Integrality Gaps of Linear and Semi-Definite Programming Relaxations for Knapsack

In this paper, we study the integrality gap of the Knapsack linear program in the Sherali-Adams and Lasserre hierarchies. First, we show that an integrality gap of 2 - e persists up to a linear number of rounds of Sherali-Adams, despite the fact that Knapsack admits a fully polynomial time approximation scheme [24, 30]. Second, we show that the Lasserre hierarchy closes the gap quickly. Specifically, after t rounds of Lasserre, the integrality gap decreases to t/(t - 1). This answers the open question in [9]. Also, to the best of our knowledge, this is the first positive result that uses more than a small number of rounds in the Lasserre hierarchy. Our proof uses a decomposition theorem for the Lasserre hierarchy, which may be of independent interest.

[1]  EDWARD A. HIRSCH,et al.  COMPLEXITY OF SEMIALGEBRAIC PROOFS , 2003 .

[2]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[3]  Gloria Pérez,et al.  An O(n log n) procedure for identifying facets of the knapsack polytope , 2003, Oper. Res. Lett..

[4]  Sandy Irani,et al.  The Power of Quantum Systems on a Line , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[5]  Eden Chlamtác,et al.  Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[6]  Laurence A. Wolsey,et al.  Valid inequalities for 0-1 knapsacks and mips with generalised upper bound constraints , 1990, Discret. Appl. Math..

[7]  Michel X. Goemans,et al.  When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures? , 2001, Math. Oper. Res..

[8]  Uriel Feige,et al.  On the optimality of the random hyperplane rounding technique for MAX CUT , 2002, Random Struct. Algorithms.

[9]  H IbarraOscar,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975 .

[10]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[11]  Levent Tunçel,et al.  Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra , 2008, Discret. Appl. Math..

[12]  Peter L. Hammer,et al.  Facet of regular 0–1 polytopes , 1975, Math. Program..

[13]  Eitan Zemel,et al.  Easily Computable Facets of the Knapsack Polytope , 1989, Math. Oper. Res..

[14]  Claire Mathieu,et al.  Sherali-adams relaxations of the matching polytope , 2009, STOC '09.

[15]  Peter L. Hammer,et al.  Computing low-capacity 0–1 knapsack polytopes , 1982, Z. Oper. Research.

[16]  Assaf Naor,et al.  . 20 24 v 2 [ cs . D S ] 18 N ov 2 00 9 A ( log n ) Ω ( 1 ) integrality gap for the Sparsest Cut SDP , 2009 .

[17]  Madhur Tulsiani.  CSP gaps and reductions in the lasserre hierarchy , 2009, STOC '09.

[18]  Kevin K. H. Cheung.  On Lov[a-acute]sz--Schrijver Lift-and-Project Procedures on the Dantzig--Fulkerson--Johnson Relaxation of the TSP , 2005, SIAM J. Optim..

[19]  Toniann Pitassi,et al.  Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures , 2009, SIAM J. Comput..

[20]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[21]  Daniel Bienstock,et al.  Approximate formulations for 0-1 knapsack sets , 2008, Oper. Res. Lett..

[22]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[23]  Stefan Dziembowski,et al.  Intrusion-Resilient Secret Sharing , 2007, FOCS 2007.

[24]  Madhur Tulsiani,et al.  Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut , 2007, STOC '07.

[25]  Eitan Zemel,et al.  The Complexity of Lifted Inequalities for the Knapsack Problem , 1992, Discret. Appl. Math..

[26]  E. Balas,et al.  Facets of the Knapsack Polytope From Minimal Covers , 1978 .

[27]  Monique Laurent,et al.  A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming , 2003, Math. Oper. Res..

[28]  Toniann Pitassi,et al.  Rank bounds and integrality gaps for cutting planes procedures , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[29]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[30]  Madhur Tulsiani,et al.  Optimal Sherali-Adams Gaps from Pairwise Independence , 2009, APPROX-RANDOM.

[31]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[32]  Gyanit Singh,et al.  Improved Approximation Guarantees through Higher Levels of SDP Hierarchies , 2008, APPROX-RANDOM.

[33]  Iannis Tourlakis,et al.  New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[34]  Paolo Toth,et al.  Algorithms and computer implementations , 1990 .

[35]  Ryan O'Donnell,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[36]  Wenceslas Fernandez de la Vega,et al.  Linear programming relaxations of maxcut , 2007, SODA '07.

[37]  Eugene L. Lawler,et al.  Parameterized Approximation Scheme for the Multiple Knapsack Problem , 2009, SIAM J. Comput..

[38]  Sanjeev Arora,et al.  Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy , 2005, STOC '05.

[39]  Assaf Naor,et al.  A $(\log n)^{\Omega(1)}$ Integrality Gap for the Sparsest Cut SDP , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[40]  Venkatesan Guruswami,et al.  MaxMin allocation via degree lower-bounded arborescences , 2009, STOC '09.

[41]  Michael Alekhnovich,et al.  Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy , 2005, STOC.

[42]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..

[43]  Konstantinos Georgiou,et al.  Vertex Cover Resists SDPs Tightened by Local Hypermetric Inequalities , 2008, IPCO.

[44]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[45]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[46]  Grant Schoenebeck,et al.  Linear Level Lasserre Lower Bounds for Certain k-CSPs , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[47]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[48]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[49]  Arist Kojevnikov,et al.  Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies , 2006, ICALP.

[50]  Daniel Bienstock,et al.  Tree-width and the Sherali-Adams operator , 2004, Discret. Optim..

[51]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[52]  Deeparnab Chakrabarty,et al.  Knapsack Problems , 2008 .

[53]  T. Pitassi,et al.  Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy , 2007, FOCS 2007.

[54]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[55]  Moses Charikar,et al.  Integrality gaps for Sherali-Adams relaxations , 2009, STOC '09.

[56]  Graciela L. Nasini,et al.  Lift and project relaxations for the matching and related polytopes , 2004, Discret. Appl. Math..

[57]  Evangelos Markakis,et al.  Integrality Gaps of Semidefinite Programs for Vertex Cover and Relations to l1 Embeddability of Negative Type Metrics , 2008, SIAM J. Discret. Math..

[58]  Toniann Pitassi,et al.  Rank Bounds and Integrality Gaps for Cutting Planes Procedures , 2006, Theory Comput..

[59]  Prasad Raghavendra,et al.  Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[60]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[61]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[62]  Rishi Saket,et al.  SDP Integrality Gaps with Local̀ 1-Embeddability Subhash Khot , 2009 .

[63]  Jean B. Lasserre,et al.  An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs , 2001, IPCO.

[64]  Avner Magen,et al.  Extending SDP Integrality Gaps to Sherali-Adams with Applications to Quadratic Programming and MaxCutGain , 2010, IPCO.