133 research outputs found
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
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
Automating Resolution is NP-Hard
We show that the problem of finding a Resolution refutation that is at most
polynomially longer than a shortest one is NP-hard. In the parlance of proof
complexity, Resolution is not automatizable unless P = NP. Indeed, we show it
is NP-hard to distinguish between formulas that have Resolution refutations of
polynomial length and those that do not have subexponential length refutations.
This also implies that Resolution is not automatizable in subexponential time
or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively
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
ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be
stated as follows: given a constraint graph , an alphabet set
and, for each , a constraint , the goal is to find an assignment
that satisfies as many constraints as possible, where a constraint is
satisfied if .
While the approximability of 2-CSPs is quite well understood when
is constant, many problems are still open when becomes super
constant. One such problem is whether it is hard to approximate 2-CSPs to
within a polynomial factor of . Bellare et al. (1993) suggested
that the answer to this question might be positive. Alas, despite efforts to
resolve this conjecture, it remains open to this day.
In this work, we separate and and ask a related but weaker
question: is it hard to approximate 2-CSPs to within a polynomial factor of
(while may be super-polynomial in )? Assuming the
exponential time hypothesis (ETH), we answer this question positively by
showing that no polynomial time algorithm can approximate 2-CSPs to within a
factor of . Note that our ratio is almost linear, which is
almost optimal as a trivial algorithm gives a -approximation for 2-CSPs.
Thanks to a known reduction, our result implies an ETH-hardness of
approximating Directed Steiner Network with ratio where is
the number of demand pairs. The ratio is roughly the square root of the best
known ratio achieved by polynomial time algorithms (Chekuri et al., 2011;
Feldman et al., 2012).
Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out
polynomial time algorithms, but also FPT algorithms parameterized by .
Similar statement applies for DSN parameterized by .Comment: 36 pages. A preliminary version appeared in ITCS'1
Parameterized Approximation Algorithms for Bidirected Steiner Network Problems
The Directed Steiner Network (DSN) problem takes as input a directed
edge-weighted graph and a set of
demand pairs. The aim is to compute the cheapest network for
which there is an path for each . It is known
that this problem is notoriously hard as there is no
-approximation algorithm under Gap-ETH, even when parametrizing
the runtime by [Dinur & Manurangsi, ITCS 2018]. In light of this, we
systematically study several special cases of DSN and determine their
parameterized approximability for the parameter .
For the bi-DSN problem, the aim is to compute a planar
optimum solution in a bidirected graph , i.e., for every edge
of the reverse edge exists and has the same weight. This problem
is a generalization of several well-studied special cases. Our main result is
that this problem admits a parameterized approximation scheme (PAS) for . We
also prove that our result is tight in the sense that (a) the runtime of our
PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists
for any generalization of bi-DSN, unless FPT=W[1].
One important special case of DSN is the Strongly Connected Steiner Subgraph
(SCSS) problem, for which the solution network needs to strongly
connect a given set of terminals. It has been observed before that for SCSS
a parameterized -approximation exists when parameterized by [Chitnis et
al., IPEC 2013]. We give a tight inapproximability result by showing that for
no parameterized -approximation algorithm exists under
Gap-ETH. Additionally we show that when restricting the input of SCSS to
bidirected graphs, the problem remains NP-hard but becomes FPT for
Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH
The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F_2, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k. Here, k is the parameter of the problem. The question of whether k-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows\u27 book (2013) as one of the "most infamous" open problems in the field of Parameterized Complexity.
In this work, we show that k-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur\u2716, Manurangsi-Raghavendra\u2716]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.\u2717]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only 0.99-satisfiable (where the parameter is the number of variables).
We also consider the parameterized k-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer k, and the goal is to determine whether the norm of the shortest vector (in the l_p norm for some fixed p) is at most k. Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any p > 1, k-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of p = 2, the inapproximability factor can be amplified to any constant
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