Approximation Algorithms problems techniques and their use in
Approximation Algorithms: problems, techniques, and their use in game theory Éva Tardos Cornell University FOCS 2002
What is approximation? Find solution for an optimization problem guaranteed to have value close to the best possible. How close? • additive error: (rare) – E. g. , 3 -coloring planar graphs is NP -complete, but 4 -coloring always possible • multiplicative error: – -approximation: finds solution for an optimization problem within an factor to the best possible. FOCS 2002 2
Why approximate? • NP-hard to find the true optimum • Just too slow to do it exactly • Decisions made on-line • Decisions made by selfish players FOCS 2002 3
Outline of talk Techniques: Problems: • Greedy • Disjoint paths • Local search • Multi-way cut and labeling • LP techniques: • • rounding Primal-dual • network design, facility location Relation to Games – local search price of anarchy – primal dual cost sharing FOCS 2002 4
Max disjoint paths problem Given graph G, n nodes, m edges, and source-sink pairs. Connect as many as possible via edge-disjoint path. t s t s FOCS 2002 t s 5
Greedy Algorithm Greedily connect s-t pairs via disjoint paths, if there is a free path using at most m½ edges: ½ 4 t m s t s t s If there is no short path at all, take a single long one. FOCS 2002 6
Greedy Algorithm Theorem: m½ –approximation. Kleinberg’ 96 Proof: One path used can block m½ better paths ½ 4 t m s t s t s Essentially best possible: m½- lower bound unless P=NP by [Guruswami, Khanna, Rajaraman, Shepherd, Yannakakis’ 99] FOCS 2002 7
Disjoint paths: open problem Connect as many as pairs possible via paths where 2 paths may share any edge t s t s • Same practical motivation • Best greedy algorithm: n½ - (and also m 1/3 -) approximation: Awerbuch, Azar, Plotkin’ 93. • No lower bound … FOCS 2002 8
Outline of talk Techniques: Problems: • Greedy • Disjoint paths • Local search • Multi-way cut and labeling • LP techniques: • • rounding Primal-dual • network design, facility location Relation to Games – local search Price of anarchy – primal dual Cost sharing FOCS 2002 9
Multi-way Cut Problem Given: – a graph G = (V, E) ; – k terminals {s 1, …, sk} – cost we for each edge e Goal: Find a partition that separates terminals, and minimizes the cost {e separated} we s 3 s 1 s 2 FOCS 2002 Separated edges s 4 10
Greedy Algorithm For each terminal in turn – Find min cut separating si from other terminals s 1 The first cut s 3 s 4 s 2 s 1 s 2 FOCS 2002 s 3 The next cut s 4 11
Theorem: Greedy is a 2 -approximation Proof: Each cut costs at most the optimum’s cut [Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis’ 94] Cuts found by algorithm: s 3 s 1 s 2 s 4 Optimum partition Selected cuts, cheaper than optimum’s cut, but each edge in optimum is counted twice. FOCS 2002 12
Multi-way cuts extension Given: – graph G = (V, E), we 0 for e E – Labels L={1, …, k} – Lv L for each node v Objective: Find a labeling of nodes such that each node v assigned to a label in Lv and it minimizes cost {e separated} we part 1 part 2 FOCS 2002 part 3 Separated edges part 4 13
gr cheap medium expensive e en e gr Re d or or s 1 Blu ee n Example s 2 Red or blue s 3 Does greedy work? For each terminal in turn – Find min cut separating si from other terminals FOCS 2002 14
Greedy doesn’t work Greedy For each terminal in turn – Find min cut separating si from other terminals gr e s 2 FOCS 2002 n ee gr Re d or or s 1 Blu ee n The first two cuts: Red or blue s 3 Remaining part not valid! 15
Local search [Boykov Veksler Zabih CVPR’ 98] 2 -approximation 1. Start with any valid labeling. 2. Repeat (until we are tired): a. Choose a color c. b. Find the optimal move where a subset of the vertices can be recolored, but only with the color c. (We will call this a c-move. ) FOCS 2002 16
A possible -move Thm [Boykov, Vekler, Zabih] The best -move can be found via an (s, t) min-cut FOCS 2002 17
Idea of the flow network for finding a -move s = all other terminals: retain current color G sc = change color to c = FOCS 2002 18
Theorem: local optimum is a 2 -approximation Partition found by algorithm: Cuts used by optimum The parts in optimum each give a possible local move: FOCS 2002 19
Theorem: local optimum is a 2 -approximation Partition found by algorithm: Possible move using the optimum Changing partition does not help current cut cheaper Sum over all colors: Each edge in optimum counted twice FOCS 2002 20
Metric labeling classification open problem Given: – – graph G = (V, E); we 0 for e E k labels L subsets of allowed labels Lv a metric d(. , . ) on the labels. Objective: Find labeling f(v) Lv for each node v to minimize e=(v, w) we d(f(v), f(w)) Best approximation known: O(ln k ln ln k) Kleinberg-T’ 99 FOCS 2002 21
Outline of talk Techniques: Problems: • Greedy • Disjoint paths • Local search • Multi-way cut and labeling • LP techniques: • • rounding Primal-dual • network design, facility location Relation to Games – local search Price of anarchy – primal dual Cost sharing FOCS 2002 22
Using Linear Programs for multi-way cuts Using a linear program = fractional cut probabilistic assignment of nodes to parts ? Label ? as : ½ + ½ Idea: Find “optimal” fractional labeling via linear programming FOCS 2002 23
Fractional Labeling Variables: 0 xva 1 p=node, a=label in Lv – xva fraction of label a used on node v Constraints: a Lv xva = 1 for all nodes v V – each node is assigned to a label cost as a linear function of x: we ½ e=(u, v) FOCS 2002 a L |xua - xva | 24
From Fractional x to multi -way cut The Algorithm (Calinescu, Karloff, Rabani, ’ 98, Kleinberg-T, ’ 99) While there are unassigned nodes • select a label a at random xva 1 u v FOCS 2002 Unassigned nodes 25
The Algorithm (Cont. ) While there are unassigned nodes – select a label a at random xva 1 u v Unassigned nodes select 0 1 at random assign all unassigned nodes v to selected label a if xva FOCS 2002 26
Why Is This Choice Good? xpa 1 p q Unassigned nodes select 0 1 at random assign all unassigned nodes v to selected label a if xva Note: • Probability of assigning node v to label a is xva • Probability of separating nodes u and v in this iteration is |xua – xva | FOCS 2002 27
From Fractional x to Multi -way cut (Cont. ) Theorem: Given a fractional x, we find multi-way cut with expected separation cost 2 (LP cost of x) Corollary: if x is LP optimum. 2 -approximation Calinescu, Karloff, Rabani, ’ 98 1. 5 approximation for multi-way cut (does not work for labeling) Karger, Klein, Stein, Thorup, Young’ 99 improved bound 1. 3438. . FOCS 2002 28
Outline of talk Techniques: Problems: • Greedy • Disjoint paths • Local search • Multi-way cut and labeling • LP techniques: • • rounding Primal-dual • network design, facility location Relation to Games – local search Price of anarchy – primal dual Cost sharing FOCS 2002 29
Metric Facility Location F is a set of facilities (servers). D is a set of clients. cij is the distance between any i and j in D F. Facility i in F has cost fi. 5 4 3 2 FOCS 2002 facility client 30
Problem Statement We need to: 1) Pick a set S of facilities to open. 2) Assign every client to an open facility (a facility in S). Goal: Minimize cost of S + p dist(p, S). 5 opened facility 4 3 2 FOCS 2002 facility client 31
What is known? All techniques can be used: • Clever greedy [Jain, Mahdian, Saberi ’ 02] • Local search [starting with Korupolu, Plaxton, and Rajaraman ’ 98], can handle capacities • LP and rounding: [starting with Shmoys, T, Aardal ’ 97] Here: primal-dual [starting with Jain -Vazirani’ 99] FOCS 2002 32
What is the primal-dual method? • Uses economic intuition from cost sharing – For each requirement, like a Lv xva = 1, someone has to pay to make it true… • Uses ideas from linear programming: – dual LP and weak duality – But does not solve linear programs FOCS 2002 33
Dual Problem: Collect Fees Client p has a fee αp (cost-share) Goal: collect as much as possible max p αp Fairness: Do no overcharge: for any subset A of clients and any possible facility i we must have p A [αp – dist(p, i)] fi amount client p would contribute to building facility i. FOCS 2002 34
Exact cost-sharing • All clients connected to a facility • Cost share αp covers connection costs for each client p • Costs are “fair” • Cost fi of selecting a facility i is covered by clients using it p αp = f(S)+ p dist(p, S) , and both facilities are fees are optimal FOCS 2002 35
Approximate cost-sharing Idea 1: each client starts unconnected, and with fee αp=0 Then it starts raising what it is willing to pay to get connected • Raise all shares evenly α Example: 4 4 = client 4 = possible facility with its cost FOCS 2002 36
Primal-Dual Algorithm (1) 4 α= 1 4 Its α =1 share could be used towards building a connection to either facility • Each client raises his fee α evenly what it is willing to pay FOCS 2002 37
Primal-Dual Algorithm (2) 4 4 α= 2 Starts contributing towards facility cost • Each client raises evenly what it is willing to pay FOCS 2002 38
Primal-Dual Algorithm (3) 4 4 α= 3 Three clients contributing • Each client raises evenly what it is willing to pay FOCS 2002 39
Primal-Dual Algorithm (4) 4 4 α= 3 Open facility clients connected to open facility Open facility, when cost is covered by contributions FOCS 2002 40
Primal-Dual Algorithm: Trouble 4 j 4 i p α= 3 Open facility Trouble: – one client p connected to facility i, but contributes to also to facility j FOCS 2002 41
Primal-Dual Algorithm (5) ghost 4 j 4 i p α= 3 Open facility Close facility j: will not open this facility. Will this cause trouble? • Client p is close to both i and j facilities i and j are at most 2α from each other. FOCS 2002 42
Primal-Dual Algorithm (6) ghost α =3 4 4 α =6 α =3 Open facility no not need to pay more than 3 Not yet connected clients raise their fee evenly Until all clients get connected FOCS 2002 43
Feasibility + fairness ? ? • All clients connected to a facility • Cost share αp covers connection costs of client p • Cost fi of opening a facility i is covered by clients connected to it • ? ? Are costs “fair” ? ? FOCS 2002 44
Are costs “fair”? ? a set of clients A, and any possible facility i we have p A [αp – dist(p, i)] fi – Why? we open facility i if there is enough contribution, and do not raise fees any further But closed facilities are ignored! and may violate fairness 4 open facility FOCS 2002 4 closed facility, ignored 45
Are costs “fair”? ? 4 open facility i 4 p cause of closing j Closed facility, ignored α’q=4 Fair till it reaches a “ghost” facility. Let α’q αq be the fee till a ghost facility is reached FOCS 2002 46
Feasibility + fairness ? ? • All clients connected to a facility • Cost share αp covers connection costs for client p • Cost αp also covers cost of selected a facilities • Costs α’p are “fair” How much smaller is α’ α ? ? 4 4 p FOCS 2002 47
How much smaller is α’ α? q client met ghost facility j j became a ghost due to client p i 4 j 4 q p p stopped raising its share first αp α’q αq Recall dist(i, j) 2 αp, so αq α’q +2 αp 3α’q FOCS 2002 48
Primal-dual approximation The algorithm is a 3 -approximation algorithm for the facility location problem [Jain-Vazirani’ 99, Mettu-Plaxton’ 00] Proof: Fairness of the α’p fees p α’p min cost [max min] cost-recovery: f(S) + p dist(p, S) = p αp α 3α’q 3 -approximation algorithm FOCS 2002 49
Outline of talk Techniques: Problems: • Greedy • Disjoint paths • Local search • Multi-way cut and labeling • LP techniques: • • rounding Primal-dual • network design, facility location Relation to Games – primal dual Cost sharing – local search Price of anarchy FOCS 2002 50
primal dual Cost sharing Dual variables αp are natural costshares: Recall: fair = no set is overcharged = core allocation p Aαp – dist(p, i) fi for all A and i. [Chardaire’ 98; Goemans-Skutella’ 00] strong connection between core costallocation and linear programming dual solutions See also Shapley’ 67, Bondareva’ 63 for other games FOCS 2002 51
Primal-Dual Cost-sharing Primal dual = for each requirement someone willing to pay to make it true Cost-sharing: only players can have shares. • Not all requirements are naturally associated with individual players. • Real players need to share the cost. FOCS 2002 52
primal dual Cost sharing Fair no subset is overcharged Stronger desirable property: population monotone (crossmonotone): Extra clients do not increase cost -shares. • Spanning-tree game: [Kent and Skorin-Kapov’ 96 and Jain Vazirani’ 01] • Facility location, single source rent-or-buy [Pal-T’ 02] FOCS 2002 53
Local search (for facility location) Local search: simple search steps to improve objective: • add(s) adds new facility s • delete(t) closes open facility t • swap(s, t) replaces open facility s by a new facility t Key to approximation bound: How bad can be a local optima? 3 -approximation [Charikar, Guha’ 00] FOCS 2002 54
Local search Price of anarchy in games Price of anarchy: facilities are operated by separate selfish agents Agents open/close facilities when it benefits their own objective. Agent’s “best response” dynamic: • Simple local steps analogous to local search. Price of anarchy: • How bad can be a stable state? • 2 -approximation in a related maximization game: [Vetta’ 02] FOCS 2002 55
Conclusions for approximation Greedy, Local search • clever greedy/local steps can lead to great results Primal-dual algorithms • Elegant combinatorial methods • Based on linear programming ideas, but fast, avoids explicitly solving large linear programs Linear programming • very powerful tool, but slow to solve Interesting connections to issues in game theory FOCS 2002 56
- Slides: 56