Incrementally Maintaining the Number of l-cliques

The main contribution of this paper is an incremental algorithm to update the number of $l$-cliques, for $l \geq 3$, in which each node of a graph is contained, after the deletion of an arbitrary node. The initialization cost is $O(n^{\omega p+q})$, where $n$ is the number of nodes, $p=\lfloor \frac{l}{3} \rfloor$, $q=l \pmod{3}$, and $\omega=\omega(1,1,1)$ is the exponent of the multiplication of two $n x n$ matrices. The amortized updating cost is $O(n^{q}T(n,p,\epsilon))$ for any $\epsilon \in [0,1]$, where $T(n,p,\epsilon)=\min\{n^{p-1}(n^{p(1+\epsilon)}+n^{p(\omega(1,\epsilon,1)-\epsilon)}),n^{p \omega(1,\frac{p-1}{p},1)}\}$ and $\omega(1,r,1)$ is the exponent of the multiplication of an $n x n^{r}$ matrix by an $n^{r} x n$ matrix. The current best bounds on $\omega(1,r,1)$ imply an $O(n^{2.376p+q})$ initialization cost, an $O(n^{2.575p+q-1})$ updating cost for $3 \leq l \leq 8$, and an $O(n^{2.376p+q-0.532})$ updating cost for $l \geq 9$. An interesting application to constraint programming is also considered

LP-Based Approximation Algorithms for Facility Location in Buy-at-Bulk Network Design

Abstract We study problems that integrate buy-at-bulk network design into the classical (connected) facility location problem. In such problems, we need to open facilities, build a routing network, and route every client demand to an open facility. Furthermore, capacities of the edges can be purchased in discrete units from K different cable types with costs that satisfy economies of scale. We extend the linear programming frame-work of Talwar [IPCO 2002] for the single-source buy-at-bulk problem to these variants and prove integrality gap upper bounds for both facility location and connected facility location buy-at-bulk problems. For the unconnected variant we prove an integrality gap bound of O(K), and for the connected version, we get an improved bound of O(1).

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Utilitarian Mechanism Design for Multiobjective Optimization

In a classic optimization problem, the complete input data is assumed to be known to the algorithm. This assumption may not be true anymore in optimization problems motivated by the Internet where part of the input data is private knowledge of independent selfish agents. The goal of algorithmic mechanism design is to provide (in polynomial time) a solution to the optimization problem and a set of incentives for the agents such that disclosing the input data is a dominant strategy for the agents. In the case of NP-hard problems, the solution computed should also be a good approximation of the optimum. In this paper we focus on mechanism design for multiobjective optimization problems. In this setting we are given a main objective function and a set of secondary objectives which are modeled via budget constraints. Multiobjective optimization is a natural setting for mechanism design as many economical choices ask for a compromise between different, partially conflicting goals. The main contribution of this paper is showing that two of the main tools for the design of approximation algorithms for multiobjective optimization problems, namely, approximate Pareto sets and Lagrangian relaxation, can lead to truthful approximation schemes. By exploiting the method of approximate Pareto sets, we devise truthful deterministic and randomized multicriteria fully polynomial-time approximation schemes (FPTASs) for multiobjective optimization problems whose exact version admits a pseudopolynomial-time algorithm, as, for instance, the multibudgeted versions of minimum spanning tree, shortest path, maximum (perfect) matching, and matroid intersection. Our construction also applies to multidimensional knapsack and multiunit combinatorial auctions. Our FPTASs compute a $(1+\varepsilon)$-approximate solution violating each budget constraint by a factor $(1+\varepsilon)$. When feasible solutions induce an independence system, i.e., when subsets of feasible solutions are feasible as well, we present a PTAS (not violating any constraint), which combines the approach above with a novel monotone way to guess the heaviest elements in the optimum solution. Finally, we present a universally truthful Las Vegas PTAS for minimum spanning tree with a single budget constraint, where one wants to compute a minimum cost spanning tree whose length is at most a given value $L$. This result is based on the Lagrangian relaxation method, in combination with our monotone guessing step and with a random perturbation step (ensuring low expected running time). This result can be derandomized in the case of integral lengths. All the mentioned results match the best known approximation ratios, which are, however, obtained by nontruthful algorithms

Direct Oral Anticoagulant Drugs: On the Treatment of Cancer-Related Venous Thromboembolism and their Potential Anti-Neoplastic Effect.

Cancer patients develop a hypercoagulable state with a four- to seven-fold higher thromboembolic risk compared to non-cancer patients. Thromboembolic events can precede the diagnosis of cancer, but they more often occur at diagnosis or during treatment. After malignancy itself, they represent the second cause of death. Low molecular weight heparins are the backbone of the treatment of cancer-associated thromboembolism. This treatment paradigm is possibly changing, as direct oral anticoagulants (DOACs) may prove to be an alternative therapeutic option. The currently available DOACs were approved during the first and second decades of the 21st century for various clinical indications. Three molecules (apixaban, edoxaban and rivaroxaban) are targeting the activated factor X and one (dabigatran) is directed against the activated factor II, thrombin. The major trials analyzed the effect of these agents in the general population, with only a small proportion of cancer patients. Two published and several ongoing studies are specifically investigating the use of DOACs in cancer-associated thromboembolism. This article will review the current available literature on the use of DOACs in cancer patients. Furthermore, we will discuss published data suggesting potential anti-cancer actions exerted by non-anticoagulant effects of DOACs. As soon as more prospective data becomes available, DOACs are likely to be considered as a potential new therapeutic option in the armamentarium for patients suffering of cancer-associated thromboembolism

Single nucleotide polymorphisms detected and in silico analysis of the 5' flanking sequence and exon 1 in the Bubalus bubalis leptin gene

In this study, we have sequenced the 5' flanking region and exon 1 of the leptin gene in buffalo, and have detected eight single nucleotide polymorphisms; we have made evidence, through in silico analysis, that many of them fall within putative binding sites for transcription factors. Starting from the bovine whole genome shotgun sequence, that encodes the complete sequence of the leptin gene, we had designed primers to amplify two amplicons, so to cover the 5' flanking and exon 1 of the leptin gene of 41 non related buffalos. The newly sequenced buffalo fragment was submitted to profile search for transcription factor binding sites, using the MATCH program, focusing on the areas where the single nucleotide polymorphisms had been detected. Our analysis shows that the majority of the identified single nucleotide polymorphisms fall into the core sequence of transcription factor binding sites that regulate the expression of target genes in many physiological processes within mammalian tissues. Because the leptin gene plays an important role in influencing economic traits in cattle, the novel detected single nucleotide polymorphisms might be used in association studies to assess their potential of being genetic markers for selection

simultaneous cycle sequencing assessment of tg m and tn tract length in cftr gene

The lengths of the dinucleotide (TG) m and mononucleotide T n repeats, both located at the intron 8/exon 9 splice acceptor site of the cystic fibrosis transmembrane conductance regulator (CFTR) gene whose mutations cause cystic fibrosis (CF), have been shown to influence the skipping of exon 9 in CFTR mRNA. This exon 9-skipped mRNA encodes a nonfunctional protein and is associated with various clinical manifestations in CF. As a result of growing interest in these repeats, several assessment methods have been developed, most of which are, however, cumbersome, multi-step, and time consuming. Here, we describe a rapid method for the simultaneous assessment of the lengths of both (TG) m and T n repeats, based on a nonradioactive cycle sequencing procedure that can be performed even without DNA extraction. This method determines the lengths of the (TG) m and T n tracts of both alleles, which in our samples ranged from TG 8 to TG 12 in the presence of T 5 , T 7 , and T 9 alleles, and also fully assesses the aplotypes. In addition, the repeats in the majority of these samples can be assessed by single-strand sequencing, with no need to sequence the other strand, thereby saving to sequence the other strand, thereby saving a considerable amount of time and effort

Autoimmune Hemolytic Anemia and Pulmonary Embolism: An Association to Consider.

Autoimmune hemolytic anemia (AIHA) is increasingly recognized as a strong risk factor for venous thrombosis. However, there are currently no guidelines on thromboembolism prevention and management during AIHA. Here, we describe the case of a patient with AIHA and pulmonary embolism and resume the current knowledge on epidemiology, risk factors, treatment, and pathophysiology of thrombosis during AIHA, as well as new therapeutic perspectives to prevent thrombus formation during AIHA

Set covering with our eyes closed

Given a universe $U$ of $n$ elements and a weighted collection $\mathscr{S}$ of $m$ subsets of $U$, the universal set cover problem is to a priori map each element $u \in U$ to a set $S(u) \in \mathscr{S}$ containing $u$ such that any set $X{\subseteq U}$ is covered by S(X)=\cup_{u\in XS(u). The aim is to find a mapping such that the cost of $S(X)$ is as close as possible to the optimal set cover cost for $X$. (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of $S(X)$ can be $\Omega(\sqrt{n})$ times larger than optimal if the set $X$ is adversarially chosen. In this paper we study the performance on average, when $X$ is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is $O(\log mn)$ times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors

All-Pairs LCA in DAGs: Breaking through the O(n2.5)O(n^{2.5}) barrier

Let $G=(V,E)$ be an $n$-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices $u$ and $v$ is a common ancestor $w$ of $u$ and $v$ such that no descendant of $w$ has the same property. In this paper, we consider the problem of computing an LCA, if any, for all pairs of vertices in a DAG. The fastest known algorithms for this problem exploit fast matrix multiplication subroutines and have running times ranging from $O(n^{2.687})$ [Bender et al.~SODA'01] down to $O(n^{2.615})$ [Kowaluk and Lingas~ICALP'05] and $O(n^{2.569})$ [Czumaj et al.~TCS'07]. Somewhat surprisingly, all those bounds would still be $\Omega(n^{2.5})$ even if matrix multiplication could be solved optimally (i.e., $\omega=2$). This appears to be an inherent barrier for all the currently known approaches, which raises the natural question on whether one could break through the $O(n^{2.5})$ barrier for this problem. In this paper, we answer this question affirmatively: in particular, we present an $\tilde O(n^{2.447})$ ($\tilde O(n^{7/3})$ for $\omega=2$) algorithm for finding an LCA for all pairs of vertices in a DAG, which represents the first improvement on the running times for this problem in the last 13 years. A key tool in our approach is a fast algorithm to partition the vertex set of the transitive closure of $G$ into a collection of $O(\ell)$ chains and $O(n/\ell)$ antichains, for a given parameter $\ell$. As usual, a chain is a path while an antichain is an independent set. We then find, for all pairs of vertices, a \emph{candidate} LCA among the chain and antichain vertices, separately. The first set is obtained via a reduction to min-max matrix multiplication. The computation of the second set can be reduced to Boolean matrix multiplication similarly to previous results on this problem. We finally combine the two solutions together in a careful (non-obvious) manner