23 research outputs found
Definability of semidefinite programming and lasserre lower bounds for CSPs
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 and Zivny [20], using different techniques
Proof Complexity Meets Algebra
We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterised algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of bounded width, or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with Sums-of-Squares refutations of sublinear degree, a fact for which we provide an alternative proof. We hence ask for the existence of a natural proof system with good behaviour with respect to reductions and simultaneously small size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovasz-Schrijver satisfies both requirements
Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer ReviewedPostprint (author's final draft
Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and
linear optimization problems for convex sets by making oracle calls to their
(weak) separation problem. We observe that the previously known method for
showing that this reduction can be done in fixed-point logic with counting
(FPC) for linear and semidefinite programs applies to any family of explicitly
bounded convex sets. We use this observation to show that the exact feasibility
problem for semidefinite programs is expressible in the infinitary version of
FPC. As a corollary we get that, for the isomorphism problem, the
Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations
collapses to the Sherali-Adams linear programming hierarchy, up to a small loss
in the degree
A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers.
It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017].
In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3
Definable inapproximability: New challenges for duplicator
AbstractWe consider the hardness of approximation of optimization problems from the point of view of definability. For many -hard optimization problems it is known that, unless , no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.</jats:p
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
On the power of symmetric linear programs
© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991) showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.Peer ReviewedPostprint (author's final draft
On the power of symmetric linear programs
We consider families of symmetric linear programs (LPs) that decide a
property of graphs (or other relational structures) in the sense that, for each
size of graph, there is an LP defining a polyhedral lift that separates the
integer points corresponding to graphs with the property from those
corresponding to graphs without the property. We show that this is equivalent,
with at most polynomial blow-up in size, to families of symmetric Boolean
circuits with threshold gates. In particular, when we consider polynomial-size
LPs, the model is equivalent to definability in a non-uniform version of
fixed-point logic with counting (FPC). Known upper and lower bounds for FPC
apply to the non-uniform version. In particular, this implies that the class of
graphs with perfect matchings has polynomial-size symmetric LPs while we obtain
an exponential lower bound for symmetric LPs for the class of Hamiltonian
graphs. We compare and contrast this with previous results (Yannakakis 1991)
showing that any symmetric LPs for the matching and TSP polytopes have
exponential size. As an application, we establish that for random, uniformly
distributed graphs, polynomial-size symmetric LPs are as powerful as general
Boolean circuits. We illustrate the effect of this on the well-studied
planted-clique problem
A Finite-Model-Theoretic View on Propositional Proof Complexity
We establish new, and surprisingly tight, connections between propositional
proof complexity and finite model theory. Specifically, we show that the power
of several propositional proof systems, such as Horn resolution, bounded-width
resolution, and the polynomial calculus of bounded degree, can be characterised
in a precise sense by variants of fixed-point logics that are of fundamental
importance in descriptive complexity theory. Our main results are that Horn
resolution has the same expressive power as least fixed-point logic, that
bounded-width resolution captures existential least fixed-point logic, and that
the polynomial calculus with bounded degree over the rationals solves precisely
the problems definable in fixed-point logic with counting. By exploring these
connections further, we establish finite-model-theoretic tools for proving
lower bounds for the polynomial calculus over the rationals and over finite
fields