13 research outputs found
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Polynomial Kernels for Deletion to Classes of Acyclic Digraphs
We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made.
In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time
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
Independent Sets near the Lower Bound in Bounded Degree Graphs
By Brook\u27s Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an independent set of size at least n/Delta+k has a kernel of size O(k)
On the hardness of learning sparse parities
This work investigates the hardness of computing sparse solutions to systems
of linear equations over F_2. Consider the k-EvenSet problem: given a
homogeneous system of linear equations over F_2 on n variables, decide if there
exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse
solution). While there is a simple O(n^{k/2})-time algorithm for it,
establishing fixed parameter intractability for k-EvenSet has been a notorious
open problem. Towards this goal, we show that unless k-Clique can be solved in
n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no
polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0.
Our work also shows that the non-homogeneous generalization of the problem --
which we call k-VectorSum -- is W[1]-hard on instances where the number of
equations is O(k log n), improving on previous reductions which produced
Omega(n) equations. We also show that for any constant eps > 0, given a system
of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a
k-sparse linear form satisfying all the equations or if every function on at
most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the
equations. In the setting of computational learning, this shows hardness of
approximate non-proper learning of k-parities. In a similar vein, we use the
hardness of k-EvenSet to show that that for any constant d, unless k-Clique can
be solved in n^{o(k)} time there is no poly(m, n)2^{o(sqrt{k}) time algorithm
to decide whether a given set of m points in F_2^n satisfies: (i) there exists
a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the
points, or (ii) any non-trivial degree d polynomial P supported on at most k
variables evaluates to zero on approx. Pr_{F_2^n}[P(z) = 0] fraction of the
points i.e., P is fooled by the set of points
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
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution