Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the problem for c=1 and 2. The time complexities of the algorithms are O((k+m)log n) for c=1 and O(m^4log n+k^3+n(k^2+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|.
@cite_10 studied the problem of finding a subgraph that has few edge crossings; given a graph (not necessarily bipartite) and its geometric drawing (i.e., every vertex is specified by a 2D point, all edges are drawn as straight line segments, and no two edges overlap or intersect at a vertex), we are asked to find a subgraph of a certain class that makes the minimum number of edge crossings. They showed that, for spanning trees, @math - @math paths, cycles, matchings of a fixed size, and 1- or 2-factors, it is NP-hard to approximate the minimum number of edge crossings within a factor of @math for any @math , where @math denotes the number of edge crossings in the given graph. They also presented fixed-parameter algorithms to decide whether there is a non-crossing subgraph of one of the above graph classes, where @math is used as the parameter.
@cite_10
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In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. 2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any . Consider a bipartite graph G = (O, D, E) where O and D are origin and destination node sets respectively (O = D = n), and E is a set of edges (i, j), i ∈ O and j ∈ D(E= m). Suppose we draw the origin nodes and the destination nodes arranged in two columns and the edges as straight line segments between origins and destinations. A noncrossing matching is a subset of edges M ⊑ E such that no two edges of M intersect (including intersections at nodes). The Maximum Non Crossing Matching (MNCM) is the problem of finding the noncrossing matching of maximum cardinality. Let p denote the cardinality of the MNCM. Problems arising in several fields can be modelled as MNCM: for example the 3-Side Switch Box in VLSI design has been presented in [2] where an O(n 2) time algorithm is proposed. The problem can be reduced to the one of finding the longest increasing subsequence in a permutation of size m [1]. An algorithm for this problem has been proposed by Fredman and slightly improved by Widmayer and Wong[7]; in our case the algorithm complexity is O(m + (m - p) log (p). In this paper some labeling algorithms for the MNCM, which work directly on the bipartite graph, will be proposed. The overall complexity is improved to O(m log log n) or to O(m + minnp, (m - p) log p). Finally the weighted case is considered. Suppose we are given a bipartite graph with vertex set X, Y, |X| = n , |Y| = N , each point in X (Y) has degree D(d) fixed, respectively, moreover, each pair of points x, x' ∈ X has at most D (log n) 3 (common) neighbours. Let t (X, Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t (X, Y) d n tends to 1 as n tends to infinity. This result has many applications: Theorem [5]. Suppose k > r > 1 are fixed, n → ∞. Then there exists a collection of (1 + o (1)) × ( n r ) ( k r ) k -subsets of an n -set so that each r -subset is contained in at least one member of the collection. Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k -sets with constant probabilities, etc. Theorem 3.3. Suppose G is a graph on v vertices and e edges and ℛ is a random graph on n vertices and edge probability e ( v 2 ). Then there exists a collection of (1 + o (1))( n 2 ) ( v 2 ) induced subgraphs of ℛ on v vertices, isomorphic to G and such that each edge (non-edge) of ℛ is covered by an edge (non-edge) of a graph in the collection. It is very well-known that there are precisely two minimal non-planar graphs: K 5 and K 3 , 3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Mobius Ladder V 10 ; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V 10 ; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V 8 . A 2-layer drawing represents a bipartite graph where each vertex is a point on one of two parallel lines, no two vertices on the same line are adjacent, and the edges are straight-line segments. In this paper we study 2-layer drawings where any two crossing edges meet at right angle. We characterize the graphs that admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is ( NP )-complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings. Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph G with edge crossings and wonder how obfuscated such drawings can be. We define obf(G), the obfuscation complexity of G, to be the maximum number of edge crossings in a drawing of G. Relating obf(G) to the distribution of vertex degrees in G, we show an efficient way of constructing a drawing of G with at least obf(G) 3 edge crossings. We prove bounds (@d(G)^2 24-o(1))n^2@?obf(G) =2. The shift complexity of G, denoted by shift(G), is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of G (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If @d(G)>=3, then shift(G) is linear in the number of vertices due to the known fact that the matching number of G is linear. However, in the case @d(G)>=2 we notice that shift(G) can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing D of a planar graph, it is NP-hard to find an optimum sequence of shifts making D crossing-free. A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity, our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity, our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well. We also consider the case where the graph has edge weights. For this case, we show that an approximation factor of 2 is possible in polynomial time for finding a k -edge connected spanning subgraph. This improves an approximation factor of 3 for k = 2, due to Frederickson and Ja´Ja´ [1981], and extends it for any k (with an increased running time though). A typical problem in network design is to find a minimum-cost sub-network H of a given network G such that H satisfies some prespecified connectivity requirements. Our focus is on approximation algorithms for designing networks that satisfy vertex connectivity requirements. Our main tool is a linear programming relaxation of the following setpair formulation due to Frank and Jordan: a setpair consists of two subsets of vertices (of the given network G); each setpair has an integer requirement, and the goal is to find a minimum-cost subset of the edges of G sucht hat each setpair is covered by at least as many edges as its requirement. We introduce the notion of skew bisupermodular functions and use it to prove that the basic solutions of the linear program are characterized by “non-crossing families” of setpairs. This allows us to apply Jain’s iterative rounding method to find approximately optimal integer solutions. We give two applications. (1) In the k-vertex connectivity problem we are given a (directed or undirected) graph G=(V,E) with non-negative edge costs, and the task is to find a minimum-cost spanning subgraph H such that H is k-vertex connected. Let n=|V|, and let e<1 be a positive number such that k≤(1−e)n. We give an @math -approximation algorithm for both problems (directed or undirected), improving on the previous best approximation guarantees for k in the range @math . (2)We give a 2-approximation algorithm for the element connectivity problem, matching the previous best approximation guarantee due to Fleischer, Jain and Williamson. We present the first deterministic 1+e approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1 e)5) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(npolylog n) bits, where @math . The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1 e)1 e ) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω(1 e) Ω(1 e) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=⌈2 e⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+e approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the ith matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand. We consider computing a maximum non-crossing matching in convex bipartite graphs. For a convex bipartite graph of n vertices and m edges, we present an O ( n log n ) time algorithm for finding a maximum non-crossing matching in the graph. The previous best algorithm takes O ( m + n log n ) time (Malucelli et?al., 1993). Since m = ? ( n 2 ) in the worst case, our result improves the previous work. In the survivable network design problem (SNDP), given an undirected graph and values r sub ij for each pair of vertices i and j, we attempt to find a minimum-cost subgraph such that there are r sub ij disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. K. (1999) propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELC-SNDP, or ELC). These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 (K. Jain, 2001), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O (log k) approximation algorithm, where k=max sub i,j r sub ij . VC-SNDP is not known to have a non-trivial approximation algorithm; however, recently L. Fleischer (2001) has shown how to extend the technique of K. Jain ( 2001) to give a 2-approximation algorithm in the case that r sub ij spl isin 0, 1, 2 . She also shows that the same techniques will not work for VC-SNDP for more general values of r sub ij . The authors show that these techniques can be extended to a 2-approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures. The complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph. For instance, the larger the maximum independent set of the pattern graph is the more efficient algorithms are known. The situation seems to be substantially different in the case of induced subgraph isomorphism for pattern graphs of fixed size. We present two results which provide evidence that no topology of an induced subgraph of fixed size can be easier to detect or count than an independent set of related size. We show that: Any fixed pattern graph that has a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on k vertices is not easier to detect than an independent set on ⌈k 2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k 2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs. For an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively. Finally, we show that the so called diamond, paw and C 4 are not easier to detect as induced subgraphs than an independent set on three vertices. Let G be a planar graph with n vertices and with a partition of the vertex set into subsets V"0,...,V"k"-"1 for some positive integer [email protected][email protected]?n. Let S be a set of n distinct points in the plane with a partition into subsets S"0,...,S"k"-"1 with |V"i|=|S"i| ([email protected][email protected]?k-1). This paper studies the problem of computing a planar polyline drawing of G, such that each vertex of V"i is mapped to a distinct point of S"i. Lower and upper bounds on the number of bends per edge are proved for any [email protected][email protected]?n. In the special case k=n, we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717-728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest. A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. In this paper, we study the problem of subgraph matching that extracts all subgraph isomorphic embeddings of a query graph q in a large data graph G. The existing algorithms for subgraph matching follow Ullmann's backtracking approach; that is, iteratively map query vertices to data vertices by following a matching order of query vertices. It has been shown that the matching order of query vertices is a very important aspect to the efficiency of a subgraph matching algorithm. Recently, many advanced techniques, such as enforcing connectivity and merging similar vertices in query or data graphs, have been proposed to provide an effective matching order with the aim to reduce unpromising intermediate results especially the ones caused by redundant Cartesian products. In this paper, for the first time we address the issue of unpromising results by Cartesian products from "dissimilar" vertices. We propose a new framework by postponing the Cartesian products based on the structure of a query to minimize the redundant Cartesian products. Our second contribution is proposing a new path-based auxiliary data structure, with the size O(|E(G)| x |V(q)|), to generate a matching order and conduct subgraph matching, which significantly reduces the exponential size O(|V(G)||V(q)|-1) of the existing path-based auxiliary data structure, where V (G) and E (G) are the vertex and edge sets of a data graph G, respectively, and V (q) is the vertex set of a query @math . Extensive empirical studies on real and synthetic graphs demonstrate that our techniques outperform the state-of-the-art algorithms by up to @math orders of magnitude. Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more).