approximation algorithms for envy-free profit-maximization problems

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.

approximation algorithms for envy-free profit-maximization problems

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