808 research outputs found
On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems
We show improved NP-hardness of approximating Ordering Constraint
Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum
Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of
and .
An OCSP is said to be approximation resistant if it is hard to approximate
better than taking a uniformly random ordering. We prove that the Maximum
Non-Betweenness Problem is approximation resistant and that there are width-
approximation-resistant OCSPs accepting only a fraction of
assignments. These results provide the first examples of
approximation-resistant OCSPs subject only to P \NP
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
We present a series of almost settled inapproximability results for three
fundamental problems. The first in our series is the subexponential-time
inapproximability of the maximum independent set problem, a question studied in
the area of parameterized complexity. The second is the hardness of
approximating the maximum induced matching problem on bounded-degree bipartite
graphs. The last in our series is the tight hardness of approximating the
k-hypergraph pricing problem, a fundamental problem arising from the area of
algorithmic game theory. In particular, assuming the Exponential Time
Hypothesis, our two main results are:
- For any r larger than some constant, any r-approximation algorithm for the
maximum independent set problem must run in at least
2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of
2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the
domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et
al., 2013)
- For any k larger than some constant, there is no polynomial time min
(k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph
pricing problem, where n is the number of vertices in an input graph. This
almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and
Blum, 2007 and an algorithm in this paper).
We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness
for polynomial-time algorithms, the k-hypergraph pricing problem admits
n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts
this problem in a rare approximability class in which approximability
thresholds can be improved significantly by allowing algorithms to run in
quasi-polynomial time.Comment: The full version of FOCS 201
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
Streaming Approximation Resistance of Every Ordering CSP
An ordering constraint satisfaction problem (OCSP) is given by a positive
integer and a constraint predicate mapping permutations on
to . Given an instance of OCSP on
variables and constraints, the goal is to find an ordering of the
variables that maximizes the number of constraints that are satisfied, where a
constraint specifies a sequence of distinct variables and the constraint is
satisfied by an ordering on the variables if the ordering induced on the
variables in the constraint satisfies . OCSPs capture natural problems
including "Maximum acyclic subgraph (MAS)" and "Betweenness".
In this work we consider the task of approximating the maximum number of
satisfiable constraints in the (single-pass) streaming setting, where an
instance is presented as a stream of constraints. We show that for every ,
OCSP is approximation-resistant to -space streaming algorithms.
This space bound is tight up to polylogarithmic factors. In the case of MAS our
result shows that for every , MAS is not
-approximable in space. The previous best
inapproximability result only ruled out a -approximation in
space.
Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and
Velusamy who show tight, linear-space inapproximability results for a broad
class of (non-ordering) constraint satisfaction problems over arbitrary
(finite) alphabets. We design a family of appropriate CSPs (one for every )
from any given OCSP, and apply their work to this family of CSPs. We show that
the hard instances from this earlier work have a particular "small-set
expansion" property. By exploiting this combinatorial property, in combination
with the hardness results of the resulting families of CSPs, we give optimal
inapproximability results for all OCSPs.Comment: 23 pages, 1 figure. Replaces earlier version with lower
bound, using new bounds from arXiv:2106.13078. To appear in APPROX'2
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)
Streaming Hardness of Unique Games
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems
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