arXiv subjects

Jakub Onufry Wojtaszczyk

Publications and source records attributed to Jakub Onufry Wojtaszczyk.

15 recordsLinked to original sources

arXiv2012

Scheduling partially ordered jobs faster than 2^n

In the SCHED problem we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. SCHED is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming algorithm solves both these problems in 2^n n^O(1) time, and whether there exists an algorithms solving SCHED in O(c^n) time for some constant c < 2 was an open problem posted in 2004 by Woeginger. In this paper we answer this question positively.

cs.DS↗

arXiv2012

Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space R^l in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into R^1 can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this particular case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices.

cs.DS↗

arXiv2011

Dominating Set is Fixed Parameter Tractable in Claw-free Graphs

We show that the dominating set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K(1,3)), the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k^2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of dominating set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of dominating set is the class of K(i,j)-free graphs, which exclude, for some fixed i,j, the complete bipartite graph K(i,j) as a subgraph. For i,i >= 2, the class of claw-free graphs and any class of K(i,j)-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of dominating set is known to be fixed-parameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the dominating set problem hard. More precisely, we show that for any t ?>= 4, the dominating set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K(1,t) as an induced subgraph. Our arguments also imply that the related connected dominating set and dominating clique problems are W[2]-hard in these graph classes. Finally, we show that for any t, the clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.

cs.DS↗

arXiv2011

Subset feedback vertex set is fixed parameter tractable

The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.

cs.DS↗

arXiv2011

On Multiway Cut parameterized above lower bounds

In this paper we consider two above lower bound parameterizations of the Node Multiway Cut problem - above the maximum separating cut and above a natural LP-relaxation - and prove them to be fixed-parameter tractable. Our results imply O*(4^k) algorithms for Vertex Cover above Maximum Matching and Almost 2-SAT as well as an O*(2^k) algorithm for Node Multiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.

cs.DS↗

arXiv2011

Solving connectivity problems parameterized by treewidth in single exponential time

For the vast majority of local graph problems standard dynamic programming techniques give c^tw V^O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw^O(tw) V^O(1) time. We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Feedback Vertex Set and Connected Dominating Set, consequently answering the question raised by Lokshtanov, Marx and Saurabh [SODA'11] in a surprising way. We also show that (under reasonable complexity assumptions) the gap cannot be breached for some problems for which Cut&Count does not work, like CYCLE PACKING. The constant c we obtain is in all cases small (at most 4 for undirected problems and at most 6 for directed ones), and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. Our results have numerous consequences in various fields, like FPT algorithms, exact and approximate algorithms on planar and H-minor-free graphs and algorithms on graphs of bounded degree. In all these fields we are able to improve the best-known results for some problems.

cs.DS↗

arXiv2010

The stubborn problem is stubborn no more (a polynomial algorithm for 3-compatible colouring and the stubborn list partition problem)

One of the driving problems in the CSP area is the Dichotomy Conjecture, formulated in 1993 by Feder and Vardi [STOC'93], stating that for any fixed relational structure G the Constraint Satisfaction Problem CSP(G) is either NP--complete or polynomial time solvable. A large amount of research has gone into checking various specific cases of this conjecture. One such variant which attracted a lot of attention in the recent years is the LIST MATRIX PARTITION problem. In 2004 Cameron et al. [SODA'04] classified almost all LIST MATRIX PARTITION variants for matrices of size at most four. The only case which resisted the classification became known as the STUBBORN PROBLEM. In this paper we show a result which enables us to finish the classification - thus solving a problem which resisted attacks for the last six years. Our approach is based on a combinatorial problem known to be at least as hard as the STUBBORN PROBLEM - the 3-COMPATIBLE COLOURING problem. In this problem we are given a complete graph with each edge assigned one of 3 possible colours and we want to assign one of those 3 colours to each vertex in such a way that no edge has the same colour as both of its endpoints. The tractability of the 3-COMPATIBLE COLOURING problem has been open for several years and the best known algorithm prior to this paper is due to Feder et al. [SODA'05] - a quasipolynomial algorithm with a n^O(log n / log log n) time complexity. In this paper we present a polynomial-time algorithm for the 3-COMPATIBLE COLOURING problem and consequently we prove a dichotomy for the k-COMPATIBLE COLOURING problem.

cs.DS↗

arXiv2009

No return to convexity

In the paper we study closures of classes of log--concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class $\mathcal{K}$. In particular we prove that if one starts from one--dimensional log--concave measures, one obtains no non--trivial uniform mesures on convex bodies.

math.FA↗

arXiv2009

A simpler proof of the negative association property for absolute values of measures tied to generalized Orlicz balls

Negative association for a family of random variables $(X_i)$ means that for any coordinate--wise increasing functions $f,g$ we have $$\E f(X_{i_1},...,X_{i_k}) g(X_{j_1},...,X_{j_l}) \leq \E f(X_{i_1},...,X_{i_k}) \E g(X_{j_1},...,X_{j_l})$$ for any disjoint sets of indices $(i_m)$, $(j_n)$. It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics, and brought to convex geometry in 2005 to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of negative association of absolute values for a wide class of measures tied to generalized Orlicz balls, including the uniform measures on generalized Orlicz balls.

math.PR↗

arXiv2009

Beyond O*(2^n) in domination-type problems

In this paper we provide algorithms faster than O*(2^n) for several NP-complete domination-type problems. More precisely, we provide: an algorithm for CAPACITATED DOMINATING SET that solves it in O(1.89^n), a branch-and-reduce algorithm solving LARGEST IRREDUNDANT SET in O(1.9657^n) time and a simple iterative-DFS algorithm for SMALLEST INCLUSION-MAXIMAL IRREDUNDANT SET that solves it in O(1.999956^n) time. We also provide an exponential approximation scheme for CAPACITATED DOMINATING SET. All algorithms require polynomial space. Despite the fact that the discussed problems are quite similar to the DOMINATING SET problem, we are not aware of any published algorithms solving these problems faster than the obvious O*(2^n) solution prior to this paper.

cs.DS↗

arXiv2008

A series whose sum range is an arbitrary finite set

In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wozniakowski obtained an example of a series the sum range of which consisted of two points, and asked whether it is possible to obtain more than two, but finitely many points. This paper answers the question positively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinitely dimensional Banach space.

math.FA↗

arXiv2008

The square negative correlation property for generalized Orlicz balls

Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper "Power-law estimates for the central limit theorem for convex sets" by B. Klartag), while for generalized Orlicz balls a much more general result is true (see "The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball" by M. Pilipczuk and J. O. Wojtaszczyk). While, however, both aforementioned papers are rather long, complicated and technical, this paper gives a simple and elementary proof of, eg., the Euclidean concentration for generalized Orlicz balls.

math.PR↗

arXiv2008

The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball

Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\ell_p^n$ balls is given.

math.PR↗

arXiv2008

Multivariate integration in C^\infty([0,1]^d) is not strongly tractable

It has long been known that the multivariate integration problem for the unit ball in $C^r([0,1]^d)$ is intractable for fixed finite $r$. H. Woźniakowski has recently conjectured that this is true even if $r=\infty$. This paper establishes a partial result in this direction. We prove that the multivariate integration problem, for infinitely differential functions all of whose variables are bounded by one, is not strongly tractable.

math.NA↗

arXiv2008

On the infimum convolution inequality

In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris.

math.PR↗