68,337 research outputs found
Lower bounds for approximation schemes for Closest String
In the Closest String problem one is given a family of
equal-length strings over some fixed alphabet, and the task is to find a string
that minimizes the maximum Hamming distance between and a string from
. While polynomial-time approximation schemes (PTASes) for this
problem are known for a long time [Li et al., J. ACM'02], no efficient
polynomial-time approximation scheme (EPTAS) has been proposed so far. In this
paper, we prove that the existence of an EPTAS for Closest String is in fact
unlikely, as it would imply that , a highly
unexpected collapse in the hierarchy of parameterized complexity classes. Our
proof also shows that the existence of a PTAS for Closest String with running
time , for any computable function
, would contradict the Exponential Time Hypothesis
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
In this paper we consider the p-Norm Hamming Centroid problem which asks to determine whether some given strings have a centroid with a bound on the p-norm of its Hamming distances to the strings. Specifically, given a set S of strings and a real k, we consider the problem of determining whether there exists a string s^* with (sum_{s in S} d^{p}(s^*,s))^(1/p) <=k, where d(,) denotes the Hamming distance metric. This problem has important applications in data clustering and multi-winner committee elections, and is a generalization of the well-known polynomial-time solvable Consensus String (p=1) problem, as well as the NP-hard Closest String (p=infty) problem.
Our main result shows that the problem is NP-hard for all fixed rational p > 1, closing the gap for all rational values of p between 1 and infty. Under standard complexity assumptions the reduction also implies that the problem has no 2^o(n+m)-time or 2^o(k^(p/(p+1)))-time algorithm, where m denotes the number of input strings and n denotes the length of each string, for any fixed p > 1. The first bound matches a straightforward brute-force algorithm. The second bound is tight in the sense that for each fixed epsilon > 0, we provide a 2^(k^(p/((p+1))+epsilon))-time algorithm. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-2 approximation algorithm for the problem
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
On the String Consensus Problem and the Manhattan Sequence Consensus Problem
In the Manhattan Sequence Consensus problem (MSC problem) we are given
integer sequences, each of length , and we are to find an integer sequence
of length (called a consensus sequence), such that the maximum
Manhattan distance of from each of the input sequences is minimized. For
binary sequences Manhattan distance coincides with Hamming distance, hence in
this case the string consensus problem (also called string center problem or
closest string problem) is a special case of MSC. Our main result is a
practically efficient -time algorithm solving MSC for sequences.
Practicality of our algorithms has been verified experimentally. It improves
upon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensus
problem for binary strings. Similarly as in Amir's algorithm we use a
column-based framework. We replace the implied general integer linear
programming by its easy special cases, due to combinatorial properties of the
MSC for . We also show that for a general parameter any instance
can be reduced in linear time to a kernel of size , so the problem is
fixed-parameter tractable. Nevertheless, for this is still too large
for any naive solution to be feasible in practice.Comment: accepted to SPIRE 201
Approximation and Parameterized Complexity of Minimax Approval Voting
We present three results on the complexity of Minimax Approval Voting. First,
we study Minimax Approval Voting parameterized by the Hamming distance from
the solution to the votes. We show Minimax Approval Voting admits no algorithm
running in time , unless the Exponential
Time Hypothesis (ETH) fails. This means that the
algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by
this, we then show a parameterized approximation scheme, running in time
, which is essentially
tight assuming ETH. Finally, we get a new polynomial-time randomized
approximation scheme for Minimax Approval Voting, which runs in time
,
almost matching the running time of the fastest known PTAS for Closest String
due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode
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