Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson, Matthew; Dawar, Anuj; Holm, Bjarki

doi:10.1109/lics.2013.23

Cited by 14 publications

(25 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now turn to prove Theorem 6, which states that the weak optimization problem for explicitly given SDPs is expressible in FPC. Our result generalizes the previous work by Anderson et al [3], [4] that established the FPC-definability of linear programming, as stated in Theorem 5. In the same fashion as in their work, the central piece of the proof is a formulation of the ellipsoid method in FPC.…”

Section: Expressing Semidefinite Programssupporting

confidence: 89%

“…We show that there is an interpretation in fixed-point logic with counting (FPC) that can, for an explicitly given semidefinite program, define its optimal solution, up to a given approximation. This result generalizes earlier work by Anderson et al [3], [4] who showed an analogous result for the case of linear programs. Extending their techniques, our proof is centered around formalizing the ellipsoid method for solving SDPs in logic.…”

Section: Introductionsupporting

confidence: 91%

“…We can now state the definability result from [3], to the effect that there is an FPC interpretation that can define solutions to linear programs. We show a generalization of the result to semidefinite programs in Theorem 6.…”

Section: Logicmentioning

confidence: 99%

See 2 more Smart Citations

Definability of semidefinite programming and lasserre lower bounds for CSPs

Dawar¹,

Wang²

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

Self Cite

View full text Add to dashboard Cite

We show that the ellipsoid method for solving semidefinite programs (SDPs) can be expressed in fixed-point logic with counting (FPC). This generalizes an earlier result that the optimal value of a linear program can be expressed in this logic. As an application, we establish lower bounds on the number of levels of the Lasserre hierarchy required to solve many optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs). In particular, we establish a dichotomy on the number of levels of the Lasserre hierarchy that are required to solve the problem exactly. We show that if a finite-valued constraint problem is not solved exactly by its basic linear programming relaxation, it is also not solved exactly by any sub-linear number of levels of the Lasserre hierarchy. The lower bounds are established through logical undefinability results. We show that the SDP corresponding to any fixed level of the Lasserre hierarchy is interpretable in a VCSP instance by means of FPC formulas. Our definability result of the ellipsoid method then implies that the solution of this SDP can be expressed in this logic. Together, these results give a way of translating lower bounds on the number of variables required in counting logic to express a VCSP into lower bounds on the number of levels required in the Lasserre hierarchy to eliminate the integrality gap. As a special case, we obtain the same dichotomy for the class of MAXCSP problems, generalizing earlier Lasserre lower bound results by Schoenebeck [17]. Recently, and independently of the work reported here, a similar linear lower bound in the Lasserre hierarchy for general-valued CSPs has also been announced by Thapper andŽivný [20], using different techniques. This work was partly carried out during the Logical Structures in Computation programme at the Simons Institute for the Theory of Computing.

show abstract

Section: Expressing Semidefinite Programssupporting

confidence: 89%

Section: Introductionsupporting

confidence: 91%

See 1 more Smart Citation

Definability of semidefinite programming and lasserre lower bounds for CSPs

Dawar¹,

Wang²

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note that this proof is similar to the "lemma on a quadrangle" from[Dinitz et al 1976], but that proof does not immediately go through because C ∩ C may not be a (s, t)-cut. 7 Note that the conference version[Anderson et al 2013] of the present paper includes an independent proof of this theorem.Journal of the ACM, Vol. V, No.…”

mentioning

confidence: 99%

Solving Linear Programs without Breaking Abstractions

Anderson

Dawar

Holm

2015

J. ACM

Self Cite

View full text Add to dashboard Cite

We show that the ellipsoid method for solving linear programs can be implemented in a way that respects the symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make choices, between variables or constraints in the program unless they are distinguished by properties definable from the program. In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable in FPC. We use this to show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [Blass et al. 1999]. On the way to defining a suitable separation oracle for the maximum matching program, we provide FPC formulas defining canonical maximum flows and minimum cuts in undirected capacitated graphs.

show abstract

“…In the case of 3-colourability, there is also a construction that deploys bijection games (as in [7]) directly to show that this problem is not decidable by such families. These lower bounds should be contrasted with the fact that the existence of perfect matchings in graphs is definable in the logic FPC [2] and therefore is decidable by polynomial-size symmetric threshold circuits.…”

Section: Cai-fürer Immerman Graphsmentioning

confidence: 99%