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a b s t r a c tWe consider a proof (more accurately, refutation) system based on the Sherali-Adams (SA) operator associated with integer linear programming. If F is a CNF contradiction that admits a Resolution refutation of width k and size s, then we prove that the SA rank of F is ≤ k and the SA size of F is ≤ (k + 1)s + 1. We establish that the SA rank of both the Pigeonhole Principle PHP n n−1 and the Least Number Principle LNP n is n − 2. Since the SA refutation system rank-simulates the refutation system of Lovász-Schrijver without semidefinite cuts (LS), we obtain as a corollary linear rank lower bounds for both of these principles in LS.

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advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details.

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. RANK COMPLEXITY GAP FOR LOVÁSZ-SCHRIJVER AND SHERALI-ADAMS PROOF SYSTEMS Stefan Dantchev and Barnaby MartinAbstract. We prove a dichotomy theorem for the rank of propositional contradictions, uniformly generated from first-order sentences, in both the Lovász-Schrijver (LS) and Sherali-Adams (SA) refutation systems. More precisely, we first show that the propositional translations of firstorder formulae that are universally false, i.e. fail in all finite and infinite models, have LS proofs whose rank is constant, independent of the size of the (finite) universe. In contrast to that, we prove that the propositional formulae that fail in all finite models, but hold in some infinite structure, require proofs whose SA rank grows polynomially with the size of the universe. Until now, this kind of so-called "complexity gap" theorem has been known for tree-like Resolution and, in somehow restricted forms, for the Resolution and Nullstellensatz systems. As far as we are aware, this is the first time the Sherali-Adams lift-and-project method has been considered as a propositional refutation system (since the conference version of this paper, SA has been considered as a refutation system in several further papers). An interesting feature of the SA system is that it simulates LS, the Lovász-Schrijver refutation system without semidefinite cuts, in a rank-preserving fashion.

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We consider the problem of minimizing average flow time on multiple machines when each job can be assigned only to a specified subset of the machines. This is a special case of scheduling on unrelated machines and we show that no online algorithm can have a bounded competitive ratio. We provide an O(log P )-approximation algorithm by modifying the single-source unsplittable flow algorithm of Dinitz et.al. Here P is the ratio of the maximum to the minimum processing times.We establish an Ω(log P )-integrality gap for our LPrelaxation and use this to show an Ω(log P/ log log P ) lower bound on the approximability of the problem. We then extend the hardness results to the problem of minimizing flow time on parallel machines and establish the first non-trivial lower bounds on the approximability; we show that the problem cannot be approximated to within Ω( log P/ log log P ).