Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds

Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of $k$-ary instances where $k$ is a constant, i.e., each agent has at most $k$ different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.Comment: 23 pages. A preliminary version appeared at AAAI 202

On Fair and Efficient Allocations of Indivisible Public Goods

We study fair allocation of indivisible public goods subject to cardinality (budget) constraints. In this model, we have n agents and m available public goods, and we want to select k ? m goods in a fair and efficient manner. We first establish fundamental connections between the models of private goods, public goods, and public decision making by presenting polynomial-time reductions for the popular solution concepts of maximum Nash welfare (MNW) and leximin. These mechanisms are known to provide remarkable fairness and efficiency guarantees in private goods and public decision making settings. We show that they retain these desirable properties even in the public goods case. We prove that MNW allocations provide fairness guarantees of Proportionality up to one good (Prop1), 1/n approximation to Round Robin Share (RRS), and the efficiency guarantee of Pareto Optimality (PO). Further, we show that the problems of finding MNW or leximin-optimal allocations are NP-hard, even in the case of constantly many agents, or binary valuations. This is in sharp contrast to the private goods setting that admits polynomial-time algorithms under binary valuations. We also design pseudo-polynomial time algorithms for computing an exact MNW or leximin-optimal allocation for the cases of (i) constantly many agents, and (ii) constantly many goods with additive valuations. We also present an O(n)-factor approximation algorithm for MNW which also satisfies RRS, Prop1, and 1/2-Prop

New Algorithms for the Fair and Efficient Allocation of Indivisible Chores

We study the problem of fairly and efficiently allocating indivisible chores among agents with additive disutility functions. We consider the widely-used envy-based fairness properties of EF1 and EFX, in conjunction with the efficiency property of fractional Pareto-optimality (fPO). Existence (and computation) of an allocation that is simultaneously EF1/EFX and fPO are challenging open problems, and we make progress on both of them. We show existence of an allocation that is - EF1+fPO, when there are three agents, - EF1+fPO, when there are at most two disutility functions, - EFX+fPO, for three agents with bivalued disutilities. These results are constructive, based on strongly polynomial-time algorithms. We also investigate non-existence and show that an allocation that is EFX+fPO need not exist, even for two agents.Comment: 43 pages. Accepted to IJCAI 2023. This version corrects a typo in Theorem

Does "ichthyosis uteri" have malignant potential? : A case report of squamous cell carcinoma of endometrium associated with extensive ichthyosis uteri

Ichthyosis uteri is a rare condition in which the entire surface of the endometrium is replaced by stratified squamous epithelium. Though the condition often is considered as benign, anaplastic and dysplastic changes have been reported. We describe herein a rare case of low-grade squamous cell carcinoma of endometrium associated with extensive ichthyosis uteri with dysplasia. The cervix showed moderate to severe dysplastic changes while the right fallopian tube showed extensive squamous metaplasia with dysplastic changes. We conclude that squamous cell carcinoma could develop into pre-existing ichthyosis uteri

Landscaping capacity-building initiatives in epidemiology in India: bridging the demand–supply gap

India, the second most populous country in the world, has 17% of the world’s population but its total share of global disease burden is 21%. With epidemiological transition, the challenge of the public health system is to deal with a high burden of noncommunicable diseases, while still continuing the battle against communicable diseases. To combat this progression, public health capacity-building initiatives for the health workforce are necessary to develop essential skills in epidemiology and competencies in other related fields of public health. This study is an effort to systematically explore the training programmes in epidemiology in India and to understand the demand–supply dynamics of epidemiologists in the country. A systematic, predefined approach, with three parallel strategies, was used to collect and assemble the data regarding epidemiology training in India and assess the demand–supply of epidemiologists in the country. The programmes offering training in epidemiology included degree and diploma courses offered by departments of preventive and social medicine/community medicine in medical colleges and 19 long-term academic programmes in epidemiology, with an estimated annual output of 1172 per year. The demand analysis for epidemiologists estimated that there is need for at least 3289 epidemiologists to cater for the demand of various institutions in the country. There is a wide gap in demand–supply of epidemiologists in the country and an urgent need for further strengthening of epidemiology training in India. More capacity-building and training initiatives in epidemiology are therefore urgently required to promote research and address the public health challenges confronting the country

A Case Report: Mucinous Tubular and Spindle Cell Carcinoma of Kidney with Spindle Cell Predominance Mimicking Mesenchymal Tumour

Mucinous tubular and spindle cell carcinoma (MTSCC) of kidney is a rare variant of renal cell carcinoma which was first described in the 2004 World Health Organization classification of tumours of the kidney. Morphologically, MTSCC is composed of tubules merging with bland-appearing spindle cells in a myxoid/mucinous stroma. Diverse morphological patterns have been reported in MTSCC; however, a spindle cell predominant morphology mimicking a mesenchymal tumour is rare and only two cases have been reported so far. We report a case of MTSCC with spindle cell predominance in kidney which was a diagnostic challenge. Though MTSCC usually shows an indolent course, there have been cases showing aggressive behaviour even with bland morphology. Hence, a thorough histopathological evaluation with ancillary studies are required to differentiate spindle cell predominant MTSCC from its mimics. Our case was a 40-year-old female who was incidentally found to have a well-defined hypodense lesion measuring around 2 cm in the upper pole of the right kidney. Right partial nephrectomy was performed which showed a 2.7 × 2.5 × 2 cm well-defined grey tan tumour without necrosis or haemorrhage, limited to kidney. Histopathological examination showed sheets of bland-appearing spindle cells mimicking a mesenchymal tumour. The tumour was extensively sampled, revealing small foci of tubule formation and mucinous stroma. Tumour cells were positive for CK7, AMACR, and PAX8. A final diagnosis of MTSCC was made. Hereby, we discuss ways of differentiating MTSCC from other spindle cell tumours of the kidney

A Rapidly-progressing Outbreak of Cholera in a Shelter-home for Mentally-retarded Females, Amta-II Block, Howrah, West Bengal, India

On 13 May 2010, a cluster of diarrhoeal disease cases was reported among the inmates of a shelter-home for mentally-retarded females in Parbaksi village of Howrah district in West Bengal, India. The outbreak was investigated to identify the aetiological agent and source of infection and to propose recommendations. A suspected case of cholera was defined as an acute onset of >3 loose watery stools in a female resident of the shelter-home since 1 May 2010. The demographic and clinical details were collected from the suspected case-patients, and the outbreak was described by time, place, and person. A retrospective cohort study was conducted to identify the risk factors associated with the illness. Of the 101 inmates, 91 (90%) developed diarrhoea, and three patients died (case fatality−3%). Four of the five stool specimens were positive for Vibrio cholerae O1 Ogawa. Drinking of water from the pond-connected tubewell (adjusted odds ratio=25.7, 95% confidence interval 2.7-236.4) was associated with the illness. Relocation of the pond-connected tubewell away from the groundwater tubewell, colour-coding of the tubewells meant for drinking purposes, and regular disinfection of the tubewells were recommended

Dynamic Vector Bin Packing for Online Resource Allocation in the Cloud

Several cloud-based applications, such as cloud gaming, rent servers to execute jobs which arrive in an online fashion. Each job has a resource demand and must be dispatched to a cloud server which has enough resources to execute the job, which departs after its completion. Under the `pay-as-you-go' billing model, the server rental cost is proportional to the total time that servers are actively running jobs. The problem of efficiently allocating a sequence of online jobs to servers without exceeding the resource capacity of any server while minimizing total server usage time can be modelled as a variant of the dynamic bin packing problem (DBP), called MinUsageTime DBP. In this work, we initiate the study of the problem with multi-dimensional resource demands (e.g. CPU/GPU usage, memory requirement, bandwidth usage, etc.), called MinUsageTime Dynamic Vector Bin Packing (DVBP). We study the competitive ratio (CR) of Any Fit packing algorithms for this problem. We show almost-tight bounds on the CR of three specific Any Fit packing algorithms, namely First Fit, Next Fit, and Move To Front. We prove that the CR of Move To Front is at most $(2\mu+1)d +1$, where $\mu$ is the ratio of the max/min item durations. For $d=1$, this significantly improves the previously known upper bound of $6\mu+7$ (Kamali & Lopez-Ortiz, 2015). We then prove the CR of First Fit and Next Fit are bounded by $(\mu+2)d+1$ and $2\mu d+1$, respectively. Next, we prove a lower bound of $(\mu+1)d$ on the CR of any Any Fit packing algorithm, an improved lower bound of $2\mu d$ for Next Fit, and a lower bound of $2\mu$ for Move To Front in the 1-D case. All our bounds improve or match the best-known bounds for the 1-D case. Finally, we experimentally study the average-case performance of these algorithms on randomly generated synthetic data, and observe that Move To Front outperforms other Any Fit packing algorithms.Comment: 24 pages, to appear at SPAA 202