Tree Nash Equilibria in the Network Creation Game

In the network creation game with n vertices, every vertex (a player) buys a set of adjacent edges, each at a fixed amount {\alpha} > 0. It has been conjectured that for {\alpha} >= n, every Nash equilibrium is a tree, and has been confirmed for every {\alpha} >= 273n. We improve upon this bound and show that this is true for every {\alpha} >= 65n. To show this, we provide new and improved results on the local structure of Nash equilibria. Technically, we show that if there is a cycle in a Nash equilibrium, then {\alpha} < 65n. Proving this, we only consider relatively simple strategy changes of the players involved in the cycle. We further show that this simple approach cannot be used to show the desired upper bound {\alpha} < n (for which a cycle may exist), but conjecture that a slightly worse bound {\alpha} < 1.3n can be achieved with this approach. Towards this conjecture, we show that if a Nash equilibrium has a cycle of length at most 10, then indeed {\alpha} < 1.3n. We further provide experimental evidence suggesting that when the girth of a Nash equilibrium is increasing, the upper bound on {\alpha} obtained by the simple strategy changes is not increasing. To the end, we investigate the approach for a coalitional variant of Nash equilibrium, where coalitions of two players cannot collectively improve, and show that if {\alpha} >= 41n, then every such Nash equilibrium is a tree

Collective fast delivery by energy-efficient agents

We consider k mobile agents initially located at distinct nodes of an undirected graph (on n nodes, with edge lengths) that have to deliver a single item from a given source node s to a given target node t. The agents can move along the edges of the graph, starting at time 0 with respect to the following: Each agent i has a weight w_i that defines the rate of energy consumption while travelling a distance in the graph, and a velocity v_i with which it can move. We are interested in schedules (operating the k agents) that result in a small delivery time T (time when the package arrives at t), and small total energy consumption E. Concretely, we ask for a schedule that: either (i) Minimizes T, (ii) Minimizes lexicographically (T,E) (prioritizing fast delivery), or (iii) Minimizes epsilon*T + (1-epsilon)*E, for a given epsilon, 0<epsilon<1. We show that (i) is solvable in polynomial time, and show that (ii) is polynomial-time solvable for uniform velocities and solvable in time O(n + k log k) for arbitrary velocities on paths, but in general is NP-hard even on planar graphs. As a corollary of our hardness result, (iii) is NP-hard, too. We show that there is a 3-approximation algorithm for (iii) using a single agent.Comment: In an extended abstract of this paper [MFCS 2018], we erroneously claimed the single agent approach for variant (iii) to have approximation ratio

An Algorithmic View on OVSF Code Assignment

Orthogonal Variable Spreading Factor (OVSF) codes are used in UMTS to share the radio spectrum among several connections of possibly different bandwidth requirements. The combinatorial core of the OVSF code assignment problem is to assign some nodes of a complete binary tree of height h (the code tree) to n simultaneous connections, such that no two assigned nodes (codes) are on the same root-to-leaf path. A connection that uses a 2-d fraction of the total bandwidth requires some code at depth d in the tree, but this code assignment is allowed to change over time. Requests for connections that would exceed the total available bandwidth are rejected. We consider the one-step code assignment problem: Given an assignment, move the minimum number of codes to serve a new request. Minn and Siu propose the so-called DCA algorithm to solve the problem optimally. In contrast, we show that DCA does not always return an optimal solution, and that the problem is NP-hard. We give an exact nO(h)-time algorithm, and a polynomial-time greedy algorithm that achieves approximation ratio Θ(h). A more practically relevant version is the online code assignment problem, where future requests are not known in advance. Our objective is to minimize the overall number of code reassignments. We present a Θ(h)-competitive online algorithm, and show that no deterministic online algorithm can achieve a competitive ratio better than 1.5. We show that the greedy strategy (minimizing the number of reassignments in every step) is not better than Ω(h) competitive. We give a 2-resource augmented online algorithm that achieves an amortized constant number of (re-)assignments. Finally, we show that the problem is fixed-parameter tractabl

Relaxed Agreement Forests

There are multiple factors which can cause the phylogenetic inference process to produce two or more conflicting hypotheses of the evolutionary history of a set X of biological entities. That is: phylogenetic trees with the same set of leaf labels X but with distinct topologies. This leads naturally to the goal of quantifying the difference between two such trees T_1 and T_2. Here we introduce the problem of computing a 'maximum relaxed agreement forest' (MRAF) and use this as a proxy for the dissimilarity of T_1 and T_2, which in this article we assume to be unrooted binary phylogenetic trees. MRAF asks for a partition of the leaf labels X into a minimum number of blocks S_1, S_2, ... S_k such that for each i, the subtrees induced in T_1 and T_2 by S_i are isomorphic up to suppression of degree-2 nodes and taking the labels X into account. Unlike the earlier introduced maximum agreement forest (MAF) model, the subtrees induced by the S_i are allowed to overlap. We prove that it is NP-hard to compute MRAF, by reducing from the problem of partitioning a permutation into a minimum number of monotonic subsequences (PIMS). Furthermore, we show that MRAF has a polynomial time O(log n)-approximation algorithm where n=|X| and permits exact algorithms with single-exponential running time. When at least one of the two input trees has a caterpillar topology, we prove that testing whether a MRAF has size at most k can be answered in polynomial time when k is fixed. We also note that on two caterpillars the approximability of MRAF is related to that of PIMS. Finally, we establish a number of bounds on MRAF, compare its behaviour to MAF both in theory and in an experimental setting and discuss a number of open problems.Comment: 14 pages plus appendi

Improved bounds on equilibria solutions in the network design game

ISSN:0020-7276ISSN:1432-127

Snakes and Ladders: a Treewidth Story

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the ladders contained in $G$ are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees

Discovery of network properties with all-shortest-paths queries

We consider the problem of discovering properties (such as the diameter) of an unknown network G(V,E) with a minimum number of queries. Initially, only the vertex set V of the network is known. Information about the edges and non-edges of the network can be obtained by querying nodes of the network. A query at a node q∈V returns the union of all shortest paths from q to all other nodes in V. We study the problem as an online problem - an algorithm does not initially know the edge set of the network, and has to decide where to make the next query based on the information that was gathered by previous queries. We study how many queries are needed to discover the diameter, a minimal dominating set, a maximal independent set, the minimum degree, and the maximum degree of the network. We also study the problem of deciding with a minimum number of queries whether the network is 2-edge or 2-vertex connected. We use the usual competitive analysis to evaluate the quality of online algorithms, i.e., we compare online algorithms with optimum offline algorithms. For all properties except maximal independent set and 2-vertex connectivity we present and analyze online algorithms. Furthermore we show, for all the aforementioned properties, that "many" queries are needed in the worst case. As our query model delivers more information about the network than the measurement heuristics that are currently used in practise, these negative results suggest that a similar behavior can be expected in realistic settings, or in more realistic models derived from the all-shortest-paths query model