Given an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with vertex weights $(w(u))_{u\in\mathcal{V}}$, vertex values $(α(u))_{u\in\mathcal{V}}$, a knapsack size $s$, and a target value $d$, the \vcknapsack problem is to determine if there exists a subset $\mathcal{U}\subseteq\mathcal{V}$ of vertices such that $\mathcal{U}$ forms a vertex cover, $w(\mathcal{U})=\sum_{u\in\mathcal{U}} w(u) \le s$, and $α(\mathcal{U})=\sum_{u\in\mathcal{U}} α(u) \ge d$. In this paper, we closely study the \vcknapsack problem and its variations, such as \vcknapsackbudget, \minimalvcknapsack, and \minimumvcknapsack, for both general graphs and trees. We first prove that the \vcknapsack problem belongs to the complexity class \NPC and then study the complexity of the other variations. We generalize the problem to \setc and \hs versions and design polynomial time $H_g$-factor approximation algorithm for the \setckp problem and d-factor approximation algorithm for \hstp using primal dual method. We further show that \setcks and \hsmb are hard to approximate in polynomial time. Additionally, we develop a fixed parameter tractable algorithm running in time $8^{\mathcal{O}({\rm tw})}\cdot n\cdot {\sf min}\{s,d\}$ where ${\rm tw},s,d,n$ are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items for the \minimalvcknapsack problem.

The Euclidean Steiner Minimal Tree problem takes as input a set P of points in the Euclidean plane and finds the minimum length network interconnecting all the points of P. In this paper, in continuation to the works of [Du et al., 1987] and [Weng and Booth, 1995], we study Euclidean Steiner Minimal Tree when P is formed by the vertices of a pair of regular, concentric and parallel n-gons. We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for P. We also consider point sets P of size n where the number of input points not on the convex hull of P is f(n) ≤ n. We give an exact algorithm with running time 2^𝒪(f(n) log n) for such input point sets P. Note that when f(n) = 𝒪(n/(log n)), our algorithm runs in single-exponential time, and when f(n) = o(n) the running time is 2^o(n log n) which is better than the known algorithm in [Hwang et al., 1992]. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P = NP [Garey et al., 1977]. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets [Scott Provan, 1988]. In this paper, we show that if the number of input points in P not belonging to the convex hull of P is 𝒪(log n), then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any ε ∈ (0,1], when there are Ω(n^ε) points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P = NP.

Abstract In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that $$\mathcal {O}(f(k) \log ^{\mathcal {O}(1)} n)$$ O ( f ( k ) log O ( 1 ) n ) space is enough, where f is an arbitrary computable function depending only on the parameter k. However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require $$\Omega (n)$$ Ω ( n ) bits of storage for any streaming algorithm; e.g. Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal, Triangle Deletion or the more general $$\mathcal{F}$$ F -Subgraph Deletion when parameterized by solution size k. Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. In this work, we consider the four most well-studied streaming models: the Ea, Dea, Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like $$\mathcal{F}$$ F -Subgraph Deletion and $$\mathcal{F}$$ F -Minor Deletion.

Abstract. We study exact algorithms for Metric TSP in [Formula: see text]. In the early 1990s, algorithms with [Formula: see text] running time were presented for the planar case, and some years later an algorithm with [Formula: see text] running time was presented for any [Formula: see text]. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Metric TSP, except for a lower bound stating that the problem admits no [Formula: see text] algorithm unless ETH fails. In this paper we settle the complexity of Metric TSP, up to constant factors in the exponent and under ETH, by giving an algorithm with running time [Formula: see text].

Winning lottery tickets refer to sparse subgraphs of deep neural networks which have classification accuracy close to the original dense networks. Resilient connectivity properties of such sparse networks play an important role in their performance. The attempt is to identify a sparse and yet well-connected network to guarantee unhindered information flow. Connectivity in a graph is best characterized by its spectral expansion property. Ramanujan graphs are robust expanders which lead to sparse but highly-connected networks, and thus aid in studying the winning tickets. A feedforward neural network consists of a sequence of bipartite graphs representing its layers. We analyze the Ramanujan graph property of such bipartite layers in terms of their spectral characteristics using the Cheeger’s inequality for irregular graphs. It is empirically observed that the winning ticket networks preserve the Ramanujan graph property and achieve a high accuracy even when the layers are sparse. Accuracy and robustness to noise start declining as many of the layers lose the property. Next we find a robust winning lottery ticket by pruning individual layers while retaining their respective Ramanujan graph property. This strategy is observed to improve the performance of existing network pruning algorithms.

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery , it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm.

A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the R ectilinear S teiner T ree problem, the input is a set K ={ z 1 , z 2 ,…, z n } of n points in the Euclidean plane (R 2 ), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the R ectilinear S teiner A rborescence problem, the input is a set K of n points in R 2 , and a root r ∈ K , and the task is to find a rectilinear Steiner arborescence for K , rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2 O (√ n log n ) time.

Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring $$c:V(G) \rightarrow \{1,2,\ldots ,q\}$$ such that for each vertex $$v \in V(G)$$ there is a vertex in N(v) that is uniquely colored from the rest of the vertices in N(v). When we replace N(v) by the closed neighborhood N[v], then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both $$(q \ge 2)$$ -ONCF-coloring and $$(q \ge 3)$$ -CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless $$\mathsf {NP \subseteq coNP/poly}$$ . On the other hand, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude the study with some combinatorial results. Denote $$\chi _{ON}(G)$$ and $$\chi _{CN}(G)$$ to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on $$\chi _{CN}(G)$$ with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on $$\chi _{ON}(G)$$ with respect to minimum vertex cover size was known. We provide tight bounds for $$\chi _{ON}(G)$$ with respect to minimum vertex cover size. Also, we provide the first upper bounds on $$\chi _{ON}(G)$$ with respect to minimum feedback vertex set size and treewidth.

We study exact algorithms for Euclidean TSP in Rd. In the early 1990s algorithms with nO(√n)running time were presented for the planar case, and some years later an algorithm with nO(n1-1/d)running time was presented for any d ≥ 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no 2O(n1-1/d-ε) algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a 2O(n1-1/d)algorithm and by showing that a 2o(n1-1/d)algorithm does not exist unless ETH fails.

In this paper, we study the query complexity of parameterized decision and optimization versions of Hitting-Set. We also investigate the query complexity of Packing. In doing so, we use generalizations to hypergraphs of an earlier query model, known as BIS introduced by Beame et al. in ITCS'18. The query models considered are the GPIS and GPISE oracles. The GPIS and GPISE oracles are used for the decision and optimization versions of the problems, respectively. We use color coding and queries to the oracles to generate subsamples from the hypergraph, that retain some structural properties of the original hypergraph. We use the stability of the sunflowers in a non-trivial way to do so.

Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph $H=(U, \mathcal{F})$, a conflict-free coloring of $H$ refers to a vertex coloring where every hyperedge has a vertex with a unique color, distinct from all other vertices in the hyperedge. In this paper, we initiate a study of a natural maximization version of this problem, namely, Max-CFC: For a given hypergraph $H$ and a fixed $r\geq 2$, color the vertices of $U$ using $r$ colors so that the number of hyperedges that are conflict-free colored is maximized. By previously known hardness results for conflict-free coloring, this maximization version is NP-hard. We study this problem in the context of both exact and parameterized algorithms. In the parameterized setting, we study this problem with respect to a natural parameter---the solution size. In particular, the question we study is the following: p-CFC: For a given hypergraph, can we conflict-free color at least $k$ hyperedges with at most $r$ colors, the parameter being the solution size $k$. We show that this problem is fixed parameter tractable by designing an algorithm with running time $2^{\mathcal{O}(k \log \log k + k \log r)}(n+m)^{\mathcal{O}(1)}$ using a novel connection to the Unique Coverage problem and applying the method of color coding in a nontrivial manner. For the special case for hypergraphs induced by graph neighborhoods we give a polynomial kernel. Finally, we give an exact algorithm for Max-CFC running in $\mathcal{O}(2^{n+m})$ time. All our algorithms, with minor modifications, work for a stronger version of conflict-free coloring, Unique Maximum Coloring.

Given a 1.5-dimensional terrain T , also known as an x -monotone polygonal chain, the T errain G uarding problem seeks a set of points of minimum size on T that guards all of the points on T . Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T . The T errain G uarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm (SODA 2005). However, only in 2010 King and Krohn (SODA 2010) finally showed that T errain G uarding is NP-hard. In spite of the remarkable developments in approximation algorithms for T errain G uarding , next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether, if parameterized by the size k of a solution guard set, it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this article, we answer the first question affirmatively by developing an n O (√ k ) -time algorithm for both D iscrete T errain G uarding and C ontinuous T errain G uarding . We also make non-trivial progress with respect to the second question: we show that D iscrete O rthogonal T errain G uarding , a well-studied special case of T errain G uarding , is fixed-parameter tractable.

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.

In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in \mathbb{R}^d and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in general position with respect to a subsystem of hyperplanes in \mathbb{R}^d if no d+1 points lie on the same hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations. When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h=n-k, or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic assumptions. We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.

Given a graph G and a pair (\mathcal{F}_1,\mathcal{F}_2) of graph families, the function {\sf GDISJ}_{G,{\cal F}_1,{\cal F}_2} takes as input, two induced subgraphs G_1 and G_2 of G, such that G_1 \in \mathcal{F}_1 and G_2 \in \mathcal{F}_2 and returns 1 if V(G_1)\cap V(G_2)=\emptyset and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we obtain nuanced upper bounds on the communication complexity of GDISJ_G,\cal F_1,\cal F_2. A concept related to communication protocols is that of a (\mathcal{F}_1,\mathcal{F}_2)-separating family of a graph G. A collection \mathcal{F} of subsets of V(G) is called a (\mathcal{F}_1,\mathcal{F}_2)-separating family} for G, if for any two vertex disjoint induced subgraphs G_1\in \mathcal{F}_1,G_2\in \mathcal{F}_2, there is a set F \in \mathcal{F} with V(G_1) \subseteq F and V(G_2) \cap F = \emptyset. Given a graph G on n vertices, for any pair (\mathcal{F}_1,\mathcal{F}_2) of hereditary graph families with sublinear communication complexity for GDISJ_G,\cal F_1,\cal F_2, we give an enumeration algorithm that finds a subexponential sized (\mathcal{F}_1,\mathcal{F}_2)-separating family. In fact, we give an enumeration algorithm that finds a 2^{o(k)}n^{\Oh(1)} sized (\mathcal{F}_1,\mathcal{F}_2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V(G)\setminus S has a bipartition (V_1,V_2) with G[V_1] \in {\cal F}_1 and G[V_2]\in {\cal F}_2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G\S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C_{odd}, families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C_{odd} are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12^{k}*n^{O(1)} for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.