46 research outputs found
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
The Strongish Planted Clique Hypothesis and Its Consequences
We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^?(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent).
We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k.
Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis
FPT Inapproximability of Directed Cut and Connectivity Problems
Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number k of terminals and the size p of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gap-instances" under stronger hypotheses such as the (Gap-)Exponential Time Hypothesis (ETH). Formally, we show the following results:
Cutting paths between a set of terminal pairs, i.e., Directed Multicut: Pilipczuk and Wahlstrom [TOCT \u2718] showed that Directed Multicut is W[1]-hard when parameterized by p if k=4. We complement this by showing the following two results:
- Directed Multicut has a k/2-approximation in 2^{O(p^2)}* n^{O(1)} time (i.e., a 2-approximation if k=4),
- Under Gap-ETH, Directed Multicut does not admit an (59/58-epsilon)-approximation in f(p)* n^{O(1)} time, for any computable function f, even if k=4.
Connecting a set of terminal pairs, i.e., Directed Steiner Network (DSN): The DSN problem on general graphs is known to be W[1]-hard parameterized by p+k due to Guo et al. [SIDMA \u2711]. Dinur and Manurangsi [ITCS \u2718] further showed that there is no FPT k^{1/4-o(1)}-approximation algorithm parameterized by k, under Gap-ETH. Chitnis et al. [SODA \u2714] considered the restriction to special graph classes, but unfortunately this does not lead to FPT algorithms either: DSN on planar graphs is W[1]-hard parameterized by k. In this paper we consider the DSN_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for DSN_Planar:
- DSN_Planar has no (2-epsilon)-approximation in FPT time parameterized by k, under Gap-ETH. This answers in the negative a question of Chitnis et al. [ESA \u2718].
- DSN_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for DSN_Planar in f(k,p,epsilon)* n^{o(k+sqrt{p+1/epsilon})} time for any computable function f.
Pairwise connecting a set of terminals, i.e., Strongly Connected Steiner Subgraph (SCSS): Guo et al. [SIDMA \u2711] showed that SCSS is W[1]-hard parameterized by p+k, while Chitnis et al. [SODA \u2714] showed that SCSS remains W[1]-hard parameterized by p, even if the input graph is planar. In this paper we consider the SCSS_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for SCSS_Planar:
- SCSS_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for SCSS_Planar in f(k,p,epsilon)* n^{o(sqrt{k+p+1/epsilon})} time for any computable function f.
Previously, the only known FPT approximation results for SCSS applied to general graphs parameterized by k: a 2-approximation by Chitnis et al. [IPEC \u2713], and a matching (2-epsilon)-hardness under Gap-ETH by Chitnis et al. [ESA \u2718]
Parameterized approximation algorithms for bidirected steiner network problems
The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s -> t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k.
For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu 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 k. 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_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k.
One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon>0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k
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
{On Subexponential Running Times for Approximating Directed Steiner Tree and Related Problems}
This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate, the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1-d)ln n, for a given parameter 0= 2^{n^{c d}}, for some constant 0= exp((1+o(1)){log^{d-c}n}), for any c>0, unless the ETH is false. Our result follows by analyzing the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for CST
Baby PIH: Parameterized Inapproximability of Min CSP
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP
theorem in the world of parameterized complexity. It asserts that no FPT
algorithm can distinguish a satisfiable 2CSP instance from one which is only
-satisfiable (where the parameter is the number of variables)
for some constant .
We consider a minimization version of CSPs (Min-CSP), where one may assign
values to each variable, and the goal is to ensure that every constraint is
satisfied by some choice among the pairs of values assigned to its
variables (call such a CSP instance -list-satisfiable). We prove the
following strong parameterized inapproximability for Min CSP: For every , it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even
-list-satisfiable. We refer to this statement as "Baby PIH", following the
recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the
combinatorial arguments underlying the Baby PCP theorem, overcoming some basic
obstacles that arise in the parameterized setting. Furthermore, our reduction
runs in time polynomially bounded in both the number of variables and the
alphabet size, and thus implies the Baby PCP theorem as well
-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm
In the Directed Steiner Tree (DST) problem we are given an -vertex
directed edge-weighted graph, a root , and a collection of terminal
nodes. Our goal is to find a minimum-cost arborescence that contains a directed
path from to every terminal. We present an -approximation algorithm for DST that runs in
quasi-polynomial-time. By adjusting the parameters in the hardness result of
Halperin and Krauthgamer, we show the matching lower bound of
for the class of quasi-polynomial-time
algorithms. This is the first improvement on the DST problem since the
classical quasi-polynomial-time approximation algorithm by
Charikar et al. (The paper erroneously claims an approximation due
to a mistake in prior work.)
Our approach is based on two main ingredients. First, we derive an
approximation preserving reduction to the Label-Consistent Subtree (LCST)
problem. The LCST instance has quasi-polynomial size and logarithmic height. We
remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all
prior work on DST achieves a reduction to a tree instance of the related Group
Steiner Tree (GST) problem of similar height, however losing a logarithmic
factor in the approximation ratio. Our second ingredient is an LP-rounding
algorithm to approximately solve LCST instances, which is inspired by the
framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a
proper LP relaxation of LCST. Our rounding algorithm proceeds level by level
from the root to the leaves, rounding and conditioning each time on a proper
subset of label variables. A small enough (namely, polylogarithmic) number of
Sherali-Adams lifting levels is sufficient to condition up to the leaves
Parameterized Inapproximability Hypothesis under ETH
The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed
parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance,
parameterized by the number of variables, from one where every assignment fails
to satisfy an fraction of constraints for some absolute constant
. PIH plays the role of the PCP theorem in parameterized
complexity. However, PIH has only been established under Gap-ETH, a very strong
assumption with an inherent gap.
In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This
is the first proof of PIH from a gap-free assumption. Our proof is
self-contained and elementary. We identify an ETH-hard CSP whose variables take
vector values, and constraints are either linear or of a special parallel
structure. Both kinds of constraints can be checked with constant soundness via
a "parallel PCP of proximity" based on the Walsh-Hadamard code