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On the Complexity of Allocation Problems with Probabilistic Players

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

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On the Complexity of Allocation Problems with Probabilistic Players

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  1. On the Complexity of Allocation Problems with Probabilistic Players RishabNithyanandResearch Proficiency ExaminationSummer 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

  2. 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

  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. The Password Allocation Problem • 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? On the Complexity of Allocation Problems with Probabilistic Players

  6. The Password Allocation Problem • 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. On the Complexity of Allocation Problems with Probabilistic Players

  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

  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 • aihas value vi and compromise probability qi = (1-pi) • i.e., compromise probability is independent of password strength • Compromise of one account 2PWj) compromise of all accounts 2PWj • 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. The PA Problem: Mathematical Formulation • 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. On the Complexity of Allocation Problems with Probabilistic Players

  13. Complexity of PA Theorem: PA with 2 passwords PA22 NP Complete. Proof: Part I: Formulating the Decision Version (PA2) • Instance: • P = {p1,…,pn} where pi2 (0,1) • V = {v1,…,vn} • r • Is there a partition of N ={1,…,n} into S1 and S2 such that: • Clearly PA22 NP. On the Complexity of Allocation Problems with Probabilistic Players

  14. Complexity of PA Theorem: PA with 2 passwords PA22 NP Complete. Proof: Part II: Finding the known hard problem • The Partition Problem: • Instance: Q = {q1, …, qn}, qi2 Z+ • Is there a partition of Q into Q1and Q2 such that: On the Complexity of Allocation Problems with Probabilistic Players

  15. Complexity of PA Theorem: PA with 2 passwords PA22 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 PA22 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 PA Theorem: PA with 2 passwords PA22 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 PA Theorem: PA with 2 passwords PA22 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 )PA22 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. Efficiently Solvable Cases The case of identical accounts We 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. On the Complexity of Allocation Problems with Probabilistic Players

  21. A Special Case a1 ai aj ak an EG(PWm) = (vj + …) pi ….. EG(PWl) = (vi + vk + …) pipk ….. PW1 PWl PWm PWk EG’(PWl) = (vi + vj + …) pipj ….. EG’(PWm) = (vk + …)pk ….. If we have: pi > pj > pk , vi > vj > vk, and pi/qi > vj/vi then… EG(PWl) + EG(PWm) < EG’(PWl) + EG’(PWm) On the Complexity of Allocation Problems with Probabilistic Players The Case of Correlated Values and Probabilities We have: and where . Property of Optimal Solution: Proof (Sketch):

  22. 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

  23. The Single Round WTA Problem: Definition • Problem Instance: • n targets: t1, t2, …, tn • k weapons: w1, w2, …, wk • widestroys 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

  24. 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

  25. The SWTA Problem: Mathematical Formulation • 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. On the Complexity of Allocation Problems with Probabilistic Players

  26. Complexity of SWTA Theorem: SWTA with 2 targets (SWTA2) 2 NP Complete. Proof: Part I: Formulating the Decision Version • Instance: • P = {pij} where pij2 (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 SWTA22 NP. On the Complexity of Allocation Problems with Probabilistic Players

  27. 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}, qi2 (0,1) • Is there a partition of N={1, …, n} into S1and S2 such that: On the Complexity of Allocation Problems with Probabilistic Players

  28. 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

  29. Complexity of SWTA Theorem: 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 iffwhich is a yes instance of RPD. • SWTA22NP Complete On the Complexity of Allocation Problems with Probabilistic Players

  30. Efficiently Solvable Cases The Case of Identical Weapons and Targets We have all weapon-target pairs with same survival probability pi.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

  31. If dividing k weapons evenly is not optimal. Then: Target i Target j xj = d+xi weapons 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

  32. Efficiently Solvable Cases The Case of Equal Weapons We 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., pixi) 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

  33. Assume Xk is the optimal solution for k weapons to n targets · Xk = <x1, x2, …, xn> Zk = <z1, z2, …, zj-1, …, zn> Let Zk be the same solution with one less weapon for target j. · Since xj < zj ± = pjxj£ (pj-1) ± = pj(zj-1)£ (pj-1) ± = pmxm£ (pm-1) Since Zk+1Xk+1, there is a j where zk+1(j) > xk+1(j) ¸xk (j) Let X*k+1 be Xk with one more weapon allocated to target j. · · X*k+1 = <x1, x2, …, xj+1, …, xn> Xk+1 = <x1, x2, …, xm+1, …, xn> Zk+1 = <z1, z2, …, zn> Let Xk+1 be the solution returned for k+1 weapons to n targets Let Zk+1 be any other solution Where ±m·±i8i2 {1, …, n} On the Complexity of Allocation Problems with Probabilistic Players

  34. Efficiently Solvable Cases The Case of One Weapon per Target We have each of the n targets getting at most one weapon – i.e., As a result: is true since xij2 {0,1} is true since there is only one xij = 1 for each target. Therefore: On the Complexity of Allocation Problems with Probabilistic Players

  35. 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

  36. SWTA Approximation Heuristics Three 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

  37. 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

  38. Conclusions and Future Work Variability of passwords 2 NP Complete 2 ? 2 P Equal accounts and PWs Equal #accounts and PWs Correlated accounts Variability of accounts • 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 On the Complexity of Allocation Problems with Probabilistic Players

  39. Conclusions and Future Work 2 NP Complete All equal targets Variability of weapons One weapon per target Equal weapons and targets 2 P All equal weapons Variability of targets • 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. On the Complexity of Allocation Problems with Probabilistic Players

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