33 research outputs found
Counting Independent Sets and Colorings on Random Regular Bipartite Graphs
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph
The Complexity of Approximately Counting Tree Homomorphisms
We study two computational problems, parameterised by a fixed tree H.
#HomsTo(H) is the problem of counting homomorphisms from an input graph G to H.
#WHomsTo(H) is the problem of counting weighted homomorphisms to H, given an
input graph G and a weight function for each vertex v of G. Even though H is a
tree, these problems turn out to be sufficiently rich to capture all of the
known approximation behaviour in #P. We give a complete trichotomy for
#WHomsTo(H). If H is a star then #WHomsTo(H) is in FP. If H is not a star but
it does not contain a certain induced subgraph J_3 then #WHomsTo(H) is
equivalent under approximation-preserving (AP) reductions to #BIS, the problem
of counting independent sets in a bipartite graph. This problem is complete for
the class #RHPi_1 under AP-reductions. Finally, if H contains an induced J_3
then #WHomsTo(H) is equivalent under AP-reductions to #SAT, the problem of
counting satisfying assignments to a CNF Boolean formula. Thus, #WHomsTo(H) is
complete for #P under AP-reductions. The results are similar for #HomsTo(H)
except that a rich structure emerges if H contains an induced J_3. We show that
there are trees H for which #HomsTo(H) is #SAT-equivalent (disproving a
plausible conjecture of Kelk). There is an interesting connection between these
homomorphism-counting problems and the problem of approximating the partition
function of the ferromagnetic Potts model. In particular, we show that for a
family of graphs J_q, parameterised by a positive integer q, the problem
#HomsTo(H) is AP-interreducible with the problem of approximating the partition
function of the q-state Potts model. It was not previously known that the Potts
model had a homomorphism-counting interpretation. We use this connection to
obtain some additional upper bounds for the approximation complexity of
#HomsTo(J_q)
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Approximate Counting CSP Seen from the Other Side
In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS\u2702]. Building on the work of Grohe [JACM\u2707] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS\u2704].
We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors
Counting Constraint Satisfaction Problems
This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter
The Complexity of Approximately Counting Retractions
Let be a graph that contains an induced subgraph . A retraction from
to is a homomorphism from to that is the identity function on
. Retractions are very well-studied: Given , the complexity of deciding
whether there is a retraction from an input graph to is completely
classified, in the sense that it is known for which this problem is
tractable (assuming ). Similarly, the complexity of
(exactly) counting retractions from to is classified (assuming
). However, almost nothing is known about
approximately counting retractions. Our first contribution is to give a
complete trichotomy for approximately counting retractions to graphs of girth
at least . Our second contribution is to locate the retraction counting
problem for each in the complexity landscape of related approximate
counting problems. Interestingly, our results are in contrast to the situation
in the exact counting context. We show that the problem of approximately
counting retractions is separated both from the problem of approximately
counting homomorphisms and from the problem of approximately counting list
homomorphisms --- whereas for exact counting all three of these problems are
interreducible. We also show that the number of retractions is at least as hard
to approximate as both the number of surjective homomorphisms and the number of
compactions. In contrast, exactly counting compactions is the hardest of all of
these exact counting problems