Non-decreasing Submodular Function Research Articles

An approximation algorithm for the -prize-collecting multicut problem in trees with submodular penalties

AbstractLet $T=(V,E)$ be a tree in which each edge is assigned a cost; let $\mathcal{P}$ be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut $M\subseteq E$ such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is $(\frac{8}{3}+\frac{4}{3}\kappa+\varepsilon)$ , where $\kappa$ is the total curvature of the submodular function and $\varepsilon$ is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.

Fair caching networks

We consider caching networks in which the routing cost for serving a content request can be reduced by caching the requested content item in cache nodes closer to the users. We refer to the cost reduction enabled by caching as the caching gain, and the product of the caching gain of a content request and its request rate as caching gain rate. We aim to study fair content allocation strategies through a utility-driven framework, where each request achieves a utility of its caching gain rate, and consider a family of α-fair utility functions to capture different degrees of fairness. The resulting problem is an NP-hard problem with a non-decreasing submodular objective function. Submodularity allows us to devise a deterministic allocation strategy with an optimality guarantee factor arbitrarily close to 1−1∕e. When 0<α≤1, we further propose a randomized strategy that attains an improved optimality guarantee, (1−1∕e)1−α, in expectation. Through extensive simulations over synthetic and real-world network topologies, we evaluate the performance of our proposed strategies and discuss the effect of fairness.

An exact method for constrained maximization of the conditional value-at-risk of a class of stochastic submodular functions

We consider a class of risk-averse submodular maximization problems (RASM) where the objective is the conditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We propose valid inequalities and an exact general method for solving RASM under the assumption that we have an efficient oracle that computes the CVaR of the random function. We demonstrate the proposed method on a stochastic set covering problem that admits an efficient CVaR oracle for the random coverage function.

FPT approximation schemes for maximizing submodular functions

We investigate the existence of approximation algorithms for maximization of submodular functions, that run in a fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function v:2X→R the goal is to select a subset S of K elements from X such that v(S) is maximized. We identify three properties of set functions, referred to as p-separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, p-separable functions, with low values of the parameter p. We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as p and K.

The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems

We study the matroid secretary problems with submodular valuation functions. In these problems, the elements arrive in random order. When one element arrives, we have to make an immediate and irrevocable decision on whether to accept it or not. The set of accepted elements must form an independent set in a predefined matroid. Our objective is to maximize the value of the accepted elements. In this paper, we focus on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function. We introduce a general algorithm for such submodular matroid secretary problems. In particular, we obtain constant competitive algorithms for the cases of laminar matroids and transversal matroids. Our algorithms can be further applied to any independent set system defined by the intersection of a constant number of laminar matroids, while still achieving constant competitive ratios. Notice that laminar matroids generalize uniform matroids and partition matroids. On the other hand, when the underlying valuation function is linear, our algorithm achieves a competitive ratio of 9.6 for laminar matroids, which significantly improves the previous result.

Frequency Domain Packet Scheduling with MIMO for 3GPP LTE Downlink

In this paper, we formalize a general Frequency Domain Packet Scheduling (FDPS) problem for 3GPP LTE Downlink (DL). The DL FDPS problem incorporates the Single-User Multiple Input Multiple Output (SU-MIMO) technique, and can express various scheduling policies, including the Proportional-Fair metric, the MaxWeight scheduling, etc. For LTE DL SU-MIMO, the constraint of selecting only one MIMO mode (transmit diversity or spatial multiplexing) per user in each transmission time interval (TTI) increases the hardness of the FDPS problem. We prove the problem is MAX SNP-hard, which implies approximation algorithms with constant approximation ratios are the best we can expect. Subsequently, we propose an approximation algorithm of polynomial runtime. The solution is based on a greedy method for maximizing a non-decreasing submodular function over a matroid. The algorithm can solve the general DL FDPS problem with an approximation ratio of 4. We implement the proposed algorithm and compare its performance with other well-known schedulers.

Polymatroid Optimization, Submodularity, and Joint Replenishment Games

In this paper we consider the problem of maximizing a separable concave function over a polymatroid. More specifically, we study the submodularity of its optimal objective value in the parameters of the objective function. This question is interesting in its own right and is encountered in many applications. But our research has been motivated mainly by a cooperative game associated with the well-known joint replenishment model. By applying our general results on polymatroid optimization, we prove that this cooperative game is submodular (i.e., its characteristic cost function is submodular) if the joint setup cost is a normalized and nondecreasing submodular function. Furthermore, the same result holds true for a more general one-warehouse multiple retailer game, which affirmatively answers an open question posed by Anily and Haviv [Anily, S., M. Haviv. 2007. The cost allocation problem for the first order interaction joint replenishment model. Oper. Res. 55(2) 292–302].

ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

Submodular Returns and Greedy Heuristics for Queueing Scheduling Problems

We consider a range of controlled stochastic systems satisfying conservation laws together with a reducibility property that says that appropriate laws continue to hold when access to the system is restricted to a subset of all possible demand (job, customer) types. We show that for linear objectives, the optimum system-wide performance is a nondecreasing submodular (or supermodular) function of the subset chosen and that these properties are inherited from the geometry of the performance space concerned. These results are of considerable interest in their own right, but they also motivate the use of greedy heuristics for the solution of a range of job selection and scheduling problems which have hitherto seemed intractable. Computational experience suggests that such heuristics perform very well.

Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

For the problem maxlcub;Z(S): S is an independent set in the matroid Xrcub;, it is well-known that the greedy algorithm finds an optimal solution when Z is an additive set function (Rado-Edmonds theorem). Fisher, Nemhauser and Wolsey have shown that, when Z is a nondecreasing submodular set function satisfying Z(∅)=0, the greedy algorithm finds a solution with value at least half the optimum value. In this paper we show that it finds a solution with value at least 1/(1 + α) times the optimum value, where α is a parameter which represents the ‘total curvature’ of Z. This parameter satisfies 0≤α≤1 and α=0 if and only if the set function Z is additive. Thus the theorems of Rado-Edmonds and Fisher-Nemhauser-Wolsey are both contained in the bound 1/(1 + α). We show that this bound is best possible in terms of α. Another bound which generalizes the Rado-Edmonds theorem is given in terms of a ‘greedy curvature’ of the set function. Unlike the first bound, this bound can prove the optimality of the greedy algorithm even in instances where Z is not additive. A third bound, in terms of the rank and the girth of X, unifies and generalizes the bounds (e−1)/e known for uniform matroids and 12 for general matroids. We also analyze the performance of the greedy algorithm when X is an independence system instead of a matroid. Then we derive two bounds, both tight: The first one is [1−(1−α/K)k]/α where K and k are the sizes of the largest and smallest maximal independent sets in X respectively; the second one is 1/(p+α) where p is the minimum number of matroids that must be intersected to obtain X.

An analysis of the greedy algorithm for the submodular set covering problem

We consider the problem: min $$\{ \mathop \Sigma \limits_{j \in s} f_j :z(S) = z(N),S \subseteqq N\} $$ wherez is a nondecreasing submodular set function on a finite setN. Whenz is integer-valued andz(O)=0, it is shown that the value of a greedy heuristic solution never exceeds the optimal value by more than a factor $$H(\mathop {\max }\limits_j z(\{ j\} ))$$ where $$H(d) = \sum\limits_{i = 1}^d {\frac{1}{i}} $$ . This generalises earlier results of Dobson and others on the applications of the greedy algorithm to the integer covering problem: min {fy: Ay ≧b, y e {0, 1}} wherea ij ,b i } ≧ 0 are integer, and also includes the problem of finding a minimum weight basis in a matroid.