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1. Reconfiguration of Connected Graph Partitions via Recombination

   https://doi.org/10.1007/978-3-030-75242-2_4  

   Akitaya, Hugo A.; Korman, Matias; Korten, Oliver; Souvaine, Diane L.; Toth, Csaba D. (May 2021, Algorithms and Complexity (CIAC 2021))

   null (Ed.)

   Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph G. A partition of V(G) is connected if every part induces a connected subgraph. In many applications, it is desirable to obtain parts of roughly the same size, possibly with some slack s. A Balanced Connected k-Partition with slack s, denoted (k, s)-BCP, is a partition of V(G) into k nonempty subsets, of sizes n1,…,nk with |ni−n/k|≤s , each of which induces a connected subgraph (when s=0 , the k parts are perfectly balanced, and we call it k-BCP for short). A recombination is an operation that takes a (k, s)-BCP of a graph G and produces another by merging two adjacent subgraphs and repartitioning them. Given two k-BCPs, A and B, of G and a slack s≥0 , we wish to determine whether there exists a sequence of recombinations that transform A into B via (k, s)-BCPs. We obtain four results related to this problem: (1) When s is unbounded, the transformation is always possible using at most 6(k−1) recombinations. (2) If G is Hamiltonian, the transformation is possible using O(kn) recombinations for any s≥n/k , and (3) we provide negative instances for s≤n/(3k) . (4) We show that the problem is PSPACE-complete when k∈O(nε) and s∈O(n1−ε) , for any constant 0<ε≤1 , even for restricted settings such as when G is an edge-maximal planar graph or when k≥3 and G is planar. 

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2. Higher-Order Cheeger Inequality for Partitioning with Buffers

   Makarychev, Konstantin; Makarychev, Yury; Shan, Liren; Vijayaraghavan, Aravindan (January 2024, SODA 2024)

   We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph G=(V,E). The buffered expansion of a set S⊆V with a buffer B⊆V∖S is the edge expansion of S after removing all the edges from set S to its buffer B. An ε-buffered k-partitioning is a partitioning of a graph into disjoint components Pi and buffers Bi, in which the size of buffer Bi for Pi is small relative to the size of Pi: |Bi|≤ε|Pi|. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets Pi with buffers Bi. Let hk,εG be the buffered expansion of the optimal ε-buffered k-partitioning, then for every δ>0, hk,εG≤Oδ(1)⋅(logkε)⋅λ⌊(1+δ)k⌋, where λ⌊(1+δ)k⌋ is the ⌊(1+δ)k⌋-th smallest eigenvalue of the normalized Laplacian of G. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for k=2). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion. 

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3. Higher-Order Cheeger Inequality for Partitioning with Buffers

   https://doi.org/10.1137/1.9781611977912.80  

   Makarychev, Konstantin; Makarychev, Yury; Shan, Liren; Vijayaraghavan, Aravindan (January 2024, Proceedings of the Symposium on Discrete Algorithms)

   We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph G = (V, E). The buffered expansion of a set S ⊆ V with a buffer B ⊆ V∖S is the edge expansion of S after removing all the edges from set S to its buffer B. An ε-buffered k-partitioning is a partitioning of a graph into disjoint components P_i and buffers B_i, in which the size of buffer B_i for P_i is small relative to the size of P_i: |B_i| ≤ ε|P_i|. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets P_i with buffers B_i. Let h^{k,ε}_G be the buffered expansion of the optimal ε-buffered k-partitioning, then for every δ>0, h^{k,ε}_G ≤ O(1)⋅(log k) ⋅λ_{⌊(1+δ)k⌋} / ε, where λ_{⌊(1+δ)k⌋} is the ⌊(1+δ)k⌋-th smallest eigenvalue of the normalized Laplacian of G. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for k=2). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion. 

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4. On Simultaneous Two-Player Combinatorial Auctions

   Braverman, Mark; Mao, Jieming; Weinberg, S. Matthew (January 2018, Symposium on Discrete Algorithms)

   We consider the following communication problem: Alice and Bob each have some valuation functions $$v_1(\cdot)$$ and $$v_2(\cdot)$$ over subsets of $$m$$ items, and their goal is to partition the items into $$S, \bar{S}$$ in a way that maximizes the welfare, $$v_1(S) + v_2(\bar{S})$$. We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with $poly(m)$ communication, a tight 3/4-approximation is known for both [Fei06,DS06]. For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and $$\log m$$ additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show: 1) There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least $3/4$ of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication. 2) For all $$\varepsilon > 0$$, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is $$\geq 1$$ or $$\leq 3/4 - 1/108+\varepsilon$$ correctly with probability $> 1/2 + 1/ poly(m)$ requires exponential communication. This provides a separation between the attainable approximation guarantees via interactive ($3/4$) versus simultaneous ($$\leq 3/4-1/108$$) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication. We further discuss the implications of our results for the design of truthful combinatorial auctions in general, and extensions to general XOS valuations. In particular, our protocol for the allocation problem implies a new style of truthful mechanisms. 

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5. Max-Cut with Multiple Cardinality Constraints

   https://doi.org/10.4230/lipics.approx/random.2025.13  

   Makarychev, Yury; Pittu, Madhusudhan Reddy; Vakilian, Ali (September 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ene, Alina; Chattopadhyay, Eshan (Ed.)

   We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph G = (V, E), a partition of the vertices into c disjoint parts V₁, …, V_c, and cardinality parameters k₁, …, k_c, the goal is to select a set S ⊆ V such that |S ∩ V_i| = k_i for each i ∈ [c], maximizing the total weight of edges crossing S (i.e., edges with exactly one endpoint in S).\r\nBy designing an approximate kernel for Constrained Max-Cut and building on the correlation rounding technique of Raghavendra and Tan (2012), we present a (0.858 - ε)-approximation algorithm for the problem when c = O(1). The algorithm runs in time O(min{k/ε, n}^poly(c/ε) + poly(n)), where k = ∑_{i∈[c]} k_i and n = |V|. This improves upon the (1/2 + ε₀)-approximation of Feige and Langberg (2001) for Max-Cut_k (the special case when c = 1, k₁ = k), and generalizes the (0.858 - ε)-approximation of Raghavendra and Tan (2012), which only applies when min{k,n-k} = Ω(n) and does not handle multiple constraints.\r\nWe also establish that, for general values of c, it is NP-hard to determine whether a feasible solution exists that cuts all edges. Finally, we present a 1/2-approximation algorithm for Max-Cut under an arbitrary matroid constraint. 

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