Approximation Algorithms for Envy-free Profit-maximization problems Chaitanya Swamy University of...

Approximation Algorithms for Envy-free Profit-

maximization problems

Chaitanya SwamyUniversity of Waterloo

Joint work with Maurice Cheung Cornell University

Profit-maximization pricing problems

• seller with m indivisible non-identical items

• items available in limited supply or capacity

• n customers wanting subset(s) of items

Profit-maximization problem: set prices on items and allocate items to customers so that– capacity constraints are respected– each customer can afford her allotted subset (value ≥ price)

GOAL: maximize seller profit = total price paid by customers

Envy-free (EF) profit maximization: also require that– customer is allotted set with maximum utility (=

value – price)

Why envy-freeness? •Economic motivation – models a fair, “equilibrium” outcome– Fairness: seller is not biased towards any

specific customer– Equilibrium: each customer is maximally

happy, no incentive to deviate from/dispute the allocation (given the prices)

•In settings where customers may lie about valuations, envy-free problem used as a metric for comparing profit-maximization truthful mechanisms

[Here: consider setting where valuations are known]

With arbitrary (set-based) customer valuation functions {vi(.)}, envy-free problem becomes very hard: •complexity issues in describing the valuation

functions

•even deciding if a given solution (pricing + allocation) is feasible is coNP-hard (even given a value oracle for computing vi(S) given set S)

•even structured cases are not well understood Focus on a more structured setting – the single-minded setting

The single-minded problem (SMEFP)

•m non-identical items: item e has supply ue (possibly )

•n customers: customer i desires a single subset Si of itemshas valuation vi = max amount she will pay for Si

Set prices {pe} on items, choose a set W of winners s.t.

– capacity constraints: |{ iW: e Si }| ≤ ue for all items e

– every winner can afford her set: vi ≥ ∑ eSi pe

for all iW

– envy-freeness: vi ≤ ∑ eSi pe for all iW

GOAL: maximize profit = ∑ iW ∑ eSi pe

= ∑ e pe.|{ iW: e Si }|

5 5 5

88

item

valuation viset Si

ue= 2 for all items




for all iW



= ∑ e pe.|{ iW: e Si }|

price pe

5 5 5

88

35 5ue= 2 for all items




for all iW



= ∑ e pe.|{ iW: e Si }|

winner

price pe

5 5 5

88





for all iW



= ∑ e pe.|{ iW: e Si }|

envy-free solution with profit = 2(3+5)+5 = 21

winner

price pe

5 5 5

88





for all iW



= ∑ e pe.|{ iW: e Si }|

envy-free solution with profit = 2(5+3+3) = 22

winner

5 5 5

88





for all iW



= ∑ e pe.|{ iW: e Si }|

NOT an envy-free solution

5 5 5

88





for all iW



= ∑ e pe.|{ iW: e Si }|

Two special cases

• Tollbooth problem: items are edges of a graph G, each set Si is a path of G– problem is APX-hard even when G is a star, all vi = 1, all ue = (Guruswami et al. (G+05))

• Highway problem: the graph G is a path sets Si intervals– problem is NP-hard even when the intervals are nested, unlimited supply: all ue = (Briest-Krysta)

Approximation Algorithm

Hard to solve the single-minded problem exactly – even very specialized cases are NP-hard.Settle for approximate solutions. Give polytime algorithm that always finds near-optimal solutions.

A is a -approximation algorithm if,

•A runs in polynomial time,

•A(I) ≥ OPT(I)/ on all instances I ( ≥ 1).

is called the approximation ratio of A.

Related Work• Guruswami et al. (G+05) introduced the envy-free

problem– also introduced the structured case of unit-demand

customers

• NO previous approx. results for SMEFP (with limited supply) or even its special cases, e.g., tollbooth, highway problems

• Previous settings considered– unlimited supply problem: logarithmic

approximation bounds; G+05, Briest-Krysta (BK05), Balcan-Blum (BB05)

– non-envy-free limited supply problem: quasi- or pseudo-polytime exact algorithms/approx. schemes for restricted SM instances; G+05, BK05, BB05, Grigoriev et al., Elbassioni et al.

– non-EF problem with submodular+ valuations: Dobzinski et al., Balcan et al.

Techniques do not extend to the envy-free problem.

Related Work (contd.)

•Hardness results:

– general SM problem: m½-inapproximability even when ue = 1 e; log

c m-inapproximability (c < 1) with unlimited-supply (Demaine et al.)

– specialized cases are also APX- or NP-hard (G+05, BK05)

Our Results• Give the first approximation algorithms for single-minded

envy-free profit-maximization (SMEFP) with limited supply– for any class of single-minded problems, given LP-based -

approx. algorithm for finding the max-value allocation, find an EF solution with Profit ≥ O(OPTvalue/(.log umax)) O(.log umax)-approx.

– O(m½ . log umax)-approx. for general SMEFP

– O(log umax)-approx. for tollbooth problem on trees

[“Often” -inapprox. for max-value problem -inapprox. for SMEFP]

• Reduction shows – concrete, explicit connection b/w OPTvalue and optimum profit

– ratio of profit obtained by non-EF and EF solutions = O(.log umax)

Social-welfare-maximization (SWM)

problemChoose an allocation, i.e., winner-set W, with maximum total value that satisfies capacity constraints: |{ iW: eSi }| ≤ ue e

LP relaxation: xi : indicates if i is chosen as a winner

Maximize ∑i vixi

subject to, ∑i:eSi xi ≤ ue

for all e0 ≤ xi ≤ 1. for all i.LP-optimum is an upper bound on optimum profit.

Will use the LP to determine winner-set W, and will compare the profit achieved against the LP-optimum

But how does the LP help in setting prices?

OPT := max ∑i vixi (P)

s.t. ∑i:eSi xi ≤ ue e

0 ≤ xi ≤ 1.i.

OPT := max ∑i vixi (P)= min ∑e ueye + ∑i zi

(D)s.t. ∑i:eSi

xi ≤ ue e s.t.

∑eSi ye + zi ≥ vi i

0 ≤ xi ≤ 1.i. ye, zi ≥ 0 e,

i

Key insight: the dual variables (ye) furnish envy-free prices

By complementary slackness, at optimality,

• if xi > 0 then ∑eSi ye + zi = vi ∑eSi

ye ≤ vi

• if xi < 1 then zi = 0 ∑eSi ye ≥ vi

• if ye > 0 then ∑i:eSi xi = ue if x is an integer optimal soln. to (P), (y, z) is opt.

soln. to (D), then x along with prices {ye} is a feasible soln. with profit ∑e ueye

x (P) need not have an integer optimal solutionx ∑e ueye could be much smaller than the optimum profit

Highway problemm edges on a path, edge e has capacity ue

n customers, customer i has valuation vi for subpath Si

if x is an integer optimal soln. to (P), (y, z) is opt. soln. to (D), then alloc’n. x + prices {ye} is a feasible soln. with profit ∑e ueye

x (P) need not have an integer optimal solution(P) always has an integer optimal soln. – follows from total-unimodularity

OPT := max ∑i vixi (P) = min ∑e ueye + ∑i zi (D)

s.t. ∑i:eSi xi ≤ ue e s.t. ∑eSi

ye + zi

≥ vi i

0 ≤ xi ≤ 1 i ye, zi ≥ 0e, i

x ∑e ueye could be much smaller than the optimum profit with unit capacities ue = 1 e, there is an optimal soln. to (D) with zi = 0 for all i get Profit = OPT

What about higher capacities?



ye + zi

≥ vi i

0 ≤ xi ≤ 1 i ye, zi ≥ 0e, i

ue= 2 for all evi = 1 for all i

(a)

In every optimal soln. to (D), have ∑e ye ≤ 1

since (a) is a winner, so Profit = ∑e ueye ≤ 2,

BUT setting price = 1 for all e yields optimal profit = n – 1

Idea: lowering capacities can increase profitAbove: if we set ue = 1 for all e, then there is an optimal soln. with ye = 1 e get optimal profit

Key technical lemma: can always find a capacity-vector u' ≤ u s.t. there exists an optimal dual soln. with capacities {u'e} with ∑e

u'eye ≥ OPT/O(log umax)

if we solve (P) and (D) with capacities u' to get allocation and prices, then get soln. with Profit ≥ OPT/O(log umax)

ue= 2 for all evi = 1 for all i

(a)

The AlgorithmConsider uniform capacities ue = U for simplicity

(Pk), (Dk): primal, dual LPs with ue = k,

OPT(k) : common optimal value of (Pk) and (Dk)

1.For k = 1,2,…,U, find optimal soln. (y(k), z(k)) to (Dk) that maximizes ∑e k ye.

2.Choose c ≤ U that maximizes ∑e c ye(c).

3.Return {ye(c)} as prices, optimal soln. to (Pc)

as allocation.Can be made polytime by considering k = powers of (1+).

AnalysisOPT(k) := max ∑i vixi (Pk) = min

∑e k ye + ∑i zi (Dk)

s.t. ∑i:eSi xi ≤ k e s.t. ∑eSi

ye + zi

≥ vi i

0 ≤ xi ≤ 1 i ye, zi ≥ 0e, i

Lemma: OPT(.) is a concave f’n.OPT(.) is linear b/w k and k' iff common soln. (y, z) that is optimal for both (Dk), (Dk')

Why? If c = k+(1–)k', opt. soln. to (Dc) is feasible for (Dk), (Dk')

1

OPT(1)

OPT(U)

U

Let bk = break pt. of OPT(.) before k

Lemma: ∑e k ye(k) ≥ k.

k – bk

OPT(k)-OPT(bk)

k

OPT(k)

bk

...

..

Let bk = break pt. of OPT(.) before k


k – bk

OPT(k)-OPT(bk)

Proof: Let (y, z) be common optimal solution to (Dk), (Dbk

).

RHS = ∑e k ye ≤ ∑e k ye(k).

k

OPT(k)

bk

...

..


k – bk

OPT(k)-OPT(bk)OPT(U)2.HU

Theorem: Return Profit P* ≥

k

OPT(k)

bk

...

..

OPT(U)2.HU


Proof: We have P* ≥ ∑e k ye(k) k.

P*(U – bU)/U ≥ OPT(U) – OPT(bU)

P*(k– bk)/k ≥ OPT(k) – OPT(bk)

P* ≥ OPT(1) [b1 = 0]

...

...


k – bk

OPT(k)-OPT(bk)

Suppose first that bk = k-1 k.

k

OPT(k)

bk

...

..

OPT(U)2.HU



P*/U = P*(U – bU)/U ≥ OPT(U) – OPT(bU)

P*/k = P*(k– bk)/k ≥ OPT(k) – OPT(bk)

P* = P* ≥ OPT(1) [b1 = 0]

...

...


k – bk

OPT(k)-OPT(bk)

Suppose first that bk = k-1 k.

P*.HU ≥ OPT(U)

k

OPT(k)

bk

...

..

OPT(U)2.HU




P*/k ≥ [OPT(bk+1) – OPT(k)]/[bk+1 – k]


P* ≥ OPT(1)[b1

= 0]

...

...

k+1

bk+

1


k – bk

OPT(k)-OPT(bk)

May assume that bk [k-1,k) k.

k

OPT(k)

bk

...

..


k – bk

OPT(k)-OPT(bk)

OPT(U)2.HU

Theorem: Return profit P* ≥



P*(bk+1 – k)/k ≥ OPT(bk+1) – OPT(k)


P* ≥ OPT(1) [b1 = 0]

...

...


k+1

bk+

1

k

OPT(k)

bk

...

..


k – bk

OPT(k)-OPT(bk)

OPT(U)2.HU

Theorem: Return profit P* ≥


P*/U ≥ P*(U – bU)/U ≥ OPT(U) – OPT(bU)

P*/k ≥ P*(bk+1 – k)/k ≥ OPT(bk+1) – OPT(k)

P*/k ≥ P*(k– bk)/k ≥ OPT(k) – OPT(bk)

P* ≥ P*(b2 – 1) ≥ OPT(b2) – OPT(1)

P* ≥ P* ≥ OPT(1) [b1 = 0]

...

...

2P*.HU ≥ OPT(U)


k+1

bk+

1

k

OPT(k)

bk

...

..

Remark: Can prove that all break pts. bk are integersi.e., dual soln. changes only at integer values of k can take bk = k-1 k and save factor of 2

Proof heavily uses total-unimodularity of constraint matrix

Open Question:

•What does this integer-breakpoint property mean?

–Implications about structure of polytope? Applications in combinatorial optimization (CO)? How does it relate to other concepts in CO?–Are there other interesting classes of problems with (“approx.”) integer-breakpoint property?

The general problem (SMEFP)

Recall complementary slackness: at optimality,• if xi > 0 then ∑eSi

ye + zi = vi ∑eSi ye ≤

vi

• if xi < 1 then zi = 0 ∑eSi ye ≥ vi

• if ye > 0 then ∑i:eSi xi = ue

x (P) need not have an integer optimal solutionif we have a winner-set W s.t. {i: xi = 1} W {i: xi > 0}, and ue ≥ |{ iW: e Si }| ≥ ∑i:eSi

xi / for every e,

then(W, {ye}) is a feasible soln. with Profit ≥ ∑e

ueye/



ye + zi

≥ vi i

0 ≤ xi ≤ 1 i ye, zi ≥ 0e, i

Can use an LP-based -approx. algorithm for SWM-problem to obtain W with desired properties– W {i: xi = 1}

– decompose (remaining fractional soln.) / into convex combination of integer solns. (Carr-Vempala, Lavi-S)

Key technical lemma: can always find a capacity-vector u' ≤ u s.t. there exists an optimal dual soln. with capacities {u'e} with ∑e

u'eye ≥ OPT/O(log umax)

if we solve (Du') to get prices, round opt. soln. to (Pu') to get W, then get soln. with Profit ≥ OPT/O(.log umax)

How to deal with non-uniform capacities?

Similar approach: obtain a bound on max. profit achievable with an optimal dual soln. with capacities {u'e}

Leverage this to get a telescoping-sum argument

BUT, OPT(.) is now a multivariate function – makes both steps more difficult

Need to define and analyze breakpoints, slopes of OPT(.) along suitable directions.

Algorithm for (general) SMEFP

1.For suitable vectors k = k1, k2,…,kr , (r:

polynomial) find optimal soln. (y(k), z(k)) to (Dk) that maximizes ∑e keye.

2.Choose vector c{k1, k2,…,kr} that maximizes ∑e ceye

(c).

3.Return {ye(c)} as prices, round optimal soln.

to (Pc) to obtain W.

(Pk), (Dk): primal, dual LPs with ue = ke e,

OPT(k) : common optimal value of (Pk) and (Dk)

Summary of Results•Give the first approx. algorithms for single-minded envy-

free profit-maximization problems with limited supply– primal LP for SWM-problem can be rounded to get

allocation; dual LP furnishes envy-free prices

– can find capacity-vector u' ≤ u and opt. dual soln. (y, z) for (Du') s.t. ∑e u'eye ≥ OPT/O(log umax)

– so LP-based -approx. for SWM-problem O(.log umax)-approx. for envy-free problem

•Same guarantees when customers desire multiple disjoint multisets, and for non-EF versions of these problems

•Envy-freeness hurts seller by at most O(.log umax)-factor

Open Questions• Results for (more) general set-based valuation

functions, say, given a demand-oracle for each customer. Need a new upper bound – OPTSWM can be >> opt. profit (Blum)

• Improved results for structured SM problems. Constant-factor for tollbooth problem? PTAS for highway problem?

• Better understanding of the integer-breakpoint property.– Implications about structure of polytope?

Applications in combinatorial optimization (CO)? How does it relate to other concepts in CO?

– Are there other interesting classes of problems with (“approx.”) integer-breakpoint property?

Thank You.

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