21 research outputs found
Limits of Preprocessing
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a
complexity theoretic assumption, none of the considered problems can be reduced
by polynomial-time preprocessing to a problem kernel whose size is polynomial
in a structural problem parameter of the input, such as induced width or
backdoor size. Our results provide a firm theoretical boundary for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: This is a slightly longer version of a paper that appeared in the
proceedings of AAAI 201
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
Limits of Preprocessing for Single-Server PIR
We present a lower bound for the static cryptographic data structure problem of single-server private information retrieval (PIR). PIR considers the setting where a server holds a database of entries and a client wishes to privately retrieve the -th entry without revealing the index to the server. In our work, we focus on PIR with preprocessing where an -bit hint may be computed in a preprocessing stage and stored by the server to be used to perform private queries in expected time . We consider the public preprocessing setting of Beimel et al. [JoC, 2004] where the hint is publicly available to everyone including the adversary.
We prove that for any single-server computationally secure PIR with preprocessing it must be that when . If , we show that . Our lower bound holds even when the scheme errs with probability and the adversary’s distinguishing advantage is . Our work improves upon the lower bound of Beimel et al. [JoC, 2004].
We prove our lower bound in a variant of the cell probe model where only accesses to the memory are charged cost and computation and accesses to the hint are free. Our main technical contribution is a novel use of the cell sampling technique (also known as the incompressibility technique) used to obtain lower bounds on data structures. In previous works, this technique only leveraged the correctness guarantees to prove lower bounds even when used for cryptographic primitives. Our work combines the cell sampling technique with the privacy guarantees of PIR to construct a powerful, polynomial-time adversary that is critical to proving our higher lower bounds
Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming
The NP-hard general factor problem asks, given a graph and for each vertex a
list of integers, whether the graph has a spanning subgraph where each vertex
has a degree that belongs to its assigned list. The problem remains NP-hard
even if the given graph is bipartite with partition U+V, and each vertex in U
is assigned the list {1}; this subproblem appears in the context of constraint
programming as the consistency problem for the extended global cardinality
constraint. We show that this subproblem is fixed-parameter tractable when
parameterized by the size of the second partite set V. More generally, we show
that the general factor problem for bipartite graphs, parameterized by |V|, is
fixed-parameter tractable as long as all vertices in U are assigned lists of
length 1, but becomes W[1]-hard if vertices in U are assigned lists of length
at most 2. We establish fixed-parameter tractability by reducing the problem
instance to a bounded number of acyclic instances, each of which can be solved
in polynomial time by dynamic programming.Comment: Full version of a paper that appeared in preliminary form in the
proceedings of IPEC'1
Fine-grained Search Space Classification for Hard Enumeration Variants of Subset Problems
We propose a simple, powerful, and flexible machine learning framework for
(i) reducing the search space of computationally difficult enumeration variants
of subset problems and (ii) augmenting existing state-of-the-art solvers with
informative cues arising from the input distribution. We instantiate our
framework for the problem of listing all maximum cliques in a graph, a central
problem in network analysis, data mining, and computational biology. We
demonstrate the practicality of our approach on real-world networks with
millions of vertices and edges by not only retaining all optimal solutions, but
also aggressively pruning the input instance size resulting in several fold
speedups of state-of-the-art algorithms. Finally, we explore the limits of
scalability and robustness of our proposed framework, suggesting that
supervised learning is viable for tackling NP-hard problems in practice.Comment: AAAI 201
Complexity and Approximability of Parameterized MAX-CSPs
International audienceWe study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance.We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a (1−ϵ)-approximation in time f(k,ϵ)⋅poly(n). 3. There is no FPTAS unless FPT=W[1].For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results
Small Circuits Imply Efficient Arthur-Merlin Protocols
The inner product function ? x,y ? = ?_i x_i y_i mod 2 can be easily computed by a (linear-size) AC?(?) circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on the bottom most layer (closest to the input)? Namely, can the inner product function be computed by an AC? circuit composed with a single layer of parity gates? This seemingly simple question is an important open question at the frontier of circuit lower bound research.
In this work, we focus on a minimalistic version of the above question. Namely, whether the inner product function cannot be approximated by a small DNF augmented with a single layer of parity gates. Our main result shows that the existence of such a circuit would have unexpected implications for interactive proofs, or more specifically, for interactive variants of the Data Streaming and Communication Complexity models. In particular, we show that the existence of such a small (i.e., polynomial-size) circuit yields:
1) An O(d)-message protocol in the Arthur-Merlin Data Streaming model for every n-variate, degree d polynomial (over GF(2)), using only O?(d) ?log(n) communication and space complexity. In particular, this gives an AM[2] Data Streaming protocol for a variant of the well-studied triangle counting problem, with poly-logarithmic communication and space complexities.
2) A 2-message communication complexity protocol for any sparse (or low degree) polynomial, and for any function computable by an AC?(?) circuit. Specifically, for the latter, we obtain a protocol with communication complexity that is poly-logarithmic in the size of the AC?(?) circuit