299 research outputs found
On the Subexponential Time Complexity of CSP
A CSP with n variables ranging over a domain of d values can be solved by
brute-force in d^n steps (omitting a polynomial factor). With a more careful
approach, this trivial upper bound can be improved for certain natural
restrictions of the CSP. In this paper we establish theoretical limits to such
improvements, and draw a detailed landscape of the subexponential-time
complexity of CSP.
We first establish relations between the subexponential-time complexity of
CSP and that of other problems, including CNF-Sat. We exploit this connection
to provide tight characterizations of the subexponential-time complexity of CSP
under common assumptions in complexity theory. For several natural CSP
parameters, we obtain threshold functions that precisely dictate the
subexponential-time complexity of CSP with respect to the parameters under
consideration.
Our analysis provides fundamental results indicating whether and when one can
significantly improve on the brute-force search approach for solving CSP
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
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
Obtaining lower bounds for NP-hard problems has for a long time been an
active area of research. Recent algebraic techniques introduced by Jonsson et
al. (SODA 2013) show that the time complexity of the parameterized SAT()
problem correlates to the lattice of strong partial clones. With this ordering
they isolated a relation such that SAT() can be solved at least as fast
as any other NP-hard SAT() problem. In this paper we extend this method
and show that such languages also exist for the max ones problem
(MaxOnes()) and the Boolean valued constraint satisfaction problem over
finite-valued constraint languages (VCSP()). With the help of these
languages we relate MaxOnes and VCSP to the exponential time hypothesis in
several different ways.Comment: This is an extended version of Relating the Time Complexity of
Optimization Problems in Light of the Exponential-Time Hypothesis, appearing
in Proceedings of the 39th International Symposium on Mathematical
Foundations of Computer Science MFCS 2014 Budapest, August 25-29, 201
Fast counting with tensor networks
We introduce tensor network contraction algorithms for counting satisfying
assignments of constraint satisfaction problems (#CSPs). We represent each
arbitrary #CSP formula as a tensor network, whose full contraction yields the
number of satisfying assignments of that formula, and use graph theoretical
methods to determine favorable orders of contraction. We employ our heuristics
for the solution of #P-hard counting boolean satisfiability (#SAT) problems,
namely monotone #1-in-3SAT and #Cubic-Vertex-Cover, and find that they
outperform state-of-the-art solvers by a significant margin.Comment: v2: added results for monotone #1-in-3SAT; published versio
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
Why are CSPs Based on Partition Schemes Computationally Hard?
Many computational problems arising in, for instance, artificial intelligence can be realized as infinite-domain constraint satisfaction problems (CSPs) based on partition schemes: a set of pairwise disjoint binary relations (containing the equality relation) whose union spans the underlying domain and which is closed under converse. We first consider partition schemes that contain a strict partial order and where the constraint language contains all unions of the basic relations; such CSPs are frequently occurring in e.g. temporal and spatial reasoning. We identify three properties of such orders which, when combined, are sufficient to establish NP-hardness of the CSP. This result explains, in a uniform way, many existing hardness results from the literature. More importantly, this result enables us to prove that CSPs of this kind are not solvable in subexponential time unless the exponential-time hypothesis (ETH) fails. We continue by studying constraint languages based on partition schemes but where relations are built using disjunctions instead of unions; such CSPs appear naturally when analysing first-order definable constraint languages. We prove that such CSPs are NP-hard even in very restricted settings and that they are not solvable in subexponential time under the randomised ETH. In certain cases, we can additionally show that they cannot be solved in O(c^n) time for any c >= 0
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