View
21
Download
0
Category
Tags:
Preview:
DESCRIPTION
On the Complexity of Allocation Problems with Probabilistic Players. Rishab Nithyanand Research Proficiency Examination Summer 2012. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A. Presentation Outline. Introduction - PowerPoint PPT Presentation
Citation preview
On the Complexity of Allocation Problems with Probabilistic Players
Rishab Nithyanand
Research Proficiency ExaminationSummer 2012
On the Complexity of Allocation Problems with Probabilistic Players 2
Presentation Outline
• Introduction
• The Password Allocation Problem
• The Weapon-Target Allocation Problem
• Conclusions and Future Work
3
Traditional Allocation Problems
• Given:– resources (r1, r2, …, rn) and tasks (t1, t2, …, tk)– objective function F
• Goal:– Find allocation for which F is optimal
• Constraint:– at most one task per resource
On the Complexity of Allocation Problems with Probabilistic Players
4
Allocation Problems with Probabilistic Players
• Given:– resources (r1, r2, …, rn) and tasks (t1, t2, …, tk)
– resource ri completes task tj with probability pij
– objective function F
• Goal:– Find allocation for which E[F] is optimal
• Constraint:– at most one task per resource
On the Complexity of Allocation Problems with Probabilistic Players
5
• Users have a large set of accounts– some are very valuable
– and some are less valuable
• Passwords are hard to remember– [Vu, 2006]: Average users remember upto 6 unique passwords.
• [Perito, 2011]: Internet accounts are easily linkable by pseudonyms.– Compromise of one account ) compromise of all accounts allocated the same password.– Some accounts (eg., email) are gateway accounts.
• Problem: – What allocation results in minimum expected loss?
The Password Allocation Problem
On the Complexity of Allocation Problems with Probabilistic Players
On the Complexity of Allocation Problems with Probabilistic Players 6
• I don’t care. I’m super secure, phish-proof, and use 40 char long passwords!– People do stupid things!
• July 12, 2012: Yahoo lost 45000 unhashed passwords.– All passwords are equal.
• Compromise probability is only server dependent.
• June 5, 2012: 6.5 million hashed passwords stolen.– Some passwords are uncrackable.
• Compromise probability is server and password dependent.
The Password Allocation Problem
7
PA as a Parallel Job Allocation Problem
• Given a set of programs to be executed and a (smaller) set of machines.
• Each program may cause a system failure with some probability.– This may be machine independent (i.e., all machines are the same).
• Parallel Processing Constraint: Failure of one of the programs ) failure of all programs on the system.
• Problem:– How should programs be allocated to machines to maximize expected
throughput?On the Complexity of Allocation Problems with Probabilistic Players
8
The Weapon-Target Allocation Problem
• Military offense allocation problem.
• Given a set of weapons and a set of enemy targets.
• Not all weapons destroy their targets– Enemy interception– Mechanical failures
• Probability of failure depends on the weapon-target pair– Placement of defenses against weapons– Distance from allocated weapon
• Problem: – What allocation maximizes expected damage to the enemy targets?
On the Complexity of Allocation Problems with Probabilistic Players
9
The Weapon-Target Allocation Problem
Research Timeline:
• Formulation: Allan Manne (Stanford) [1958]
• NP-Completeness: Lloyd and Witsenhausen (Bell Labs) [1988]
• Analysis, Variants: Hosein (MIT) [1987-1992], Athans (Bell Labs) [1989-1992]
• Approximation (heuristics): 1977 – today
• Best approximations: Ahuja (UF), Orlin (MIT) [2007]
• (Existence of) Constant-factor approximations: ??
On the Complexity of Allocation Problems with Probabilistic Players
On the Complexity of Allocation Problems with Probabilistic Players 10
Presentation Outline
• Introduction
• The Password Allocation Problem
• The Weapon-Target Allocation Problem
• Conclusions and Future Work
On the Complexity of Allocation Problems with Probabilistic Players 11
The PA Problem: Definition
• Problem Instance:– n accounts: a1, a2, …, an
– k passwords: PW1, PW2, …, PWk
– ai has value vi and compromise probability qi = (1-pi)• i.e., compromise probability is independent of password strength
• Compromise of one account 2 PWj ) compromise of all accounts 2 PWj
• Constraint: Every account receives exactly one password.
• Goal: Minimize expected loss through password compromise– Equivalent to maximizing expected survival value (or, Expected Gain (EG)).
On the Complexity of Allocation Problems with Probabilistic Players 12
• Allocation matrix: X = {xij}– xij = 1 ) account aj is allocated password PWi
– xij = 0, otherwise
• Objective Function (Expected Gain): [to be maximized]
• Constraint: – Every account is allocated exactly one password.
The PA Problem: Mathematical Formulation
kX
i=1
0@nY
j =1((pj )xi j )
nX
j =1(xi j vj )
1A
kX
i=1
0@nY
j =1((pj )xi j )
nX
j =1(xi j vj )
1A (1)xi j =
½ 1 if account j is allocated to password i0 otherwise
maximize : F =kX
i=1
0@nY
j =1((pj )x i j )
nX
j =1(xi j vj )
1A
subj ect to :kX
i=1xi j =1; 8j 2 f1;:::;ng
F is theexpected gain (EG).
F = P ki=1³ Q n
j =1((pj )xi j )P nj =1(xi j vj )
´F = P ki=1
³ Q nj =1((pj )xi j )
P nj =1(xi j vj )
´
On the Complexity of Allocation Problems with Probabilistic Players 13
Complexity of PA
Theorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part I: Formulating the Decision Version (PA2)
• Instance: – P = {p1,…,pn} where pi 2 (0,1)
– V = {v1,…,vn}– r
• Is there a partition of N ={1,…,n} into S1 and S2 such that:
• Clearly PA2 2 NP.
On the Complexity of Allocation Problems with Probabilistic Players 14
Complexity of PA
Theorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part II: Finding the known hard problem
• The Partition Problem:– Instance: Q = {q1, …, qn}, qi 2 Z+
– Is there a partition of Q into Q1 and Q2 such that:
On the Complexity of Allocation Problems with Probabilistic Players 15
Complexity of PA
Theorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part III: Making the Transformation
• Convert Partition instance to PA2 instance in poly-time.
• Given: Q = {q1, …, qn}
• Construct PA2 instance as follows:
• What is x?– For now, just a rational 2 (0,1)
On the Complexity of Allocation Problems with Probabilistic Players 16
Complexity of PA
Theorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part IV: Why it works• Solving equations:
• Gives us the following solutions:
On the Complexity of Allocation Problems with Probabilistic Players 17
Complexity of PATheorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part IV: Why it works• We will eliminate the solutions where VS1
VS2.
– As a result our solver will return that the constructed PA2 instance is a yes instance iff the Partition instance is a yes instance.
• Eliminating solution 1: – Recall our transformation:
– When we have:• Since x < 1
– Therefore, solution 1 can never occur.
On the Complexity of Allocation Problems with Probabilistic Players 18
Complexity of PATheorem: PA with 2 passwords PA2 2 NP Complete.Proof: Part IV: Why it works• Eliminating solution 2:
– Recall our transformation:
– We need to ensure that when :
– We will find an x such that
– Therefore, when , solution 2 can never occur ) PA2 2 NP Complete.
On the Complexity of Allocation Problems with Probabilistic Players 19
Efficiently Solvable Cases
The case of n = k• Optimal Strategy: Allocate exactly one account to each
password.
• Proof of optimality: – Since and , we have:
– This means an account contributes the most to the EG when it has its own password.
On the Complexity of Allocation Problems with Probabilistic Players 20
The case of identical accountsWe have The problem reduces to:
where xi = number of accounts allocated to pi
• Optimal Strategy: Assign accounts (sequentially) to the password for which the EG increases the most.
• Proof of Optimality: Greedy argument – we always stay on par or ahead of any feasible solution.
Efficiently Solvable Cases
On the Complexity of Allocation Problems with Probabilistic Players 22
A Special Case
The Case of Correlated Values and ProbabilitiesWe have:
and where .
• Property of Optimal Solution:
• Proof (Sketch):
a1 ai aj ak an
PW1 PWl PWm PWk
EG(PWl) = (vi + vk + …) pi pk ….. EG(PWm) = (vj + …) pi …..
If we have: pi > pj > pk , vi > vj > vk, and pi/qi > vj/vi then…
EG’(PWl) = (vi + vj + …) pi pj ….. EG’(PWm) = (vk + …) pk …..
EG(PWl) + EG(PWm) < EG’(PWl) + EG’(PWm)
On the Complexity of Allocation Problems with Probabilistic Players 23
Presentation Outline
• Introduction
• The Password Allocation Problem
• The Weapon-Target Allocation Problem
• Conclusions and Future Work
On the Complexity of Allocation Problems with Probabilistic Players 24
The Single Round WTA Problem: Definition
• Problem Instance:– n targets: t1, t2, …, tn
– k weapons: w1, w2, …, wk
– wi destroys tj with probability qij
• i.e., kill probability is weapon and target dependent
• Constraint: Each weapon is allocated to exactly one target.
• Goal: Minimize expected survival of enemy targets– Equivalent to maximizing expected damage to enemy targets.
On the Complexity of Allocation Problems with Probabilistic Players 25
SWTA Problem Assumptions
• Given kill probabilities are independent of other pairs.– Isolates problem from geometric and geographic factors.
• Either a target is destroyed completely or survives completely.– qij (kill probability) = 1-pij (survival probability)
• Damage is surveyed after weapons are fired.– Models short battles with limited ammunition – does not consider enemy
retreats
• No fractional allocations may be made.– A weapon can only be allocated to a single target
On the Complexity of Allocation Problems with Probabilistic Players 26
• Allocation matrix: X = {xij}– xij = 1 ) weapon wj is allocated to target ti
– xij = 0, otherwise
• Objective Function (Survival Value): [to be minimized]
• Constraint: – Every weapon is allocated to exactly one target.
The SWTA Problem: Mathematical Formulation
kX
i=1
0@nY
j =1((pj )xi j )
nX
j =1(xi j vj )
1A
kX
i=1
0@nY
j =1((pj )xi j )
nX
j =1(xi j vj )
1A (1)xi j =
½ 1 if account j is allocated to password i0 otherwise
maximize : F =kX
i=1
0@nY
j =1((pj )x i j )
nX
j =1(xi j vj )
1A
subj ect to :kX
i=1xi j =1; 8j 2 f1;:::;ng
F is theexpected gain (EG).
F = P ki=1³ Q n
j =1((pj )xi j )P nj =1(xi j vj )
´F = P ki=1
³ Q nj =1((pj )xi j )
P nj =1(xi j vj )
´
On the Complexity of Allocation Problems with Probabilistic Players 27
Complexity of SWTA
Theorem: SWTA with 2 targets (SWTA2) 2 NP Complete.Proof: Part I: Formulating the Decision Version
• Instance: – P = {pij} where pij 2 (0,1) – r
• Is there a 0-1 matrix X such that:– The sum of the survival probabilities of the 2 targets is less than r
– and every weapon is allocated to at least one target.
• Clearly SWTA2 2 NP.
On the Complexity of Allocation Problems with Probabilistic Players 28
Complexity of SWTA
Theorem: SWTA with 2 targets (SWTA2) 2 NP Complete.Proof: Part II: Finding the known hard problem
• The Rational Product Dichotomy (Fractional Subset Product):– Instance: Q = {q1, …, qn}, qi 2 (0,1)– Is there a partition of N={1, …, n} into S1 and S2 such that:
On the Complexity of Allocation Problems with Probabilistic Players 29
Complexity of SWTA
Theorem: SWTA with 2 targets (SWTA2) 2 NP Complete.Proof: Part III: Making the Transformation
• Convert RPD instance to SWTA2 instance in poly-time.
• Given: Q = {q1, …, qn}
• Construct SWTA2 instance as follows:
Where pij is the survival probability of target i after a strike by weapon j.
On the Complexity of Allocation Problems with Probabilistic Players 30
Complexity of SWTATheorem: SWTA with 2 targets (SWTA2) 2 NP Complete.Proof: Part IV: Why it works• Our SWTA2 solver will return yes iff
Where qi is the ith rational in the given RPD instance.
• By AGMI: – Therefore, can never occur.– SWTA2 solver returns yes iff
• By AGMI: – SWTA2 solver returns yes iff
which is a yes instance of RPD.
• SWTA2 2 NP Complete
On the Complexity of Allocation Problems with Probabilistic Players 31
Efficiently Solvable Cases
The Case of Identical Weapons and TargetsWe have all weapon-target pairs with same survival probability p
i.e.,The problem reduces to:
subject towhere xi is the number of weapons allocated to target i.
• Optimal Strategy: Divide weapons as evenly as possible.
On the Complexity of Allocation Problems with Probabilistic Players 32
If dividing k weapons evenly is not optimal. Then:
Target i Target j
xi weapons xj = d+xi weapons
But, switching one of the weapons target gives us:
1+xi weapons xj = d-1+xi weapons
Since p2(0,1) and xi < d+xi - 1
Therefore, switching targets strictly decreases the net survival value ) solution is not optimal
On the Complexity of Allocation Problems with Probabilistic Players 33
Efficiently Solvable Cases
The Case of Equal WeaponsWe have one type of weapon – so all weapons destroy target i
with the same probability – pi.Problem reduces to:
• Optimal Strategy: Assign weapons to the target for which the objective function (i.e., pi xi) decreases the most.
• Proof of Optimality: By induction.– When allocating one weapon to n weapons trivially true.
On the Complexity of Allocation Problems with Probabilistic Players 34
Assume Xk is the optimal solution for k weapons to n targets
Xk = <x1, x2, …, xn>
Let Xk+1 be the solution returned for k+1 weapons to n targets
Xk+1 = <x1, x2, …, xm+1, …, xn>
Where ±m · ±i 8 i 2 {1, …, n}
± =
p m x m
£ (p
m-1
)
Zk+1 = <z1, z2, …, zn>
Let Zk+1 be any other solution
Since Zk+1 Xk+1, there is a j where zk+1(j) > xk+1(j) ¸ xk (j)
Zk = <z1, z2, …, zj-1, …, zn>
Let Zk be the same solution with one less weapon for target j.
± =
p j (zj -1
) £ (p
j-1)
X*k+1 = <x1, x2, …, xj+1, …, xn>
Let X*k+1 be Xk with one more weapon allocated to target j.
± = pj xj £ (p
j -1)
·
· Since xj < zj
··
On the Complexity of Allocation Problems with Probabilistic Players 35
Efficiently Solvable Cases
The Case of One Weapon per TargetWe have each of the n targets getting at most one weapon – i.e.,
As a result:
(1) is true since xij 2 {0,1}(2) is true since there is only one xij = 1 for each target.
Therefore:
On the Complexity of Allocation Problems with Probabilistic Players 36
Efficiently Solvable Cases
The Case of One Weapon per Target• This can now be written as:
• Which is the transportation problem with:– costij = -qij
– k supply nodes with supply = 1– n demand nodes with demand = 1
On the Complexity of Allocation Problems with Probabilistic Players 37
SWTA Approximation HeuristicsThree main techniques:• Integer constraint relaxation
– Allow fractional allocations of weapons to targets.– Solve resulting LP .– Use randomized rounding to obtain approximate solution to integer problem.
• Modeling as network flow problems– Create a graph of weapons and targets.– Each edge between a weapon and target has a cost approximately equal to the change
in objective function.• Approximate due to non-linear nature
– Set appropriate constraints (eg., supply/demand, capacity).– Solve network flow problem using MCMF, MF, TP algorithms (as is appropriate).
• Localized search– Start with a feasible solution of reasonable quality.– Perform swaps and multi-swaps yielding better solutions.
On the Complexity of Allocation Problems with Probabilistic Players 38
Presentation Outline
• Introduction
• The Password Allocation Problem
• The Weapon-Target Allocation Problem
• Conclusions and Future Work
On the Complexity of Allocation Problems with Probabilistic Players 39
Conclusions and Future Work• The Password Allocation Problem
– Also models parallel processing allocation problems.
– NP Complete even when all passwords are equal
– Has several efficiently solvable cases
• Analysis for cases with varying passwords.
• Approximation Techniques – Heuristics– Boundable algorithms (??)
• Online version of the problem
Varia
bilit
y of
pas
swor
dsVariability of accounts
Equal accounts and PWs
Equal #accounts and PWs
Correlated accounts
2 P
2 NP Complete2 ?
On the Complexity of Allocation Problems with Probabilistic Players 40
Conclusions and Future Work• The Weapon-Target Allocation
Problem– NP Complete even for the single-
round version.– There are special poly-time solvable
cases.– General approaches to making
approximate solutions.• Most current work ignores
analysis – too much focus on heuristics (unboundable)!– Existence of constant-factor
bounds?– Almost no analysis for multi-round
variant.
Varia
bilit
y of
wea
pons
Variability of targets
All equal targets
Equal weapons and targets
All equal weapons
One weapon per target
2 P
2 NP Complete
Recommended