Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory , and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies—that is, a centrally planned routing that minimizes the average cost of agents—is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

Binary similarity involves determining whether two binary programs exhibit similar functionality with applications in vulnerability detection, malware analysis, and copyright detection. However, variations in compiler settings, target architectures, and deliberate code obfuscations significantly complicate the similarity measurement by effectively altering the syntax, semantics, and structure of the underlying binary. To address these challenges, we propose VexIR2Vec , a robust, architecture-neutral approach based on VEX-IR to solve binary similarity tasks. VexIR2Vec consists of three key components: a peephole extractor, a normalization engine ( VexINE ), and an embedding model ( VexNet ). The process to build program embeddings starts with the extraction of sequences of basic blocks, or peepholes , from control-flow graphs via random walks, capturing structural information. These generated peepholes are then normalized using VexINE , which applies compiler-inspired transformations to reduce architectural and compiler-induced variations. Embeddings of peepholes are generated using representation learning techniques, avoiding Out-of-Vocabulary (OOV) issues. These embeddings are then fine-tuned with VexNet , a feed-forward Siamese network that maps functions into a high-dimensional space for diffing and searching tasks in an application-independent manner. We evaluate VexIR2Vec against five baselines—BinDiff, DeepBinDiff, SAFE, BinFinder, and histograms of opcodes—on a dataset comprising 2.7 M functions and 15.5 K binaries from 7 projects compiled across 12 compilers targeting x86 and ARM architectures. The experiments span four adversarial settings—cross-optimization, cross-compilation, cross-architecture, and obfuscations—that are typically exploited by malware and vulnerabilities. In diffing experiments, VexIR2Vec outperforms the nearest baseline in these four scenarios by \(40\%\) , \(18\%\) , \(21\%\) , and \(60\%\) , respectively. In the searching experiment, VexIR2Vec achieves a mean average precision of 0.76, the nearest baseline, by \(46\%\) . Our framework is highly scalable and is built as a lightweight, multi-threaded, parallel library using only open source tools. VexIR2Vec is \(\approx 3.1\) – \(3.5\times\) faster than the closest baselines and orders-of-magnitude faster than other tools.

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies - that is, a centrally planned routing that minimizes the average cost of agents - is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

We present a hypergraph coloring based approach to pliable index coding (PICOD). We represent the given PICOD problem using a hypergraph consisting ofmmessages as vertices and the request-sets of thenclients as hyperedges. Aconflict-free coloringof a hypergraph is an assignment of colors to its vertices so that each hyperedge contains a uniquely colored vertex. We show that various parameters arising out of conflict-free colorings (and some new variants) of the PICOD hypergraph result in new upper bounds for the optimal PICOD length. Using these new upper bounds, we show the existence of single-request PICOD schemes with lengthO(log2Γ), where Γ is the maximum number of hyperedges overlapping with any hyperedge. For thet-request PICOD scenario, we show the existence of PICOD schemes of length max(O(log Γ logm),O(tlogm)), under some mild conditions on the graph parameters. These results improve upon earlier work in general. We also show that our achievable lengths in thet-request case are asymptotically optimal, up to a multiplicative factor of logt. Our existence results are accompanied by randomized constructive algorithms, which have complexity polynomial in the parameters of the PICOD problem, in expectation or with high probability.

A Conflict-Free Open Neighborhood coloring, abbreviated CFON^* coloring, of a graph G = (V,E) using k colors is an assignment of colors from a set of k colors to a subset of vertices of V(G) such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^* coloring using k colors is called the CFON^* chromatic number of G, denoted by χ_{ON}^*(G). The analogous notion for closed neighborhood is called CFCN^* coloring and the analogous parameter is denoted by χ_{CN}^*(G). The problem of deciding whether a given graph admits a CFON^* (or CFCN^*) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper. - For k ≥ 3, we show that if G is a K_{1,k}-free graph then χ_{ON}^*(G) = O(k²log Δ), where Δ denotes the maximum degree of G. Dębski and Przybyło in [J. Graph Theory, 2021] had shown that if G is a line graph, then χ_{CN}^*(G) = O(log Δ). As an open question, they had asked if their result could be extended to claw-free (K_{1,3}-free) graphs, which are a superclass of line graphs. Since it is known that the CFCN^* chromatic number of a graph is at most twice its CFON^* chromatic number, our result positively answers the open question posed by Dębski and Przybyło. - We show that if the minimum degree of any vertex in G is Ω(Δ/{log^ε Δ}) for some ε ≥ 0, then χ_{ON}^*(G) = O(log^{1+ε}Δ). This is a generalization of the result given by Dębski and Przybyło in the same paper where they showed that if the minimum degree of any vertex in G is Ω(Δ), then χ_{ON}^*(G)= O(logΔ). - We give a polynomial time algorithm to compute χ_{ON}^*(G) for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether the CFON^* chromatic number can be computed in polynomial time on interval graphs. - We explore biconvex graphs, a subclass of bipartite graphs and give a polynomial time algorithm to compute their CFON^* chromatic number. This is interesting as Abel et al. [SIDMA, 2018] had shown that it is NP-complete to decide whether a planar bipartite graph G has χ_{ON}^*(G) = k where k ∈ {1, 2, 3}.

In the pliable index coding (PICOD) problem, a server is to serve multiple clients, each of which possesses a unique subset of the complete message set as side information and requests a new message which it does not have. The goal of the server is to do this using as few transmissions as possible. This work presents a hypergraph coloring approach to the PICOD problem. A conflict-free coloring of a hypergraph is known from literature as an assignment of colors to its vertices so that each edge of the graph contains one uniquely colored vertex. For a given PICOD problem represented by a hypergraph consisting of messages as vertices and request-sets as edges, we present achievable PICOD schemes using conflict-free colorings of the PICOD hypergraph. Various graph theoretic parameters arising out of such colorings (and some new coloring variants) then give a number of upper bounds on the optimal PICOD length, which we study in this work. Our achievable schemes based on hypergraph coloring include scalar as well as vector linear PICOD schemes. For the scalar case, using the correspondence with conflict-free coloring, we show the existence of an achievable scheme which has length$O(\log^{2}\Gamma)$, where$\Gamma$refers to a parameter of the hypergraph that captures the maximum ‘incidence’ number of other edges on any edge. This result improves upon known achievability results in PICOD literature, in some parameter regimes.

The Frieze–Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. R. Williams recently asked if one can construct a partition satisfying the conditions of the Frieze–Kannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an $\tilde O(n^{\omega})$ time algorithm for constructing such a partition, where $\omega < 2.376$ is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze–Kannan regularity lemma.

Message Ferrying is a mobility assisted technique for working around the disconnectedness and sparsity of Mobile ad hoc networks. One of the importantquestions which arise in this context is to determine the routing of the ferry,so as to minimize the buffers used to store data at the nodes in thenetwork. We introduce a simple model to capture the ferry routingproblem. We characterize {\em stable} solutions of the system andprovide efficient approximation algorithms for the {\sc Min-Max Buffer Problem} for the case when the nodes are onhierarchically separated metric spaces.