁned as follows.

Markarian, C.

Online Set Cover with Happiness Costs.

DOI: 10.5220/0010738300003062

In Proceedings of the 2nd International Conference on Innovative Intelligent Industrial Production and Logistics (IN4PL 2021), pages 40-45

ISBN: 978-989-758-535-7

Deﬁnition 1. (Online Set Cover or OSC (Alon et al.,

2003)) Given a universe of elements, one arriving in

each step, and a collection of subsets of the universe.

Each subset is associated with a cost. As elements

arrive over time, the algorithm purchases subsets to

cover these elements. In each step, the algorithm

needs to ensure that at the end of the step, there is

at least one purchased subset that contains the arriv-

ing element. The goal is to minimize the total cost of

purchased subsets.

In this paper, we study a generalization of

OSC (Alon et al., 2003), which we refer to as Online

Set Cover With Happiness Costs (OSC-HC), deﬁned

as follows.

Deﬁnition 2. (Online Set Cover With Happiness

Costs or OSC-HC) Given a universe of n elements

and a collection of m subsets of the universe. Each

subset is associated with a cost. In each step, a re-

quest consisting of at most k elements arrives, such

that each element arrives only once. Each request is

associated with a happiness cost. A request is served

by either a single subset containing all of its elements

or by a number of subsets jointly containing all of its

elements. In the latter case, the algorithm needs to

pay the happiness cost associated with the request. At

the end of each step, the algorithm needs to serve the

request by previously purchased subset(s) or by pur-

chasing new subset(s). The goal is to minimize the

total cost of purchased subsets and happiness costs

paid.

OSC-HC is motivated by intrinsic service-

providing scenarios in which clients need not only be

served but are to be satisﬁed with the service. With

today’s digital transformation, maintaining good cus-

tomer relationships becomes one of the most essential

elements of a business’ success and growth. Keep-

ing clients happy by serving them with one service

provider rather than many, is one way to provide a

comfortable service and hence achieve customer sat-

isfaction. We have formalized this as the Online Set

Cover With Happiness Costs problem (OSC-HC), by

introducing happiness costs associated with requests

and incorporating them into the optimization objec-

tive. From an algorithmic point of view, this general-

ization of OSC makes sense in the presence of happi-

ness costs since otherwise elements arriving in a sin-

gle step can be treated as arriving sequentially and so

any algorithm for OSC would sufﬁce.

The Online Set Cover problem (OSC) due to

(Alon et al., 2003) is a special case of OSC-HC in

which the number k of elements for all requests is set

to 1 and all happiness costs are set to 0. There are

two lower bounds for OSC-HC resulting from lower

bounds given for OSC. (Alon et al., 2003) showed that

the best competitive ratio achievable by any determin-

istic algorithm for OSC is Ω(

lognlogm

loglogn+loglogm

), where

n is the number of elements and m is the number of

subsets. Furthermore, (Korman, 2005) showed that no

polynomial-time randomized algorithm can achieve a

competitive ratio better than Ω(lognlog m), under the

assumption that BPP 6= NP.

Results & Techniques. We develop the ﬁrst on-

line algorithm for OSC-HC, which we show has an

asymptotically optimal O(logd log n)-competitive ra-

tio, matching the lower bound in (Korman, 2005),

where:

– d is the maximum number of subsets an element

belongs to (which is at most m)

– n is the number of elements

Our algorithm is randomized and based on:

– Formulating a given instance of OSC-HC as

an online directed edge-weighted graph problem

with connectivity requirements

– Applying the techniques of constructing a frac-

tional solution and randomized rounding (Ragha-

van and Tompson, 1987) to achieve a feasible so-

lution for the underlying graph problem

– Mapping the graph solution to output a feasible

solution for OSC-HC

Outline. The rest of the paper is structured as fol-

lows. In Section 2, we give an overview of works

related to OSC-HC. In Section 3, we give a graph for-

mulation of OSC-HC. In Section 4, we present our on-

line algorithm and prove its competitive ratio in Sec-

tion 5. We dedicate Section 6 to some thoughts about

future work.

2 RELATED WORK

The Set Cover problem (SC) has been extensively

studied in the online setting. (Alon et al., 2003) intro-

duced the Online Set Cover problem (OSC) and pro-

posed an online deterministic algorithm with nearly

optimal O(log nlog m)-competitive ratio, where n is

the number of elements and m is the number of sub-

sets.

(Alon et al., 2005) gave an O(lognlog m)-

competitive randomized algorithm for a generaliza-

tion of OSC in which elements are repeated and the

algorithm needs to cover a repeating element each

time by a different subset. Moreover, a wide range

of covering problems related to OSC were studied

Online Set Cover with Happiness Costs

in the context of primal-dual algorithms (Buchbinder

and Naor, 2005; Azar et al., 2016).

Another related problem appears in (Bhawalkar

et al., 2014), in which requests comprise a set of el-

ements and subsets have capacities forming packing

constraints. The authors provide a randomized algo-

rithm with nearly optimal competitive ratio. Note that

although in this model, a request contains a set of el-

ements, as in OSC-HC, the problem is substantially

different.

Many other online models for SC were intro-

duced, such as (Gupta et al., 2017b; Abshoff et al.,

2016; Markarian and Kassar, 2020). (Gupta et al.,

2017b) studied an online dynamic model in which el-

ements that need to be covered change over time and

the goal is to construct a solution while making as

few changes per timestep as possible. (Abshoff et al.,

2016; Markarian and Kassar, 2020) studied an online

leasing model in which subsets are not purchased but

leased for different durations and prices, and elements

need to be covered only at the step they arrive.

3 GRAPH FORMULATION &

SOLUTION MAPPING

In this section, we formulate the Online Set Cover

With Happiness Costs problem (OSC-HC) as an on-

line directed edge-weighted graph problem with con-

nectivity requirements.

Given an instance of OSC-HC. The algorithm ini-

tially knows the universe of elements, the subsets, and

the subset costs. In each step, the adversary reveals to

the algorithm a request comprising a set of elements

and a happiness cost. Before the appearance of any

request, the algorithm constructs the following nodes

and edges.

Before Any Request Arrives. The algorithm con-

structs for each element a so-called element node and

for each subset, two nodes, called a subset node and

a duplicate subset node, respectively. For the edges,

the algorithm adds a directed edge from each subset

node to its corresponding duplicate subset node, of

weight equal to the cost of the associated subset. The

algorithm also adds a directed edge from each dupli-

cate subset node to each element node it contains, of

weight equal to 0.

Upon the arrival of a new request, the algorithm

constructs the following nodes and edges.

Whenever a Request Arrives. The algorithm con-

structs a request node for the request and a so-called

Figure 1: OSC-HC graph formulation of an instance of ﬁve

elements, three subsets, and a request of two elements.

intermediary node. For the edges, the algorithm adds

a directed edge from the request node to each subset

node that contains all of its elements, of weight equal

to 0. It also adds a directed edge from the request

node to the intermediary node, of weight equal to the

happiness cost associated with the request. Moreover,

it adds a directed edge from the intermediary node to

each subset node that contains at least one element of

the request but does not contain all of the elements, of

weight equal to 0.

An example graph for an OSC-HC instance of ﬁve

elements, three subsets, and a request of two elements

is illustrated in Figure 1. The problem can now be de-

scribed as follows. When a new request arrives, the

algorithm needs to ﬁnd a directed path from the re-

quest node to each element node comprising the re-

quest. The solution paths purchased by the algorithm

would either consist of the intermediary node or not.

This is translated as either a single subset covering

all the elements of the request, or multiple subsets

needed to cover the elements of the request. In the

former case, the happiness cost, represented as the

weight on the edge from the request node to the in-

termediary node, is paid, since this edge belongs to

the solution paths. Finally, the algorithm will pur-

chase the subsets that correspond to the edges in the

solution paths purchased. That is, the weights of the

edges from the subset nodes to the corresponding du-

plicate subset nodes that belong to the solution paths

represent the subset costs incurred by the algorithm.

4 ONLINE ALGORITHM

In this section, we present an online randomized algo-

rithm for OSC-HC based on the online graph problem

described earlier.

Following the example in Figure 1, request {a, c}

arrives. The algorithm needs to ﬁnd two paths: one

from the request node to the element node a, and an-

other from the request node to the element node c.

Every edge of weight equal to a subset cost repre-

IN4PL 2021 - 2nd International Conference on Innovative Intelligent Industrial Production and Logistics

sents the corresponding subset. Thus, if such an edge

belongs to the solution paths, then the subset is pur-

chased by the algorithm. Similarly, every edge of

weight equal to a happiness cost represents the re-

quest’s happiness cost. Hence, the algorithm pays the

happiness cost if the corresponding edge is in the so-

lution paths.

Upon the arrival of a new request, the nodes and

the edges associated with the request are formed,

as described in the previous section. The algorithm

assigns a fractional value v

to each edge e in the

graph. These values are set to 0 when the edges are

formed and increase over time. The maximum ﬂow

between two nodes is deﬁned as the smallest total val-

ues of edges which if removed would disconnect the

two nodes. These edges form a minimum cut. Be-

fore the algorithm receives its ﬁrst request, it gen-

erates a random number r chosen as the minimum

among 2

logn

independent random variables dis-

tributed uniformly in the interval [0, 1], where n is the

total number of elements.

Given a request and its elements. For each el-

ement node, the algorithm calls the so-called Find-

Directed-Path function, that returns a set of edges

forming a directed path from the request node to the

element node. We denote the weight of an edge e by

. The algorithm’s steps describing its reaction to a

new request are depicted in Algorithm 1 below.

Algorithm 1: Online Algorithm for OSC-HC.

For each element node j:

If there already is a directed path to the element

node j in the current solution, do nothing. Other-

wise,

- Run Find-Directed-Path( j).

- Buy the subsets and pay the happiness costs

corresponding to the edges outputted by Find-

Directed-Path( j).

Find-Directed-Path( j)

i. While the maximum ﬂow from the request node

to j is less than 1:

- Generate a minimum cut K from the request

node to j and then increase the value v

of each

edge e ∈ K according to the following equation:

← v

(1 +

) +

|K | · w

ii. Output edge e if its value v

≥ r.

iii. If there is no directed path to the element

node j in the current solution, output the edges

of a smallest-weight directed path from the request

node to the element node j.

Feasibility. Upon the arrival of a new request, the

algorithm breaks down the request into parts, one for

each element in the request, after forming the graph

nodes and edges associated with the request. Accord-

ing to the graph formulation, the only way a request

node is connected to an element node is through a di-

rected path containing a subset node and its duplicate

subset node which represent a subset containing the

element in the original instance. The weight of the

edge in between these nodes is what the algorithm

pays in terms of subset costs. If the solution path does

not contain the intermediary node, then the edge as-

sociated with a happiness cost is not in this path as

per the graph construction. In this case, the algorithm

does not pay a happiness cost. Note that, at some

point in a given time step, if the algorithm decides to

purchase a subset that contains all the elements of the

request, then all the remaining elements of the request

are covered and the step ends. On the other hand,

even if the algorithm pays the happiness cost at some

point in a given time step, it may eventually decide

to purchase a subset that contains all the elements of

the request. In such a case, the algorithm, in practice,

does not have to pay for the happiness cost, since its

decisions are revocable within a time step. However,

as we will see in the next section, this does not affect

our competitive analysis of the algorithm.

5 COMPETITIVE ANALYSIS

In this section, we show that the online algorithm pre-

sented has an O(logd log n)-competitive ratio, where

d is the maximum number of subsets an element be-

longs to and n is the number of elements.

Let S be the collection of edges outputted by the

algorithm excluding the edges outputted in Step iii.

We denode by C

the total weight of these edges. Let

Opt be the cost of an optimal ofﬂine solution.

The function Find-Directed-Path outputs an edge

if its fractional value exceeds the random number r

(recall that r is generated before the execution of the

algorithm).

Fix edge e and i : 1 ≤ i ≤ 2

logn

. We denote

by X

e,i

the indicator variable of the event that e is

outputted by Find-Directed-Path. C

can then be ex-

pressed using the following sum.

∑

e∈S

logn

∑

i=1

· Exp[X

e,i

] (1)

= 2

logn

∑

e∈S

(2)

Online Set Cover with Happiness Costs

We compare now this sum to the optimal ofﬂine

solution, as follows. Every time a minimum cut is

constructed, there is at least one edge that belongs to

the optimal ofﬂine solution, due to the deﬁnition of

minimum cut and the fact that the optimal solution

must also ﬁnd a directed path to each element node.

It remains to count the number of times the function

Find-Directed-Path constructs a minimum cut.

Lemma 1. The total number of times Find-Directed-

Path constructs a minimum cut is at most O(Opt ·

log|K|), where |K| is the size of the largest minimum

cut.

Proof. The proof is based on observing the edges in

the optimal solution. Each edge in the optimal so-

lution can appear in a bounded number of minimum

cuts, after which its fractional value becomes 1 and

it can’t appear in any future minimum cut, as per

the while-loop condition of the Find-Directed-Path

function. This bound can be achieved by using the

equation in the algorithm and counting the number

of times needed for the fractional value of any edge

in the optimal solution to become 1. The bound de-

pends on the weight of the corresponding edge and

the maximum size of the minimum cut: O(w

log|K|).

This bound holds true for every edge in the opti-

mal solution. Moreover, each minimum cut con-

structed must contain at least one edge from the op-

timal solution. Hence, the number of times the func-

tion Find-Directed-Path constructs a minimum cut is

O(Opt · log |K|).

We also have that the size of any minimum cut

constructed does not exceed d, the maximum num-

ber of subsets an element belongs to, since there are

at most d directed paths from the request node to

the element node. Hence, |K| ≤ d. Next, we show

that the total increase in the fractional values of the

edges associated with each minimum cut can be up-

per bounded by 2.

Lemma 2. The total increase in the fractional values

of the edges associated with each minimum cut is at

most 2.

Proof. Fix any minimum cut K. Edge e in K incurs

an increase of w



·w



. The maximum ﬂow

is less than 1 before the increase is made, that is,

∑

e∈K

< 1. The same can be said about all the edges

in the cut. Hence, the following holds true:

∑

e∈K



· w



< 2

As a result of Lemma 1 and Lemma 2, we con-

clude that

∑

e∈S

≤ O(Opt · log d). Thus,

≤ O(Opt · log n · log d) (3)

It remains to upper bound the cost of the algorithm

in Step iii, which we prove affects the competitive ra-

tio of the algorithm by a negligible factor.

We deﬁne the ﬂow of a path to be the minimum

value among the edge values of the path. To calculate

the cost incurred in Step iii, we need to observe the

probability that there is no directed path outputted for

the element node prior to this step. This probability is

at most the probability that r exceeds the ﬂow of each

directed path to the element node. We ﬁx a minimum

cut K constructed at the end of Step i. Before per-

forming Step ii, Find-Directed-Path guarantees that

the sum of ﬂows of all paths to the element node is at

least 1. Hence, the probability that there is no directed

path to the element is:

∏

e∈K

(1 − v

) ≤ e

−

∑

e∈K

≤

Computing for all i: 1 ≤ i ≤ 2

logn

, the proba-

bility that there is no directed path to the element node

is at most

. When there is no directed path to the el-

ement node, the algorithm buys the smallest-weight

path to the element node, which is a lower bound on

Opt. Thus, the cost of the algorithm incurred in Step

iii is at most n ·

Opt

, since there are n elements and

each element may arrive only once.

Hence, we conclude the following theorem.

Theorem 1. There is an O(logd log n)-competitive

randomized algorithm for the Online Set Cover with

Happiness Costs problem, where d is the maximum

number of subsets an element belongs to and n is the

number of elements.

6 CONCLUDING THOUGHTS

We have proposed in this paper a new framework

for optimization problems, motivated by service-

providing scenarios in which the goal is to minimize

serving costs while taking into account the happiness

of clients. We have considered in our model one hap-

piness prespective, in which clients prefer to be served

through one service provider. Clearly, there is much

one can explore, since the satisfaction of clients can

be viewed from various perspectives.

Another research direction is to study other op-

timization problems with different objectives, using

this framework, such as the Online Facility Loca-

tion with Service Installation Costs problem, in which

IN4PL 2021 - 2nd International Conference on Innovative Intelligent Industrial Production and Logistics

the distances between clients and the facilities they

are served by are also minimized (Markarian, 2021;

Markarian and Khallouf, 2021).

Each element in our model arrives only once. This

is in fact needed to achieve the competitive ratio of

the algorithm. It would be interesting to extend our

model to include repetition of elements as in (Alon

et al., 2005).

In our model, the algorithm is not given any in-

formation about future requests. In fact, it might be

possible to make some assumptions about the future

and to use this information to improve decisions, by

considering, for instance, various probability distribu-

tions for the request arrival.

Implementing the proposed algorithm on a sim-

ulated or real environment is an interesting next step.

This would allow us to understand the difﬁculty of the

problem as well as the performance of the algorithm

and its effectiveness in practical applications.

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