Oblivious Rounding and the Integrality Gap URIEL FEIGE, WEIZMANN - PowerPoint PPT Presentation

Oblivious Rounding and the Integrality Gap URIEL FEIGE, WEIZMANN MICHAL FELDMAN, TEL-AVIV U. INBAL TALGAM-COHEN, HEBREW & TEL-AVIV U. WORK DONE AT MSR, HERZLIYA 1 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Setting: A Maximization Problem (𝑊, 𝑌) EXAMPLE: MAX-CUT IN COMPLETE IN GENERAL WEIGHTED GRAPHS OF 𝑜 VERTICES 𝑌 = set of feasible solutions 𝑌 = set of cuts 𝑊 = set of linear objective functions 𝑊 = set of edge weight functions 𝑌, 𝑊 are sets of vectors of dimension 𝑜 𝑌, 𝑊 are sets of non-negative real vectors 2 (vectors in 𝑌 are 0,1 -vectors) Given 𝑤 ∈ 𝑊 , find 𝑦 ∈ 𝑌 that maximizes 𝑤 ⋅ 𝑦 2 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Oblivious Rounding Classic approach to hard maximization problem (𝑊, 𝑌) : 𝒁 Relax (𝑊, 𝑌) to (𝑊, 𝑍) where 𝑌 ⊂ 𝑍 and 𝑍 is fractional 1. 𝒀 Given 𝑤 ∈ 𝑊 find 𝑧 ∈ 𝑍 , guarantees 𝑤 ⋅ 𝑧 2. 𝒚 𝒛 3. Round 𝑧 to 𝑦 ∈ 𝑌 Rounding The approximation ratio of the rounding is 𝑤⋅𝑦 𝑤⋅𝑧 (in the worst case) If step 3 does not use 𝑤 , we call the rounding “ oblivious ”* *Not to be confused with [Young’95] 3 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Examples from the Literature OBLIVIOUS ROUNDING NON-OBLIVIOUS ROUNDING Threshold rounding for vertex cover Rounding of SDPs for CSP ◦ [Hochbaum’82] ◦ [Raghavendra- Steurer’09] Randomized rounding for set cover Facility location ◦ [Raghavan- Thompson’87] ◦ [Li’13] Random hyperplane rounding for max-cut ◦ [Goemans- Williamson’95] Welfare maximization for submodular Welfare maximization for gross substitutes valuations valuations ◦ [Feige’09, Feige - Vondrak’10] ◦ [Nisan- Segal’06] 4 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main Question For which problems and relaxations can we expect oblivious rounding to give a good approximation ratio? A question about information ◦ Rounding not restricted to be in polynomial time Answer useful for: ◦ Algorithm designers ◦ Mechanism designers (details in a few slides) 5 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main Result & Application 6 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main Result [Informal] The approximation ratio of the best oblivious rounding scheme for a given relaxation = the integrality gap of the problem’s closure The closure of problem (𝑊, 𝑌) is (𝐷(𝑊), 𝑌) ◦ where 𝐷(𝑊) is the convex closure of 𝑊 𝑤 (e.g., weights of graph edges) Corollary: If a problem is closed ( 𝑊 = 𝐷(𝑊) ) then oblivious rounding can achieve the integrality gap Closure 7 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main Result Illustration 𝒁 Integrality gap Problem Relaxation 𝒀 (𝑊, 𝑌) (𝑊, 𝑍) Oblivious rounding 𝐷 𝑊 , 𝑌 𝐷 𝑊 , 𝑍 Closure Relaxed closure Integrality gap of closure = Oblivious rounding approximation ratio Closure 8 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

An Application: Welfare Maximization Informally: Allocate 𝑛 indivisible items among buyers to maximize total value ◦ Each buyer 𝑗 has a valuation function 𝑤 𝑗 : 2 [𝑛] → ℝ ≥0 ◦ Valuations belong to classes (e.g., additive, submodular, …) More formally: ◦ 𝑤 ∈ 𝑊 = the buyers’ valuations, from class 𝑊 ◦ 𝑦 ∈ 𝑌 = an allocation ◦ 𝑧 ∈ 𝑍 = an allocation as if the items were divisible (all sets of vectors of dimension 2 𝑛 ) 9 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main thm: Approximation ratio of oblivious rounding = integrality gap of problem’s closure Welfare Max: The Relaxation Problem: Indivisible items Welfare = 5.5 $2 $3.5 Integrality gap Relaxation: Divisible items Welfare = $7 $2.5 $4.5 10 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Welfare Max: The Relaxation 𝑌 → 𝑍 The relaxation used in ≈ all welfare approximation algorithms: Configuration LP Problem: Relaxation: max 𝑗,𝑇 ∑𝑦 𝑗,𝑇 𝑤 𝑗,𝑇 max 𝑗,𝑇 ∑𝑧 𝑗,𝑇 𝑤 𝑗,𝑇 s.t. s.t. ∑ 𝑇 𝑦 𝑗,𝑇 ≤ 1 for every buyer 𝑗 ∑ 𝑇 𝑧 𝑗,𝑇 ≤ 1 for every buyer 𝑗 ∑ 𝑗,𝑇:𝑘∈𝑇 𝑦 𝑗,𝑇 ≤ 1 for every item 𝑘 ∑ 𝑗,𝑇:𝑘∈𝑇 𝑧 𝑗,𝑇 ≤ 1 for every item 𝑘 𝑦 𝑗,𝑇 ∈ {0,1} 𝑧 𝑗,𝑇 ≥ 0 11 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main thm: Approximation ratio of oblivious rounding = integrality gap of problem’s closure Welfare Max: The Closure 𝑊 → 𝐷(𝑊) Let 𝑊 be unit-demand (UD) Closure valuations: Submodular ◦ 𝑤 𝑗 𝑇 = item 𝑘∈𝑇 {𝑤(𝑘)} max Submodular Cone GS ◦ Subclass of gross substitutes (GS) GS ◦ So integrality gap of UD Coverage configuration LP = 1 Valuation classes Valuation classes 12 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main thm: Approximation ratio of oblivious rounding = integrality gap of problem’s closure Main Result Applied to Welfare Max. Theorem: For welfare maximization with unit-demand valuations, oblivious rounding of solutions to the configuration LP achieves ≤ 0.782 approximation (0.833 for 2 buyers) ◦ Despite the integrality gap of 1 for unit-demand/GS Conclusion : For welfare maximization, “ignorance is not always bliss” ◦ Need to know the valuations to round the fractional allocation Proof: The integrality gap of the configuration LP for coverage valuations is no better than 0.782 [ cf. Feige- Vondrak’10] , and coverage is the closure of unit-demand 13 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Implications for Mechanism Design Advantages of oblivious rounding for welfare maximization with strategic buyers: 1. Incentive compatibility ◦ [Duetting-Kesselheim- Tardos’15] : Can embed into a mechanism that approximately maximizes welfare in equilibrium 2. Fairness ◦ Treats buyers equally, approximation ratio holds per buyer 3. Communication ◦ Does not access exponential-sized valuations Motivates understanding the possibilities/limitations of oblivious rounding 14 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Sketch of Main Proof 15 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main thm: Approximation ratio of oblivious rounding = integrality gap of problem’s closure Approx. Ratio and Integrality Gap at 𝑧 Fix a fractional solution 𝑧 ∈ 𝑍 APPROXIMATION RATIO OF BEST INTEGRALITY GAP OF CLOSURE AT 𝑧 OBLIVIOUS ROUNDING AT 𝑧 𝑤⋅𝑦 𝑤⋅𝑦 inf 𝑤∈𝐷(𝑊) max max 𝑦∈𝐷(𝑌) inf 𝑤⋅𝑧 𝑤⋅𝑧 𝑦∈𝑌 v∈𝑊 First choose worst-case 𝑤 from the closure First choose best randomized rounding Then find the best integral solution 𝑦 Then find the worst-case 𝑤 for this rounding This is where the obliviousness comes in 16 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Main thm: Approximation ratio of oblivious rounding = integrality gap of problem’s closure Applying the Minimax Theorem Fix a fractional solution 𝑧 ∈ 𝑍 MINIMIZING “OBJECTIVE” PLAYER MAXIMIZING “ROUNDING” PLAYER 𝑤⋅𝑦 𝑤⋅𝑦 inf 𝑤∈𝐷(𝑊) max max 𝑦∈𝐷(𝑌) inf 𝑤⋅𝑧 𝑤⋅𝑧 𝑦∈𝑌 v∈𝑊 Choose minimizing mixed strategy Choose maximizing mixed strategy 17 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Summary Many commonly-used rounding schemes are oblivious ◦ Do not use the objective to round a fractional solution We study when oblivious rounding suffices for good approximation ◦ Focus on information Approximation ratio equals the integrality gap of a related problem – the closure Application to welfare maximization ◦ Oblivious rounding does not suffice for unit-demand, gross substitutes ◦ Suffices for submodular Another tool for the toolbox of algorithm and mechanism designers 18 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN

Directions for Future Research 1. Use the understanding of the potential and limitations of oblivious rounding as a guide in designing rounding schemes ◦ for problems for which tight approximation ratios not yet known ◦ e.g., when best known approximation is oblivious but the problem is not closed 2. Possibilities/limitations of polynomial-time oblivious rounding 3. Other properties of combinatorial problems predicting the success/failure of rounding techniques 19 OBLIVIOUS ROUNDING AND THE INTEGRALITY GAP FEIGE, FELDMAN, TALGAM-COHEN