1,773 research outputs found
Approximating Dense Max 2-CSPs
In this paper, we present a polynomial-time algorithm that approximates
sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within
approximation ratio for any constant .
Using this algorithm, we also achieve similar results for free games,
projection games on sufficiently dense random graphs, and the Densest
-Subgraph problem with sufficiently dense optimal solution. Note, however,
that algorithms with similar guarantees to the last algorithm were in fact
discovered prior to our work by Feige et al. and Suzuki and Tokuyama.
In addition, our idea for the above algorithms yields the following
by-product: a quasi-polynomial time approximation scheme (QPTAS) for
satisfiable dense Max 2-CSPs with better running time than the known
algorithms
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
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)
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