6 research outputs found
Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits
Initially developed for the min-knapsack problem, the knapsack cover
inequalities are used in the current best relaxations for numerous
combinatorial optimization problems of covering type. In spite of their
widespread use, these inequalities yield linear programming (LP) relaxations of
exponential size, over which it is not known how to optimize exactly in
polynomial time. In this paper we address this issue and obtain LP relaxations
of quasi-polynomial size that are at least as strong as that given by the
knapsack cover inequalities.
For the min-knapsack cover problem, our main result can be stated formally as
follows: for any , there is a -size LP relaxation with an integrality gap of at most ,
where is the number of items. Prior to this work, there was no known
relaxation of subexponential size with a constant upper bound on the
integrality gap.
Our construction is inspired by a connection between extended formulations
and monotone circuit complexity via Karchmer-Wigderson games. In particular,
our LP is based on -depth monotone circuits with fan-in~ for
evaluating weighted threshold functions with inputs, as constructed by
Beimel and Weinreb. We believe that a further understanding of this connection
may lead to more positive results complementing the numerous lower bounds
recently proved for extended formulations.Comment: 21 page
Extended formulations from communication protocols in output-efficient time
Deterministic protocols are well-known tools to obtain extended formulations,
with many applications to polytopes arising in combinatorial optimization.
Although constructive, those tools are not output-efficient, since the time
needed to produce the extended formulation also depends on the number of rows
of the slack matrix (hence, on the exact description in the original space). We
give general sufficient conditions under which those tools can be implemented
as to be output-efficient, showing applications to e.g.~Yannakakis' extended
formulation for the stable set polytope of perfect graphs, for which, to the
best of our knowledge, an efficient construction was previously not known. For
specific classes of polytopes, we give also a direct, efficient construction of
extended formulations arising from protocols. Finally, we deal with extended
formulations coming from unambiguous non-deterministic protocols
Small extended formulation for knapsack cover inequalities from monotone circuits
Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any Δ > 0, there is a (1/Δ) O(1) n O(logn) -size LP relaxation with an integrality gap of at most 2 + Δ, where n is the number of items. Previously, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our techniques are also sufficiently versatile to give analogous results for the closely related flow cover inequalities that are used to strengthen relaxations for scheduling and facility location problems.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Strengths and Limitations of Linear Programming Relaxations
Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than and , respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than for the Vertex Cover Problem, and better than for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture