Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the previous version did NOT satisfy requirement 4 of lemma 18 (the proof of Claim 20 was incorrect

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we use a recent technique of finding redundant constraints by representing them as low-degree polynomials, to obtain a kernel of bitsize O(k^(q-1) log k) for q-Coloring parameterized by Vertex Cover for any q >= 3. This size bound is optimal up to k^o(1) factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^(2-e)) for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors

Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with O(n^{d-epsilon}) bits for any epsilon > 0. For the Not-All-Equal SAT problem, a compression to size tilde-O(n^{d-1}) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n^{2-epsilon}) bits for any epsilon > 0, unless NP is contained in coNP/poly

Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations

We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS\u2712] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size k^{O(1)}. In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth eta. We prove that for each set F of connected graphs and constant eta, the F-Deletion problem parameterized by the size of a treedepth-eta modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G-X. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal F-Deletion solution from a single connected component of G-X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for F-Deletion parameterized by a vertex cover, which is a treedepth-one modulator

Approximate Turing Kernelization for Problems Parameterized by Treewidth

We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An $\alpha$-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs $c$-approximate solutions in $O(1)$ time, obtains an $(\alpha \cdot c)$-approximate solution to the considered problem, using calls to the oracle of size at most $f(k)$ for some function $f$ that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth $\ell$ has a $(1+\varepsilon)$-approximate Turing kernel with $O(\frac{\ell^2}{\varepsilon})$ vertices, answering an open question posed by Lokshtanov et al. [STOC 2017]. Furthermore, we give $(1+\varepsilon)$-approximate Turing kernels for the following graph problems parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint Triangle Packing and Connected Vertex Cover. We generalize the result for Independent Set and Vertex Cover, by showing that all graph problems that we will call "friendly" admit $(1+\varepsilon)$-approximate Turing kernels of polynomial size when parameterized by treewidth. We use this to obtain approximate Turing kernels for Vertex-Disjoint $H$-packing for connected graphs $H$, Clique Cover, Feedback Vertex Set and Edge Dominating Set

Sparsification Lower Bounds for List H-Coloring

We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ? V(G) is mapped to a vertex on its list L(v) ? V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize ?(n^(2-?)) for some ? > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ? coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs

Із зали засідань Президії НАН України

20 червня 2012 року відбулося виїзне засідання Президії Національної академії наук України на запрошення президента — генерального конструктора Державного підприємства «АНТОНОВ» академіка НАН України Д.С. Ківи

Living with myotonic dystrophy; what can be learned from couples? a qualitative study

Contains fulltext : 96062.pdf (publisher's version ) (Open Access)BACKGROUND: Myotonic dystrophy type 1 (MD1) is one of the most prevalent neuromuscular diseases, yet very little is known about how MD1 affects the lives of couples and how they themselves manage individually and together. To better match health care to their problems, concerns and needs, it is important to understand their perspective of living with this hereditary, systemic disease. METHODS: A qualitative study was carried out with a purposive sample of five middle-aged couples, including three men and two women with MD1 and their partners. Fifteen in-depth interviews with persons with MD1, with their partners and with both of them as a couple took place in the homes of the couples in two cities and three villages in the Netherlands in 2009. Results : People with MD1 associate this progressive, neuromuscular condition with decreasing abilities, describing physical, cognitive and psychosocial barriers to everyday activities and social participation. Partners highlighted the increasing care giving burden, giving directions and using reminders to compensate for the lack of initiative and avoidant behaviour due to MD1. Couples portrayed the dilemmas and frustrations of renegotiating roles and responsibilities; stressing the importance of achieving a balance between individual and shared activities. All participants experienced a lack of understanding from relatives, friends, and society, including health care, leading to withdrawal and isolation. Health care was perceived as fragmentary, with specialists focusing on specific aspects of the disease rather than seeking to understand the implications of the systemic disorder on daily life. CONCLUSIONS: Learning from these couples has resulted in recommendations that challenge the tendency to treat MD1 as a condition with primarily physical impairments. It is vital to listen to couples, to elicit the impact of MD1, as a multisystem disorder that influences every aspect of their life together. Couple management, supporting the self-management skills of both partners is proposed as a way of reducing the mismatch between health services and health needs

Endometrial scratching in women with one failed IVF/ICSI cycle-outcomes of a randomised controlled trial (SCRaTCH)

STUDY QUESTION: Does endometrial scratching in women with one failed IVF/ICSI treatment affect the chance of a live birth of the subsequent fresh IVF/ICSI cycle? SUMMARY ANSWER: In this study, 4.6% more live births were observed in the scratch group, with a likely certainty range between -0.7% and +9.9%. WHAT IS KNOWN ALREADY: Since the first suggestion that endometrial scratching might improve embryo implantation during IVF/ICSI, many clinical trials have been conducted. However, due to limitations in sample size and study quality, it remains unclear whether endometrial scratching improves IVF/ICSI outcomes. STUDY DESIGN, SIZE, DURATION: The SCRaTCH trial was a non-blinded randomised controlled trial in women with one unsuccessful IVF/ICSI cycle and assessed whether a single endometrial scratch using an endometrial biopsy catheter would lead to a higher live birth rate after the subsequent IVF/ICSI treatment compared to no scratch. The study took place in 8 academic and 24 general hospitals. Participants were randomised between January 2016 and July 2018 by a web-based randomisation programme. Secondary outcomes included cumulative 12-month ongoing pregnancy leading to live birth rate. PARTICIPANTS/MATERIALS, SETTING, METHODS: Women with one previous failed IVF/ICSI treatment and planning a second fresh IVF/ICSI treatment were eligible. In total, 933 participants out of 1065 eligibles were included (participation rate 88%). MAIN RESULTS AND THE ROLE OF CHANCE: After the fresh transfer, 4.6% more live births were observed in the scratch compared to control group (110/465 versus 88/461, respectively, risk ratio (RR) 1.24 [95% CI 0.96-1.59]). These data are consistent with a true difference of between -0.7% and +9.9% (95% CI), indicating that while the largest proportion of the 95% CI is positive, scratchin