9,498 research outputs found
The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover
We suggest the research agenda of establishing new hardness of approximation results based on the “projection games conjecture”, i.e., an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games.
We pursue this line of research by establishing a tight NP-hardness result for the Set-Cover problem. Specifically, we show that under the projection games conjecture (in fact, under a quantitative version of the conjecture that is only slightly beyond the reach of current techniques), it is NP-hard to approximate Set-Cover on instances of size N to within (1 − α)ln N for arbitrarily small α > 0. Our reduction establishes a tight trade-off between the approximation accuracy α and the time required for the approximation 2[superscript NΩ(α)], assuming Sat requires exponential time.
The reduction is obtained by modifying Feige’s reduction. The latter only provides a lower bound of 2[superscript NΩ(α/loglogN)] on the time required for (1 − α)ln N-approximating Set-Cover assuming Sat requires exponential time (note that N[superscript 1/loglogN] = N[superscript o(1)]). The modification uses a combinatorial construction of a bipartite graph in which any coloring of the first side that does not use a color for more than a small fraction of the vertices, makes most vertices on the other side have their neighbors all colored in different colors
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
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
Hamming Approximation of NP Witnesses
Given a satisfiable 3-SAT formula, how hard is it to find an assignment to
the variables that has Hamming distance at most n/2 to a satisfying assignment?
More generally, consider any polynomial-time verifier for any NP-complete
language. A d(n)-Hamming-approximation algorithm for the verifier is one that,
given any member x of the language, outputs in polynomial time a string a with
Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the
verifier. Previous results have shown that, if P != NP, then every NP-complete
language has a verifier for which there is no
(n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0.
Our main result is that, if P != NP, then every paddable NP-complete language
has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation
algorithm. That is, one cannot get even half the bits right. We also consider
natural verifiers for various well-known NP-complete problems. They do have
n/2-Hamming-approximation algorithms, but, if P != NP, have no
(n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0.
We show similar results for randomized algorithms
Improved Hardness of Approximating Chromatic Number
We prove that for sufficiently large K, it is NP-hard to color K-colorable
graphs with less than 2^{K^{1/3}} colors. This improves the previous result of
K versus K^{O(log K)} in Khot [14]
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