24 research outputs found
An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for
testing isomorphism of two graphs and indeed, the currently fastest
implementations of isomorphism solvers all follow this approach. While these
solvers are fast in practice, from a theoretical point of view, no general
lower bounds concerning the worst case complexity of these tools are known. In
fact, it is an open question whether individualization-refinement algorithms
can achieve upper bounds on the running time similar to the more theoretical
techniques based on a group theoretic approach.
In this work we give a negative answer to this question and construct a
family of graphs on which algorithms based on the individualization-refinement
paradigm require exponential time. Contrary to a previous construction of
Miyazaki, that only applies to a specific implementation within the
individualization-refinement framework, our construction is immune to changing
the cell selector, or adding various heuristic invariants to the algorithm.
Furthermore, our graphs also provide exponential lower bounds in the case when
the -dimensional Weisfeiler-Leman algorithm is used to replace the standard
color refinement operator and the arguments even work when the entire
automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
Benchmark Graphs for Practical Graph Isomorphism
The state-of-the-art solvers for the graph isomorphism problem can readily
solve generic instances with tens of thousands of vertices. Indeed, experiments
show that on inputs without particular combinatorial structure the algorithms
scale almost linearly. In fact, it is non-trivial to create challenging
instances for such solvers and the number of difficult benchmark graphs
available is quite limited. We describe a construction to efficiently generate
small instances for the graph isomorphism problem that are difficult or even
infeasible for said solvers. Up to this point the only other available
instances posing challenges for isomorphism solvers were certain incidence
structures of combinatorial objects (such as projective planes, Hadamard
matrices, Latin squares, etc.). Experiments show that starting from 1500
vertices our new instances are several orders of magnitude more difficult on
comparable input sizes. More importantly, our method is generic and efficient
in the sense that one can quickly create many isomorphism instances on a
desired number of vertices. In contrast to this, said combinatorial objects are
rare and difficult to generate and with the new construction it is possible to
generate an abundance of instances of arbitrary size. Our construction hinges
on the multipedes of Gurevich and Shelah and the Cai-F\"{u}rer-Immerman gadgets
that realize a certain abelian automorphism group and have repeatedly played a
role in the context of graph isomorphism. Exploring limits of such
constructions, we also explain that there are group theoretic obstructions to
generalizing the construction with non-abelian gadgets.Comment: 32 page
Resolution with Symmetry Rule Applied to Linear Equations
This paper considers the length of resolution proofs when using
Krishnamurthy's classic symmetry rules. We show that inconsistent linear
equation systems of bounded width over a fixed finite field with
a prime have, in their standard encoding as CNFs, polynomial length
resolutions when using the local symmetry rule (SRC-II).
As a consequence it follows that the multipede instances for the graph
isomorphism problem encoded as CNF formula have polynomial length resolution
proofs. This contrasts exponential lower bounds for
individualization-refinement algorithms on these graphs.
For the Cai-F\"urer-Immerman graphs, for which Tor\'an showed exponential
lower bounds for resolution proofs (SAT 2013), we also show that already the
global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.Comment: 18 pages, to be published in STACS 202
Rank Logic is Dead, Long Live Rank Logic!
Motivated 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. While FPR can express most of the known queries that separate FPC from PTIME, nearly nothing was known about the limitations of its expressive power.
In our first main result we show that the extensions of FPC by 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 FPR* with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR.
One important step in our proof is to consider solvability logic FPS which is the analogous extension of FPC by 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
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting.
In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances