Result diversification has many important applications in databases, operations research, information retrieval, and finance. In this paper, we study and extend a particular version of result diversification, known as max-sum diversification. More specifically, we consider the setting where we are given a set of elements in a metric space and a set valuation function f defined on every subset. For any given subset S, the overall objective is a linear combination of f (S) and the sum of the distances induced by S. The goal is to find a subset S satisfying some constraints that maximizes the overall objective.This problem is first studied by Gollapudi and Sharma in [17] for modular set functions and for sets satisfying a cardinality constraint (uniform matroids). In their paper, they give a 2-approximation algorithm by reducing to an earlier result in [20]. The first part of this paper considers an extension of the modular case to the monotone submodular case, for which the algorithm in [17] no longer applies. Interestingly, we are able to maintain the same 2-approximation using a natural, but different greedy algorithm. We then further extend the problem by considering any matroid constraint and show that a natural single swap local search algorithm provides a 2-approximation in this more general setting. This extends the Nemhauser, Wolsey and Fisher approximation result [29] for the problem of submodular function maximization subject to a matroid constraint (without the distance function component).The second part of the paper focuses on dynamic updates for the modular case. Suppose we have a good initial approx- * imate solution and then there is a single weight-perturbation either on the valuation of an element or on the distance between two elements. Given that users expect some stability in the results they see, we ask how easy is it to maintain a good approximation without significantly changing the initial set. We measure this by the number of updates, where each update is a swap of a single element in the current solution with a single element outside the current solution. We show that we can maintain an approximation ratio of 3 by just a single update if the perturbation is not too large.

Abstract. The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that n n−1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity n n−1 + n − 1, and two-sided ideals and factor-closed languages of syntactic complexity n n−2 + (n − 2)2 n−2 + 1.

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that n n−2 is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be (n − 1) n−2 + (n − 2), (n − 1) n−3 + (n − 2) n−3 + (n − 3)2 n−3 , and (n − 1) n−3 + (n − 3)2 n−3 + 1, respectively, and exhibit languages with these syntactic complexities.keyword bifix-free, factor-free, finite automaton, monoid, prefix-free, regular language, reversal, semigroup, suffix-free, syntactic complexity ⋆ If Σ is a non-empty finite alphabet, then Σ * is the free monoid generated by Σ, and Σ + is the free semigroup generated by Σ. A word is any element of Σ * , and the empty word is ε. The length of a word w ∈ Σ * is |w|. A language over Σ is any subset of Σ * . If w = uxv for some u, x, v ∈ Σ * , then u is a prefix of w, v is a suffix of w, and x is a factor of w. Both u and v are also factors of w. A proper prefix (suffix, factor) of w is a prefix (suffix, factor) of w other than w.The left quotient, or simply quotient, of a language L by a word w is the language L w = {x ∈ Σ * | wx ∈ L}. For any L ⊆ Σ * , the Nerode right congruence [17] ∼ L of L is defined as follows:x ∼ L y if and only if xv ∈ L ⇔ yv ∈ L, for all v ∈ Σ * .Clearly, L x = L y if and only if x ∼ L y. Thus each equivalence class of this right congruence corresponds to a distinct quotient of L.B sf (n) = {t ∈ T Q | 1 ∈ rng(t), nt = n, and for all j 1, 1t j = n or 1t j = it j ∀i, 1 < i < n}.Proposition 2. If L is a regular language with quotient DFA A n = (Q, Σ, δ, 1, F ) and syntactic semigroup T L , then the following hold:1. If L is bifix-free, then T L is a subset of B bf (n). 2. If ε is the only accepting quotient of L, and T L ⊆ B bf (n), then L is bifix-free.

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In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = ( V, E ) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v 1 , …, v n such that the property P holds on the induced subgraph G [ N ( v i )∩ V i ], where N ( v i ) is the neighborhood of v i and V i = { v i , …, v n }. It turns out that if we take P as a graph with maximum independent set size no greater than k , then this definition gives a natural generalization of both chordal graphs and ( k + 1)-claw-free graphs. We refer to such graphs as inductive k -independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k -independent for small k . In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3 -independent graph; furthermore, any planar graph is an inductive 3 -independent graph. For any fixed constant k , we develop simple, polynomial time approximation algorithms for inductive k -independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.