Fixed-point Logic With Counting Research Articles

Inapproximability of Unique Games in Fixed-Point Logic with Counting

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

Symmetric Circuits for Rank Logic

Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finit field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by Anderson and Dawar in 2017. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.

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.

Definable inapproximability: new challenges for duplicator

AbstractWe consider the hardness of approximation of optimization problems from the point of view of definability. For many $\textrm{NP}$-hard optimization problems it is known that, unless $\textrm{P} = \textrm{NP} $, 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.

RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

AbstractMotivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power.In our first main result we show that the extensions ofFPCby rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic${\text{FPR}}^{\text{*}}$with an operator that uniformly expresses the matrix rank over finite fields is more expressive thanFPR.One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

Solving Linear Programs without Breaking Abstractions

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.

Session details: Complexity

In the 1980's Eric Lander, Jin-Yi Cai, Martin Fürer and I began to explore the power of fixed point logic with counting (FPC). This natural class is deeply entwined with the challenge of capturing the class of polynomial-time graph properties and determining the complexity of graph isomorphism - problems that remain open. Despite that, so much more is now known about the remarkable power of FPC. Anuj Dawar explains in the following lovely survey.