In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others.In the (min, +)-convolution problem, the goal is to compute a sequence (c. This can easily be done in O(n 2 ) time, but no O(n 2−ε ) algorithm is known for ε > 0. In this paper, we undertake a systematic study of the (min, +)-convolution problem as a hardness assumption.First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min, +)-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.

Contention resolution schemes have proven to be an incredibly powerful concept which allows to tackle a broad class of problems. The framework has been initially designed to handle submodular optimization under various types of constraints, that is, intersections of exchange systems (including matroids), knapsacks, and unsplittable flows on trees. Later on, it turned out that this framework perfectly extends to optimization under uncertainty, like stochastic probing and online selection problems, which further can be applied to mechanism design. We add to this line of work by showing how to create contention resolution schemes for intersection of matroids and knapsacks when we work in the random order setting. More precisely, we do know the whole universe of elements in advance, but they appear in an order given by a random permutation. Upon arrival we need to irrevocably decide whether to take an element or not. We bring a novel technique for analyzing procedures in the random order setting that is based on the martingale theory. This unified approach makes it easier to combine constraints, and we do not need to rely on the monotonicity of contention resolution schemes.Our paper fills the gaps, extends, and creates connections between many previous results and techniques. The main application of our framework is a k + 4 + ε approximation ratio for the Bayesian multi-parameter unit-demand mechanism design under the constraint of k matroids intersection, which improves upon the previous bounds of 4k − 2 and e(k + 1). Other results include improved approximation ratios for stochastic k-set packing and submodular stochastic probing over arbitrary non-negative submodular objective function, whereas previous results required the objective to be monotone. * This is an extended version of a paper whose preliminary version appeared in

The second replicator region of the native plasmid pTAVl of Paracoccus versutus has been identified thus proving the composite nature of this replicon. The minimal replicon designated pTAV320 (43 kb) was cloned and sequenced. pTAV320 encodes three putative proteins -RepA, RepB and RepC.This replicator region shows strong structural and functional similarity t o mpABC-type rep1 icons found in several Agrobacterium and Rhizobium plasmids. The origin of replication appears to be localized within the coding sequence of the repC gene. RepC was shown t o be essential for replication. RepA and RepB were necessary for stable maintenance of the plasmid, which implies a role in active partitioning. The presence of the complete sequence of pTAV320 (in its non-replicative form) could stabilize in cis pTAV202, a rninireplicon derived from the other pTAVl replicator region. Deletions introduced into the mpC gene abolished the 'stabilizing' activity of pTAV320, suggesting that the centromere-like sequence, necessary for partitioning, might be localized within this gene. The two replicator regions of pTAV1 (pTAV320 and pTAV202) expressed incompatibility towards the parental plasmid but were compatible in trans in P. wersutus cells. The pTAV320 replicon can be maintained in several Paracoccus, Agrobacterium, Rhizobium and Rhodohcter strains in addition t o P. versutus.

We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G \ S belongs to some class H. We consider the case where graphs in H have treewidth bounded by t, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator, respectively.For the vertex deletion setting, our framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar-F Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization.For the edge deletion setting, we give improved approximation algorithms for k-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.3 The conference version of [Lee18] presents a randomized algorithm, but the journal version derandomized it.

The subject of this paper is the time complexity of approximating Knapsack, Subset Sum, Partition, and some other related problems. The main result is an O(n + 1/ε 5/3 ) time randomized FPTAS for Partition, which is derived from a certain relaxed form of a randomized FPTAS for Subset Sum. To the best of our knowledge, this is the first NP-hard problem that has been shown to admit a subquadratic time approximation scheme, i.e., one with time complexity of O((n + 1/ε) 2−δ ) for some δ > 0. To put these developments in context, note that a quadratic FPTAS for Partition has been known for 40 years.Our main contribution lies in designing a mechanism that reduces an instance of Subset Sum to several simpler instances, each with some special structure, and keeps track of interactions between them. This allows us to combine techniques from approximation algorithms, pseudopolynomial algorithms, and additive combinatorics.We also prove several related results. Notably, we improve approximation schemes for 3SUM, (min, +)-convolution, and TreeSparsity. Finally, we argue why breaking the quadratic barrier for approximate Knapsack is unlikely by giving an Ω((n + 1/ε) 2−o(1) ) conditional lower bound.

The complete nucleotide sequence of the small, cryptic plasmid pWKS1 (2697 bp) of Paracoccus pantotrophus DSM 11072 was determined. The GMC content of the sequence of this plasmid was 62 mol %. Analysis revealed that over 80 % of the plasmid genome was covered by two ORFs, ORF1 and ORF2, which were capable of encoding putative peptides of 441 and 378 kDa, respectively. Mutational analysis showed that ORF2 was crucial for plasmid replication.

The replicon of the pTAV3 megaplasmid (approx. 400 kb) of Paracoccus versutus has been localized to a 4 3 kb EcoRI restriction fragment and its entire nucleotide sequence determined. The GMC content of the entire sequence is 66 mol %, which is within the range (62-66 mol %) previously determined for P. versutus total DNA. ORF1 encodes a replication initiation protein Rep (47 2 kDa), which shares substantial similarity with putative proteins of the Coxiella burnetii plasmids QpH1 and QpDV, and the replication protein of Pseudomonas syringae plasmid pPS10. ORF2, located in the opposite transcriptional orientation to ORF1, encodes a putative protein that shares similarity to a subfamily of ATPases involved in plasmid partitioning. The highest similarity was observed with homologous proteins (RepA) encoded by the repABC family of replicons found in several plasmids of Agrobacterium, Rhizobium and Paracoccus spp. The predicted product of ORF3 was similar to AcoR, Nif and NtrC transcriptional activators. A strong incompatibility determinant (inc) was localized between ORF1 (rep) and ORF2 (parA). The origin of replication of pTAV400 contains a short AMT-rich region and several imperfect palindromic sequences. Curing experiments demonstrated that the megaplasmid bears genes required for growth in minimal media and can therefore be referred to as a mini-chromosome. Megaplasmids pTAV3 of P. versutus UW1 and pKLW2 of Paracoccus pantotrophus DSM 11073 were found to carry closely related, incompatible replicons. It has been shown that plasmid pORI6 (containing oriV of pTAV3 cloned into plasmid pABW1, which does not replicate in Paracoccus spp.) can be trans activated not only by pTAV3, but also by pKLW2. Using pORI6, it was demonstrated that replication systems related to pTAV3 are also present in the replicons of Paracoccus alcaliphilus JCM 7364, Paracoccus thiocyanatus IAM 12816 and Paracoccus methylutens DM 12.

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