Multicast Network Design Game on a Ring

In this paper we study quality measures of different solution concepts for the multicast network design game on a ring topology. We recall from the literature a lower bound of 4/3 and prove a matching upper bound for the price of stability, which is the ratio of the social costs of a best Nash equilibrium and of a general optimum. Therefore, we answer an open question posed by Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs of a potential optimizer and of an optimum, provide a construction of a lower bound, and give a computer-assisted argument that it reaches $2$ for any precision. We then turn our attention to players arriving one by one and playing myopically their best response. We provide matching lower and upper bounds of 2 for the myopic sequential price of anarchy (achieved for a worst-case order of the arrival of the players). We then initiate the study of myopic sequential price of stability and for the multicast game on the ring we construct a lower bound of 4/3, and provide an upper bound of 26/19. To the end, we conjecture and argue that the right answer is 4/3.Comment: 12 pages, 4 figure

Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H -Subgraph Hitting , where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V(G) that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques (e.g., Courcelle’s theorem) show that, for every fixed H and the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time 2O(tμ(H))⋅|V(G)|, but cannot be solved in time 2o(tμ(H))⋅|V(G)|O(1), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H -Subgraph Hitting is much richer

Most General Winning Secure Equilibria Synthesis in Graph Games

This paper considers the problem of co-synthesis in $k$-player games over a finite graph where each player has an individual $\omega$-regular specification $\phi_i$. In this context, a secure equilibrium (SE) is a Nash equilibrium w.r.t. the lexicographically ordered objectives of each player to first satisfy their own specification, and second, to falsify other players' specifications. A winning secure equilibrium (WSE) is an SE strategy profile $(\pi_i)_{i\in[1;k]}$ that ensures the specification $\phi:=\bigwedge_{i\in[1;k]}\phi_i$ if no player deviates from their strategy $\pi_i$. Distributed implementations generated from a WSE make components act rationally by ensuring that a deviation from the WSE strategy profile is immediately punished by a retaliating strategy that makes the involved players lose. In this paper, we move from deviation punishment in WSE-based implementations to a distributed, assume-guarantee based realization of WSE. This shift is obtained by generalizing WSE from strategy profiles to specification profiles $(\varphi_i)_{i\in[1;k]}$ with $\bigwedge_{i\in[1;k]}\varphi_i = \phi$, which we call most general winning secure equilibria (GWSE). Such GWSE have the property that each player can individually pick a strategy $\pi_i$ winning for $\varphi_i$ (against all other players) and all resulting strategy profiles $(\pi_i)_{i\in[1;k]}$ are guaranteed to be a WSE. The obtained flexibility in players' strategy choices can be utilized for robustness and adaptability of local implementations. Concretely, our contribution is three-fold: (1) we formalize GWSE for $k$-player games over finite graphs, where each player has an $\omega$-regular specification $\phi_i$; (2) we devise an iterative semi-algorithm for GWSE synthesis in such games, and (3) obtain an exponential-time algorithm for GWSE synthesis with parity specifications $\phi_i$.Comment: TACAS 202

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

In the quest for a logic capturing Ptime the next natural classes of structures to consider are those with bounded color class size. We present a canonization procedure for graphs with dihedral color classes of bounded size in the logic of Choiceless Polynomial Time (CPT), which then captures Ptime on this class of structures. This is the first result of this form for non-abelian color classes. The first step proposes a normal form which comprises a "rigid assemblage". This roughly means that the local automorphism groups form 2-injective 3-factor subdirect products. Structures with color classes of bounded size can be reduced canonization preservingly to normal form in CPT. In the second step, we show that for graphs in normal form with dihedral color classes of bounded size, the canonization problem can be solved in CPT. We also show the same statement for general ternary structures in normal form if the dihedral groups are defined over odd domains

An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS\u2723). It outputs a perfect matching with k\u27 red edges with the guarantee that 0.5k ? k\u27 ? 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k\u27 red edges such that k/3 ? k\u27 ? k