Undirected Multigraph Research Articles

Gauges, loops, and polynomials for partition functions of graphical models

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and optimization applications. The problem is #P-hard that is of an exponential complexity with respect to the number of variables. In this manuscript, aimed at approximating the partition function, we consider multi-graph models where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge function technique from Chertkov and Chernyak (2006 Phys. Rev. E 73 065102; 2006 J. Stat. Mech. 2006 P06009) with the technique developed in Anari and Oveis Gharan 2017 arXiv:1702.02937; Gurvits 2011 arXiv:1106.2844; Straszak and Vishnoi 2017 55th Annual Allerton Conf. on Communication, Control, and Computing, based on the recent progress in the field of real stable polynomials. We show that the Gauge function, representing a single-out term in a finite sum expression for the partition function which achieves extremum at the so-called belief-propagation gauge, has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, Gauge function can be used to recover the partition function through a sequence of transformations allowing appealing algebraic and graphical interpretations. Algebraically, one step in the sequence consists of the application of a differential operator over gauges associated with an edge. Graphically, the sequence is interpreted as a repetitive elimination/contraction of edges resulting in multi-graph models on decreasing in size (number of edges) graphs with the same partition function as in the original multi-graph model. Even though the complexity of computing factors in the sequence of the derived multi-graph models and respective Gauge functions grow exponentially with the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property. Moreover, we show that BP estimations in the sequence do not decrease, each low-bounding the partition function.

On Track Procession Assignment Problem at the Railway Network Sections

We consider the problem of assigning a “track procession,” i.e., the time for railway network maintenance, when train movements are forbidden. The railway network is presented in the form of an undirected multigraph. Movements along this multigraph are carried out with a set of conflict-free subslots, where each “subslot” is a five-tuple of the arrival and departure vertices, number of the edge used for movement, and arrival and departure times. In this work, we construct a mathematical model for carrying out transportations plan with accounting for constraints on time when trains are for departure as well as constraints on arrival time. An optimization problem on time selection for track processions assignment and trains schedule corresponds to a mixed integer linear programming problem with a criteria of the minimization of total time spend by trains in run. An applied implementation of the proposed approach was performed using ILOG CPLEX software. We present the results of a numerical experiment.

Min-max controllable risk problems

A min-max controllable risk problem, defined on combinatorial structures which are either simple paths of a directed multigraph or spanning trees of an undirected multigraph, with resource dependent risk functions of the arcs or the edges, is studied. The resource amount is limited, and the objective is to distribute it between the arcs or the edges so that the maximum risk over the arcs of a simple path or the edges of a spanning tree is minimized. Two new solution approaches are presented, which are asymptotically faster than the solution approaches suggested in the literature.

A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Subgraph Problem

We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.

Maximum cycle packing using SPR-trees

Let G = ( V , E ) be an undirected multigraph without loops. The maximum cycle packing problem is to find a collection Z * = { C 1 , ..., C s } of edge-disjoint cycles C i subset G of maximum cardinality v ( G ) . In general, this problem is NP-hard. An approximation algorithm for computing v ( G ) for 2-connected graphs is presented, which is based on splits of G . It essentially uses the representation of the 3-connected components of G by its SPR-tree. It is proved that for generalized series-parallel multigraphs the algorithm is optimal, i.e. it determines a maximum cycle packing Z * in linear time.

Multicoloring the incidentors of a weighted undirected multigraph

Multicoloring the incidentors of a weighted undirected multigraph

On [formula omitted] and [formula omitted] performance regions of multi-agent systems

This paper addresses the distributed H 2 and H ∞ control problems for multi-agent systems with linear or linearized dynamics. An undirected multigraph with loops is used to represent the communication topology of a multi-agent network. A distributed controller is designed, based on the relative states of neighboring agents and a subset of absolute states of the networked agents. The notions of H ∞ and H 2 performance regions are introduced and analyzed, respectively. A necessary and sufficient condition for the existence of a controller yielding an unbounded H ∞ performance region is derived. A multi-step procedure for suboptimal H ∞ controller synthesis is presented. It is also shown that the H ∞ performance limit of the network under the distributed controller is equal to the minimal H ∞ norm of a single agent achieved by using the state feedback controller. It is finally shown that, contrarily to the H ∞ case, the H 2 performance limit scales with the number of agents in the network.

Minimum augmentation of edge-connectivity with monotone requirements in undirected graphs

For a finite ground set V , we call a set-function r : 2 V → Z + monotone, if r ( X ′ ) ≥ r ( X ) holds for each X ′ ⊆ X ⊆ V , where Z + is the set of nonnegative integers. Given an undirected multigraph G = ( V , E ) and a monotone requirement function r : 2 V → Z + , we consider the problem of augmenting G by a smallest number of new edges, so that the resulting graph G ′ satisfies d G ′ ( X ) ≥ r ( X ) for each 0̸ ≠ X ⊂ V , where d G ( X ) denotes the degree of a vertex set X in G . This problem includes the edge-connectivity augmentation problem, and in general, it is NP-hard, even if a polynomial time oracle for r is available. In this paper, we show that the problem can be solved in O ( n 4 ( m + n log n + q ) ) time, under the assumption that each 0̸ ≠ X ⊂ V satisfies r ( X ) ≥ 2 whenever r ( X ) > 0 , where n = | V | , m = | { { u , v } ∣ ( u , v ) ∈ E } | , and q is the time required to compute r ( X ) for each X ⊆ V .

Bounds for the incidentor chromatic number of a weighted undirected multigraph

A proper incidentor coloring of an undirected weighted multigraph is called admissible if the absolute value of the difference between the colors of the incidentors of each edge is at least the weight of this edge. The minimum number of colors necessary for an admissible incidentor coloring is called the incidentor chromatic number of the multigraph. The problem of finding this number is studied in the paper. The NP-hardness of this problem is proved for Δ colors. Some upper and lower bounds are found for the incidentor chromatic number.

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

Given an undirected multigraph G=(V,E), a family \(\mathcal{W}\) of areas W⊆V, and a target connectivity k≥1, we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least k edge-disjoint paths between v and W for every pair of a vertex v∈V and an area \(W\in \mathcal{W}\) . So far this problem was shown to be NP-complete in the case of k=1 and polynomially solvable in the case of k=2. In this paper, we show that the problem for k≥3 can be solved in O(m+n(k 3+n 2)(p+kn+nlog n)log k+pkn 3log (n/k)) time, where n=|V|, m=|{{u,v}|(u,v)∈E}|, and \(p=|\mathcal{W}|\) .

Inequalities on the Number of Connected Spanning Subgraphs in a Multigraph

Consider an undirected multigraph G=(V, E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(m≥i≥n) edges, in G. Recently, we showed in [3] the validity of (m-i+1)Ni-1>(i-n+⌊3+√9+8(i-n)/2⌋)Ni for a simple graph and each i(m≥i≥n). Note that, from this inequality, (m-n)Nn/2Nn+1+Nn/(m-n+1)Nn-1≥2 is easily derived. In this paper, for a multigraph G and all i(m≥i≥n), we prove (m-i+1)Ni-1≥(i-n+2)Ni, and give a necessary and sufficient condition by which (m-i+1)Ni+1=(i-n+2)Ni. In particular, this means that (m-i+1)Ni+1>(i-n+⌊3+√9+8(i-n)/2⌋)Ni is not valid for all multigraphs, in general. Furthermore, we prove (m-n)Nn/2Nn+1+Nn/(m-n+1)Nn-1≥2, which is not straightforwardly derived from (m-i+1)Ni-1≥(i-n+2)Ni, and also introduce a necessary and sufficent condition by which (m-n)Nn/2Nn+1+Nn/(m-n+1)Nn-1=2. Moreover, we show a sufficient condition for a multigraph to have N2n>Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least [2/3(m-n)]+1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then N2n>Nn-1Nn+1.

An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem*

Given an undirected multigraph G and a subset of vertices S ⊆ V (G), the STEINER TREE PACKING problem is to find a largest collection of edge-disjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APX-hard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not even an approximation algorithm with asymptotic ratio o(n) was known despite several attempts. In this work, we present the first polynomial time constant factor approximation algorithm for the STEINER TREE PACKING problem. The main theorem is an approximate min-max relation between the maximum number of edge-disjoint trees that each connects S (S-trees) and the minimum size of an edge-cut that disconnects some pair of vertices in S (S-cut). Specifically, we prove that if every S-cut in G has at least 26k edges, then G has at least k edge-disjoint S-trees; this answers Kriesells conjecture affirmatively up to a constant multiple.

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G = ( V , E ) , a family W of sets W ⊆ V of vertices (areas), and a requirement function r : W → Z + (where Z + is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r ( W ) edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area W ∈ W . So far this problem was shown to be NP-hard in the uniform case of r ( W ) = 1 for each W ∈ W , and polynomially solvable in the uniform case of r ( W ) = r ⩾ 2 for each W ∈ W . In this paper, we show that the problem can be solved in O ( m + pn 4 ( r * + log n ) ) time, even if r ( W ) ⩾ 2 holds for each W ∈ W , where n = | V | , m = | { { u , v } | ( u , v ) ∈ E } | , p = | W | , and r * = max { r ( W ) ∣ W ∈ W } .

Sparse connectivity certificates via MA orderings in graphs

For an undirected multigraph G = ( V , E ) , let α be a positive integer weight function on V. For a positive integer k, G is called ( k , α ) -connected if any two vertices u , v ∈ V remain connected after removal of any pair ( Z , E ′ ) of a vertex subset Z ⊆ V - { u , v } and an edge subset E ′ ⊆ E such that ∑ v ∈ Z α ( v ) + | E ′ | < k . The ( k , α ) -connectivity is an extension of several common generalizations of edge-connectivity and vertex-connectivity. Given a ( k , α ) -connected graph G, we show that a ( k , α ) -connected spanning subgraph of G with O ( k | V | ) edges can be found in linear time by using MA orderings. We also show that properties on removal cycles and preservation of minimum cuts can be extended in the ( k , α ) -connectivity.

On Minimum k-Edge-Connectivity Augmentation for Specified Vertices of a Graph with Upper Bounds on Vertex-Degree

The k-edge-connectivity augmentation problem for a specified set of vertices of a graph with degree constraints, kECA-SV-DC, is defined as follows: Given an undirected multigraph G = (V, E), a specified set of vertices S ⊆ V and a function g: V→Z+∪{∞}, find a smallest set E' of edges such that (V, E ∪ E') has at least k edge-disjoint paths between any pair of vertices in S and such that, for any νe V, E' includes at most g(v) edges incident to v, where Z+ is the set of nonnegative integers. This paper first shows polynomial time solvability of kECA-SV-DC and then gives a linear time algorithm for 2ECA-SV-DC.

An Improved Analysis for Approximating the Smallest k-Edge Connected Spanning Subgraph of a Multigraph

Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] present an approximation algorithm (the KR algorithm) for finding the smallest k-edge connected spanning subgraph (k-ECSS) of an undirected multigraph. They prove the KR algorithm has an approximation ratio < 1.85. We improve this bound to $\le 1+\sqrt{1/e}<1.61$ (for odd k we modify the base case of the KR algorithm). This is the best-known performance bound for a combinatorial approximation algorithm for the smallest k-ECSS problem for arbitrary k. Our analysis also gives the best-known combinatorial performance bound for any fixed value of $k\ge 3$, e.g., for even k the approximation ratio is $\le 1+(1-{1\over k})^{k/2}$. Our analysis is based on a laminar family of sets (similar to families used in related contexts) which gives a better accounting of edges added in previous iterations of the algorithm. We also present a polynomial time implementation of the KR algorithm on multigraphs, running in the time for O(nm) maximum flow computations, where n (m) is the number of vertices (edges, not counting parallel copies), respectively. This complements the implementation of Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] which uses time O((kn)2) and is efficient for small k.

An Ear Decomposition Approach to Approximating the Smallest 3-Edge Connected Spanning Subgraph of a Multigraph

This paper gives a 3/2 approximation algorithm for the smallest 3-edge connected spanning subgraph of an undirected multigraph. The previous best algorithm of Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] has approximation ratio $5/3$. The algorithm of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528--560] achieves ratio 3/2 for simple graphs. Our approach, based on the close relationship between an ear decomposition of a 2-edge connected graph and 3-edge connected components, enables us to achieve running time $O( m \alpha(m,n) )$.

Minimal length test vectors for multiple-fault detection

A methodology for circuit testing is proposed for detecting multiple circuit faults in the course of a minimal length “guided tour” of the circuit transition structure. Deriving a test vector to guide this tour through an n state subsystem with at most I inputs possible in situ at each state, corresponds to solving an open tour multigraph version of the “Chinese Postman” problem, in which out-degrees are bounded by I. In this case, the length L of a minimal length open tour is shown to satisfy L⩽ In 2; a minimal length open tour is computable in O( n 3+ nI) steps for undirected multigraphs and O( n 3+( nI) 2log n/log I) steps for directed multigraphs, both one-time costs, using weighted matching and bipartite weighted matching, respectively. An open tour can result in a test vector as much as 1 2 shorter than the test vector associated with a closed tour, without any loss in error detection. Examples show that for a directed graph, the length of a minimal length open tour may be as great as n 3/6 for I= n, or Ω(n 2) when I is bounded, while in an undirected multigraph, a minimal length tour requires no more than n−3 repeated state transitions. This mitigates in favor of “mixed” circuits in which certain transitions are reversible and need be tested in only one direction. The practicality of this approach rests with the ability to apply it separately to small subsystems, in conjunction with symbolic testing of inter-subsystem coordination. The former is feasible with existing commercial technologies, such as electron beam scanning, while the latter is feasible with a finite-state model-checker. In summary, the proposed methodology comprises three steps: 1. decompose a circuit into subsystems sufficiently small to be model-checked exhaustively; 2. perform symbolic tests of inter-subsystem coordination and conclude that if each subsystem is correctly implemented, then the entire circuit will behave as required; 3. for each circuit subsystem, exercise every realizable transition through a minimal length (open) tour, comparing the actual transitions with those of the specification.

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

Given an undirected multigraph G = (V, E), a family W of areas W ⊆ V, and a target connectivity k ≥ 1, we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least k edge-disjoint paths between μ and W for every pair of a vertex μ ∈ V and an area W ∈ W. So far this problem was shown to be NP-complete in the case of k = 1. and polynomially solvable in the case of k = 2. In this paper, we show that the problem for k ≥ 3 can be solved in O(m + n (k3 + n2)(p + kn + n logn) logk + pkn3 log(n/k)) time, where n = ∣V∣, m = ∣{{u,v}∣(u, v) ∈ E}∣, and p = ∣ W∣ .

Source location problems considering vertex‐connectivity and edge‐connectivity simultaneously

AbstractLet G = (V, E) be an undirected multigraph, where V and E are a set of vertices and a set of edges, respectively. Let k and l be fixed nonnegative integers. This paper considers location problems of finding a minimum‐size vertex‐subset S ⊆ V such that for each vertex x ∈ V the vertex‐connectivity between S and x is greater than or equal to k and the edge‐connectivity between S and x is greater than or equal to l. For the problem with edge‐connectivity requirements, that is, k = 0, an O(L(|V|, |E|, l)) time algorithm is already known, where L(|V|, |E|, l) is the time to find all h‐edge‐connected components for h = 1, 2, … , l and O(L(|V|, |E|, l)) = O(|E| + |V|2 + |V|min{|E|, l|V|}min{l, |V|}) is known. In this paper, we show that the problem with k ≥ 3 is NP‐hard even for l = 0. We then present an O(L(|V|, |E|, l)) time algorithm for 0 ≤ k ≤ 2 and l ≥ 0. Moreover, we prove that the problem parameterized by the size of S is fixed‐parameter tractable (FPT) for k = 3 and l ≥ 0. © 2002 Wiley Periodicals, Inc.