Introduction. 1. Mathematical Preliminaries. Submodular Systems and Base Polyhedra. 2. From Matroids to Submodular Systems. Matroids. Polymatroids. Submodular Systems. 3. Submodular Systems and Base Polyhedra. Fundamental Operations on Submodular Systems. Greedy Algorithm. Structures of Base Polyhedra. Intersecting- and Crossing-Submodular Functions. Related Polyhedra. Submodular Systems of Network Type. Neoflows. 4. The Intersection Problem. The Intersection Theorem. The Discrete Separation Theorem. The Common Base Problem. 5. Neoflows. The Equivalence of the Neoflow Problems. Feasibility for Submodular Flows. Optimality for Submodular Flows. Algorithms for Neoflows. Matroid Optimization. Submodular Analysis. 6. Submodular Functions and Convexity. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions. Subgradients of Submodular Functions. The Lovasz Extensions of Submodular Functions. 7. Submodular Programs. Submodular Programs - Unconstrained Optimization. Submodular Programs - Constrained Optimization. Nonlinear Optimization with Submodular Constraints. 8. Separable Convex Optimization. Optimality Conditions. A Decomposition Algorithm. Discrete Optimization. 9. The Lexicographically Optimal Base Problem. Nonlinear Weight Functions. Linear Weight Functions. 10. The Weighted Max-Min and Min-Max Problems. Continuous Variables. Discrete Variables. 11. The Fair Resource Allocation Problem. Continuous Variables. Discrete Variables. 12. The Neoflow Problem with a Separable Convex Cost Function. References. Index.

Submodular functions and optimization

The significance of systems configurations has been increased in many applied domains (e.g., software, hardware, manufacturing systems, communication systems, supply chain systems, solving strategies, modular planning, material engineering and combinatorial chemistry). Generally, the following kinds of systems under configuration (reconfiguration) can be examined: (a) initial data/infomation (or row materials); (b) processing systems (machines, computers, algorithms); (c) resultant decisions (e.g., products, plans); (d) applied support systems (e.g., network systems, services); (e) requirements; and (f) standards. In the paper several system configuration problems are investigated: (i) searching for (selection of) a set (structure) of system components, (ii) searching for a set of compatible system components, (iii) allocation of system components, (iv) reconfiguration of a system as redesign of the system structure, (v) multi-stage design and redesign of system configuration, (vi) design or redesign of the system configuration for multi-product systems, and (vii) design of system hierarchy. Combinatorial optimization models (including multicriteria statements) are under examination: multiple choice problem, allocation problem, graph coloring problems, morphological clique problem (with compatibility of system components), multipartite clique and their modifications, spanning trees problems.

Combinatorial optimization in system configuration design

We consider the problem of placing the minimum number of monitors in a dynamic network to identify additive link metrics from path metrics measured along cycle-free paths between monitors. Our goal is robust monitor placement, i.e., the same set of monitors can maintain network identifiability under topology changes. Our main contribution is a set of monitor placement algorithms with different performance-complexity tradeoffs that can simultaneously identify multiple topologies occurring during the network lifetime. In particular, we show that the optimal monitor placement is the solution to a generalized hitting set problem, for which we provide a polynomial-time algorithm to construct the input and a greedy algorithm to select the monitors with logarithmic approximation. Although the optimal placement is NP-hard in general, we identify non-trivial special cases that can be solved efficiently. Our secondary contribution is a dynamic triconnected decomposition algorithm to compute the input needed by the monitor placement algorithms, which is the first such algorithm that can handle edge deletions. Our evaluations on mobility-induced dynamic topologies verify the efficiency and the robustness of the proposed algorithms.

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Robust and Efficient Monitor Placement for Network Tomography in Dynamic Networks

We consider the problem of simultaneous source location: selecting locations for sources in a capacitated graph such that a given set of demands can be satisfied simultaneously, with the goal of minimizing the number of locations chosen. For general directed and undirected graphs we give an O(log D)-approximation algorithm, where D is the sum of demands, and prove matching Ω(log D) hardness results assuming P ≠ NP. For undirected trees, we give an exact algorithm and show how this can be combined with a result of Racke to give a solution that exceeds edge capacities by at most O(log2n log log n), where n is the number of nodes. For undirected graphs of bounded treewidth we show that the problem is still NP-hard, but we are able to give a PTAS with at most (1 + ϵ) violation of the capacities for arbitrarily small ϵ, or a (k+1) approximation with exact capacities, where k is the treewidth.

/pdf/simultaneous-source-location-3ob34c60o8.pdf

Simultaneous source location

We consider the problem of placing the minimum number of monitors in a communication network with possible topology changes to identify additive link metrics from path metrics. The core of our solution is a suite of robust monitor placement algorithms with different performance-complexity tradeoffs that guarantee network identifiability for the multiple possible topologies. In particular, we show that the optimal (i.e., minimum) monitor placement is the solution to a generalized hitting set problem, where we provide a polynomial-time algorithm to construct the input. Although the optimal placement is NP-hard in general, we identify non-trivial special cases that can be solved efficiently. We further demonstrate how the proposed algorithms can be augmented to handle unpredictable topology changes and tradeoffs between monitor cost and adaptation cost. Our evaluations on mobility-induced dynamic topologies verify the effectiveness and robustness of the proposed algorithms.

/pdf/robust-monitor-placement-for-network-tomography-in-dynamic-242qdzmgj6.pdf

Robust monitor placement for network tomography in dynamic networks

In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ and $\hat{\kappa}$), where the source location problems are to find a minimum-cost set S⊆V in a given graph G=(V,A) with a capacity function u:A→ℝ+ such that for each vertex v∈V, the connectivity from S to v (resp., from v to S) is at least a given demand d−(v) (resp., d+(v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54–68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of cln D for some constant c, unless every problem in NP has an O(Nlog log N)-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1+ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are Θ(ln ∑v∈V(d+(v)+d−(v)))-approximable.

Minimum Cost Source Location Problems with Flow Requirements

Minimizing a monotone concave function with laminar covering constraints

Abstract
In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ and 
$\hat{\kappa}$
), where the source location problems are to find a minimum-cost set S⊆V in a given graph G=(V,A) with a capacity function u:A→ℝ+ such that for each vertex v∈V, the connectivity from S to v (resp., from v to S) is at least a given demand d−(v) (resp., d+(v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54–68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of cln D for some constant c, unless every problem in NP has an O(Nlog log N)-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1+ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are Θ(ln ∑v∈V(d+(v)+d−(v)))-approximable.


MATHEMATICAL ENGINEERING TECHNICAL REPORTS Minimum Cost Source Location Problems with Flow Requirements

Given a system $(V,f,d)$ on a finite set $V$ consisting of two set functions $f:2^V\to\mathbb{R}$ and $d:2^V\to\mathbb{R}$, we consider the problem of finding a set $R\subseteq V$ of minimum cardinality such that $f(X)\ge d(X)$ for all $X\subseteq V-R$, where the problem can be regarded as a natural generalization of the source location problems and the external network problems in (undirected) graphs and hypergraphs. We give a structural characterization of minimal deficient sets of $(V,f,d)$ under certain conditions. We show that all such sets form a tree hypergraph if $f$ is posimodular and $d$ is modulotone (i.e., each nonempty subset $X$ of $V$ has an element $v\in X$ such that $d(Y)\ge d(X)$ for all subsets $Y$ of $X$ that contain $v$) and that, conversely, any tree hypergraph can be represented by minimal deficient sets of $(V,f,d)$ for a posimodular function $f$ and a modulotone function $d$. By using this characterization, we present a polynomial-time algorithm if, in addition, $f$ is submodular and $d$ is given by either $d(X)=\max\{p(v)\mid v\in X\}$ for a function $p:V\to\RR_+$ or $d(X)=\max\{r(v,w)\mid v\in X,w\in V-X\}$ for a function $r:V^2\to\mathbb{R}_+$. Our result provides first polynomial-time algorithms for the source location problem in hypergraphs and the external network problems in graphs and hypergraphs. We also show that the problem is intractable, even if $f$ is submodular and $d\equiv\mathbf{0}$.

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Mariko Sakashita

Papers

Minimum Cost Source Location Problems with Flow Requirements

Minimizing a monotone concave function with laminar covering constraints

MATHEMATICAL ENGINEERING TECHNICAL REPORTS Minimum Cost Source Location Problems with Flow Requirements

Minimum Transversals in Posimodular Systems

MATHEMATICAL ENGINEERING TECHNICAL REPORTS Minimizing a Monotone Concave Function with Laminar Covering Constraints