Approximation algorithms for maximum cut with limited unbalance

AbstractWe consider the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B, so as to maximize the weight of the crossing edges. For B equal to 0 this problem is known as Max Bisection, whereas for B equal to the number n of nodes it is the maximum cut problem. We present polynomial time randomized approximation algorithms with non trivial performance guarantees for its solution. The approximation results are obtained by extending the methodology used by Y. Ye for Max Bisection and by combining this technique with another one that uses the algorithm of Goemans and Williamson for the maximum cut problem. When B is equal to zero the approximation ratio achieved coincides with the one obtained by Y. Ye; otherwise it is always above this value and tends to the value obtained by Goemans and Williamson as B approaches the number n of nodes

A characterization of the base-matroids of a graphic matroid

Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ turns out to be the family of independent sets of a new matroid on $E$, called base-matroid and denoted by $M_B$. In this paper we prove that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti. Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$

A characterization of the base-matroids of a graphic matroid

Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ turns out to be the family of independent sets of a new matroid on $E$, called base-matroid and denoted by $M_B$. In this paper we prove that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti. Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$

Randomized Algorithms over Finite Fields for the Exact Parity Base Problem

AbstractWe present three randomized pseudo-polynomial algorithms for the problem of finding a base of specified value in a weighted represented matroid subject to parity conditions. These algorithms, the first two being an improved version of those presented by P. M. Camerini et al. (1992, J. Algorithms13, 258–273) use fast arithmetic working over a finite field chosen at random among a set of appropriate fields. We show that the choice of a best algorithm among those presented depends on a conjecture related to the best value of the so-called Linnik constant concerning the distribution of prime numbers in arithmetic progressions. This conjecture, which we call the C-conjecture, is a strengthened version of a conjecture formulated in 1934 by S. Chowla. If the C-conjecture is true, the choice of a best algorithm is simple, since the last algorithm exhibits the best performance, either when the performance is measured in arithmetic operations, or when it is measured in bit operations and mild assumptions hold. If the C-conjecture is false we are still able to identify a best algorithm, but in this case the choice is between the first two algorithms and depends on the asymptotic growth of m with respect to those of U and n, where 2n, 2m, U are the rank, the number of elements, and the maximum weight assigned to the elements of the matroid, respectively

Note on combinatorial optimization with max-linear objective functions

AbstractWe consider combinatorial optimization problems with a feasible solution set S⊆{0,1}n specified by a system of linear constraints in 0–1 variables. Additionally, several cost functions c1,…,cp are given. The max-linear objective function is defined by f(x):=max{c1x,…,cpx: x∈S}; where cq:=(cq1,…,cqn) is for q=1,…,p an integer row vector in Rn.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S⊆{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO

The MDM2 antagonist idasanutlin in patients with polycythemia vera:results from a single-arm phase 2 study

Idasanutlin, an MDM2 antagonist, showed clinical activity and a rapid reduction in JAK2 V617F allele burden in patients with polycythemia vera (PV) in a phase 1 study. This open-label phase 2 study evaluated idasanutlin in patients with hydroxyurea (HU)-resistant/-intolerant PV, per the European LeukemiaNet criteria, and phlebotomy dependence; prior ruxolitinib exposure was permitted. Idasanutlin was administered once daily on days 1 through 5 of each 28-day cycle. The primary end point was composite response (hematocrit control and spleen volume reduction > 35%) in patients with splenomegaly and hematocrit control in patients without splenomegaly at week 32. Key secondary end points included safety, complete hematologic response (CHR), patient-reported outcomes, and molecular responses. All patients (n = 27) received idasanutlin; 16 had response assessment (week 32). Among responders with baseline splenomegaly (n = 13), 9 (69%) attained any spleen volume reduction, and 1 achieved composite response. Nine patients (56%) achieved hematocrit control, and 8 patients (50%) achieved CHR. Overall, 43% of evaluable patients (6/14) showed a ≥50% reduction in the Myeloproliferative Neoplasm Symptom Assessment Form Total Symptom Score (week 32). Nausea (93%), diarrhea (78%), and vomiting (41%) were the most common adverse events, with grade ≥ 3 nausea or vomiting experienced by 3 patients (11%) and 1 patient (4%), respectively. Reduced JAK2 V617F allele burden occurred early (after 3 cycles), with a median reduction of 76%, and was associated with achieving CHR and hematocrit control. Overall, the idasanutlin dosing regimen showed clinical activity and rapidly reduced JAK2 allele burden in patients with HU-resistant/- intolerant PV but was associated with low-grade gastrointestinal toxicity, leading to poor long-term tolerability. This trial was registered at www.clinincaltrials.gov as #NCT03287245

Anastomosis configuration and technique following ileocaecal resection for Crohn's disease: a multicentre study

A limited ileocaecal resection is the most frequently performed procedure for ileocaecal CD and different anastomotic configurations and techniques have been described. This manuscript audited the different anastomotic techniques used in a national study and evaluated their influence on postoperative outcomes following ileocaecal resection for primary CD. This is a retrospective, multicentre, observational study promoted by the Italian Society of Colorectal Surgery (SICCR), including all adults undergoing elective ileocaecal resection for primary CD from June 2018 May 2019. Postoperative morbidity within 30 days of surgery was the primary endpoint. Postoperative length of hospital stay (LOS) and anastomotic leak rate were the secondary outcomes. 427 patients were included. The side to side anastomosis was the chosen configuration in 380 patients (89%). The stapled anastomotic (n = 286; 67%), techniques were preferred to hand-sewn (n = 141; 33%). Postoperative morbidity was 20.3% and anastomotic leak 3.7%. Anastomotic leak was independent of the type of anastomosis performed, while was associated with an ASA grade ≥ 3, presence of perianal disease and ileocolonic localization of disease. Four predictors of LOS were identified after multivariate analysis. The laparoscopic approach was the only associated with a reduced LOS (p = 0.017), while age, ASA grade ≥ 3 or administration of preoperative TPN were associated with increased LOS. The side to side was the most commonly used anastomotic configuration for ileocolic reconstruction following primary CD resection. There was no difference in postoperative morbidity according to anastomotic technique and configuration. Anastomotic leak was associated with ASA grade ≥ 3, a penetrating phenotype of disease and ileo-colonic distribution of CD

National variations in perioperative assessment and surgical management of Crohn's disease: a multicentre study

Aim: Crohn's disease (CD) requires a multidisciplinary approach and surgery should be undertaken by dedicated colorectal surgeons with audited outcomes. We present a national, multicentre study, with the aim to collect benchmark data on key performance indicators in CD surgery, to highlight areas where standards of CD surgery excel and to facilitate targeted quality improvement where indicated. Methods: All patients undergoing ileocaecal or redo ileocolic resection in the participating centres for primary and recurrent CD from June 2018 to May 2019 were included. The main objective was to collect national data on hospital volume and practice variations. Postoperative morbidity was the primary outcome. Laparoscopic surgery and stoma rate were the secondary outcomes. Results: In all, 715 patients were included: 457 primary CD and 258 recurrent CD with a postoperative morbidity of 21.6% and 34.7%, respectively. Laparoscopy was used in 83.8% of primary CD compared to 31% of recurrent CD. Twenty-five hospitals participated and the total number of patients per hospital ranged from 2 to 169. Hospitals performing more than 10 primary CD procedures per year showed a higher adoption of laparoscopy and bowel sparing surgery. Conclusions: There is significant heterogeneity in the number of CD surgeries performed per year nationally in Italy. Our data suggest that high-volume hospitals perform more complex procedures, with a higher adoption of bowel sparing surgery. The rate of laparoscopy in high-volume hospitals is higher for primary CD but not for recurrent CD compared with low-volume hospitals