How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach

In this paper we study a resource allocation problem that encodes correlation between items in terms of \conflict and maximizes the minimum utility of the agents under a conflict free allocation. Admittedly, the problem is computationally hard even under stringent restrictions because it encodes a variant of the {\sc Maximum Weight Independent Set} problem which is one of the canonical hard problems in both classical and parameterized complexity. Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from the classical complexity perspective to draw the boundary between {\sf NP}-hardness and tractability for a constant number of agents. The problem was shown to be hard even for small constant number of agents and various other restrictions on the underlying graph. Notwithstanding this computational barrier, we notice that there are several parameters that are worth studying: number of agents, number of items, combinatorial structure that defines the conflict among the items, all of which could well be small under specific circumstancs. Our search rules out several parameters (even when taken together) and takes us towards a characterization of families of input instances that are amenable to polynomial time algorithms when the parameters are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)} algorithm for our problem where $m$ denotes the number of items that significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using ideas from FFT based fast polynomial multiplication; and we identify simple graph classes relevant to our problem's motivation that admit efficient algorithms

Parameterized Complexity of Conflict-Free Matchings and Paths

An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid

Application of physics-informed neural networks to inverse problems in unsaturated groundwater flow

This paper investigates the application of Physics-Informed Neural Networks (PINNs) to inverse problems in unsaturated groundwater flow. PINNs are applied to the types of unsaturated groundwater flow problems modelled with the Richards partial differential equation and the van Genuchten constitutive model. The inverse problem is formulated here as a problem with known or measured values of the solution to the Richards equation at several spatio-temporal instances, and unknown values of solution at the rest of the problem domain and unknown parameters of the van Genuchten model. PINNs solve inverse problems by reformulating the loss function of a deep neural network such that it simultaneously aims to satisfy the measured values and the unknown values at a set of collocation points distributed across the problem domain. The novelty of the paper originates from the development of PINN formulations for the Richards equation that requires training of a single neural network. The results demonstrate that PINNs are capable of efficiently solving the inverse problem with relatively accurate approximation of the solution to the Richards equation and estimates of the van Genuchten model parameters.publishedVersio

Acute pancreatitis masquerading as an ovarian torsion: a rare case report

Ovarian torsion is a common diagnostic challenge constituting 2.7% to 7.4% of all gynaecological emergencies. It commonly occurs in women of reproductive age however pre-pubertal girls and postmenopausal women can also be affected. Ovarian torsion accounts for approximately 3% of all cases of children with acute abdominal pain and requires immediate surgical intervention. Other common acute adnexal pathologies include simple ovarian cysts (OCs) with or without rupture. Owing to the nonspecific clinical presentation and poor specificity of radiologic tests, the diagnosis of OT in girls remains challenging. Here we are reporting a case of acute abdomen with large cyst on ultrasound clinically presented as torsion later on in the post-operative period diagnosed with acute pancreatitis

Blood zinc levels in children hospitalized with severe pneumonia: a case control study

A case control study was conducted in a referral and teaching hospital in North India on children aged 2 months to 5 years, to compare blood zinc levels in 50 cases of severe pneumonia and 50 age, sex and nutritional status matched controls. Mean blood Zinc levels in cases and controls was 376.1 µg/dL ± 225.73 and 538.52 µg/dL ± 228.0 respectively ( P value 0.0003). In logistic regression model severe pneumonia was associated with lower blood zinc level, use of biomass fuel and isolation of H. Influenzae from nasopharyngeal swab. Cotrimoxazole resistant S. pneumoniae were isolated from 95% of cases and 41.2% of controls (P = 0.0004). Therefore, the role of zinc in treatment of severe pneumonia should be investigated

Exact and Approximate Digraph Bandwidth

In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively. - Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree. - Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor. - Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n). - Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n)

On the (Parameterized) Complexity of Almost Stable Marriage

In the Stable Marriage problem, when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, the goal is to find a matching whose size exceeds that of a stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm

Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints

In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $\Phi$, and $k \ge 0$, and the goal is to find an assignment $\beta$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $\beta$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $\Phi$ is monotone, i.e., does not contain any negative literals. CC-MaxSAT and MaxCov are extremely well-studied problems in the approximation algorithms as well as parameterized complexity literature. Our first contribution is that the two problems are equivalent to each other in the context of FPT-Approximation parameterized by $k$ (approximation is in terms of number of clauses satisfied/elements covered). We give a randomized reduction from CC-MaxSAT to MaxCov in time $O(1/\epsilon)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-\epsilon$. Furthermore, this reduction also works in the presence of fairness and matroid constraints. Armed with this reduction, we focus on designing FPT-Approximation schemes (FPT-ASes) for MaxCov and its generalizations. Our algorithms are based on a novel combination of a variety of ideas, including a carefully designed probability distribution that exploits sparse coverage functions. These algorithms substantially generalize the results in Jain et al. [SODA 2023] for CC-MaxSAT and MaxCov for $K_{d,d}$-free set systems (i.e., no $d$ sets share $d$ elements), as well as a recent FPT-AS for Matroid-Constrained MaxCov by Sellier [ESA 2023] for frequency-$d$ set systems.Comment: Abstract shortened due to arxiv restriction

Conflict and Fairness in Resource Allocation

In the standard model of fair allocation of resources to agents, every agent has some utility for every resource, and the goal is to assign resources to agents so that the agents' welfare is maximized. Motivated by job scheduling, interest in this problem dates back to the work of Deuermeyer et al. [SIAM J. on Algebraic Discrete Methods'82]. Recent works consider the compatibility between resources and assign only mutually compatible resources to an agent. We study a fair allocation problem in which we are given a set of agents, a set of resources, a utility function for every agent over a set of resources, and a {\it conflict graph} on the set of resources (where an edge denotes incompatibility). The goal is to assign resources to the agents such that $(i)$ the set of resources allocated to an agent are compatible with each other, and $(ii)$ the minimum satisfaction of an agent is maximized, where the satisfaction of an agent is the sum of the utility of the assigned resources. Chiarelli et al. [Algorithmica'22] explore this problem from the classical complexity perspective to draw the boundary between the cases that are polynomial-time solvable and those that are \NP-hard. In this article, we study the parameterized complexity of the problem (and its variants) by considering several natural and structural parameters.Comment: arXiv admin note: substantial text overlap with arXiv:2309.0499

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the F-Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k, F-Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. We obtain the following results for the F-Contraction problem. - Clique Contraction is known to be FPT. However, unless NP ⊆ coNP/poly, it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (PSAKS). That is, it admits a (1 + ε)-approximate kernel with {O}(k^{f(ε)}) vertices for every ε > 0. - Split Contraction is known to be W[1]-Hard. We deconstruct this intractability result in two ways. Firstly, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)-FPT-approximation algorithm for Split Contraction). Furthermore, we show that, assuming Gap-ETH, there is no (5/4-δ)-FPT-approximation algorithm for Split Contraction. Here, ε, δ > 0 are fixed constants. - Chordal Contraction is known to be W[2]-Hard. We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1], there is no F(k)-FPT-approximation algorithm for Chordal Contraction. Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k)-FPT-approximation algorithm for the F-Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and |X| ≤ k, it outputs an edge set Y of size at most h(k) ⋅ k for which G/Y is in F. We find it extremely interesting that three closely related problems have different behavior with respect to FPT-approximation