LP-Based Algorithms for Scheduling in a Quantum Switch

LP-Based Algorithms for Sc heduling in a Quan tum Switc h R. Srik an t ECE, CSL, NCSA rsrik an t@illinois.edu F ebruary 2026 Abstract W e consider sc heduling in a quantum switch with sto c hastic en tanglement generation, finite quan tum memories, and decoherence. The ob jectiv e is to design a scheduling algorithm with polynomial-time computational complexity that stabilizes a non trivial fraction of the capacit y region. Scheduling in such a switch corresp onds to finding a matching in a graph sub ject to additional constraints. W e prop ose an LP-based policy , whic h finds a p oin t in the matching p olytop e, which is further implemented using a randomized decomposition into matc hings. The main challenge is that service ov er an edge is feasible only when en tanglement is simultaneously av ailable at b oth endp oint memories, so the effective service rates dep end on the steady-state av ailability induced by the sc heduling rule. T o address this, w e introduce a single-no de reference Marko v c hain and deriv e low er b ounds on ac hiev able service rates in terms of the steady-state nonemptiness probabilities. W e then use a Lyapuno v drift argument to sho w that, whenever the request arriv al rates lie within the resulting throughput region, the proposed algorithm stabilizes the request queues. W e further analyze how the achiev able throughput dep ends on entanglemen t generation rates, decoherence probabilities, and buffer sizes, and show that the throughput lo wer b ound con verges exp onen tially fast to its infinite-buffer limit as the memory size increases. Numerical results illustrate that the guaranteed throughput fraction is substan tial for parameter regimes relev an t to near-term quantum net working systems 1 In tro duction The dev elopment of a large-scale quan tum internet is widely viewed as a k ey enabling technology for both secure communication and distributed quantum computation. En tanglement-based protocols allow distan t users to establish information-theoretically secure k eys, while also enabling remote quantum op erations, telep ortation, and the interconnection of geographically distributed quantum processors [26, 17]. A central arc hitectural component in such netw orks is the quantum switch , a device that generates and stores en tangled qubits and performs entanglemen t sw apping op erations to establish end-to-end en tanglemen t b etw een user pairs. In contrast to classical pack et switches, en tanglement is a fragile resource that can b e stored only for limited time due to decoherence. As a result, scheduling decisions in a quantum switch must join tly account for stochastic entanglemen t generation, finite quantum-memory buffers, and random loss due to decoherence. Motiv ated b y these c hallenges, recen t w ork has b egun to study scheduling p olicies for quantum switches. Sp ecial cases of general quan tum switches were first studied in [25, 28, 18, 21]. The general v ersion of the quantum switch was studied in [3] where the capacit y region of the system was characterized and the throughput-optimal scheduling rules were established. These results provide an imp ortant theoretical foun- dation for the op eration of quan tum switches. How ever, the resulting optimal p olicies can be computationally exp ensiv e. In particular, the throughput-optimal p olicy requires the solution of a Marko v decision process, whose size grows exp onentially with the num b er of en tanglement-buffers, rendering exact dynamic program- ming metho ds impractical except for very small switches. A step tow ards resolving this issue has b een tak en in [4], where the authors sho w that performing one v alue iteration step at each time instant is sufficient to ac hieve throughput optimality . But ev en this could b e computationally expensive if the n umber and size of the entanglemen t buffers is large. The quantum switch problem has some similarities with scheduling problems in traditional communication net works initiated in [23], where it was sho wn that MaxW eight scheduling stabilizes an y arriv al rate vector 1 in the interior of the capacity region for a broad class of constrained queueing systems. In the context of input-queued pack et switches, [16] demonstrated that MaxW eight-based algorithms ac hieve full throughput for crossbar switches. These results laid the foundations for a large bo dy of w ork on throughput-optimal sc heduling in wireless netw orks and pac ket switches (see, e.g., [22]). At the same time, a complemen tary line of w ork has in vestigated c onstant-factor appr oximations to throughput-optimal sc heduling. In many systems, implementing the exact MaxW eight rule requires solving computationally difficult combinatorial optimization problems in ev ery time slot. This observ ation has motiv ated the study of simplified scheduling rules that achiev e a guaranteed fraction of the capacity region. F or example, tw o well-kno wn pap ers hav e established half-capacit y or constant-factor guarantees for switch scheduling algorithms [27, 6]. These results demonstrate that significan t computational sa vings can b e ac hieved while sacrificing only a b ounded fraction of the maxim um achiev able throughput. In this pap er, we inv estigate similar appro ximation guaran tees for the quantum switch. W e restrict our atten tion to the case where only bipartite en tanglements are requested. Our goal is to design a sc heduling algorithm that (i) has polynomial computational complexity and (ii) pro vides guaran tees on the fraction of the capacity region that can b e stabilized. T o this end, our con tributions are as follo ws: • W e prop ose an LP-based scheduling framework for the quantum switc h. The k ey idea is to replace the in tractable MDP with a linear programming relaxation that determines fractional edge activ ation rates. These fractional rates are then implemen ted through a randomized decomp osition into matchings. A cen tral technical c hallenge arises from the sto c hastic av ailability of entanglemen t resources. T o address this issue, w e introduce a reference en tanglement-buffer Marko v c hain whose steady-state a v ailabilit y captures the probability that entanglemen t is present at each no de. Using a coupling argument, we sho w that the ac hiev able service rates under our p olicy are low er bounded by the pro duct of these no de a v ailabilities. This leads to a c oher enc e factor that quantifies the fraction of the classical matching capacit y region ac hiev able b y the quantum switch. W e show that the algorithm stabilizes the queues if the arriv al rates are within this fraction of the capacity region using a Lyapuno v drift argumen t. • W e further analyze how this coherence factor dep ends on physical system parameters, including entan- glemen t arriv al rates, decoherence probabilities, and buffer sizes. Our analysis establishes exp onential con vergence of the ac hiev able throughput to its infinite-buffer limit and shows that the ac hiev able frac- tion of the capacity region approac hes one as decoherence decreases or entanglemen t generation rates increase. • Finally , we present a low er-complexity algorithm that omits blossom constrain ts from the LP formula- tion. Using classical polyhedral prop erties of fractional matc hings, we show that a simple scaling yields a feasible p oint in the matching p olytop e, resulting in a scheduling p olicy that stabilizes a constant fraction of the capacit y region while further reducing computational complexity . T ogether, these results provide p olynomial-time scheduling algorithms with prov able p erformance guar- an tees for quan tum switches, bridging ideas from clas sical switch scheduling, polyhedral com binatorics, and emerging quantum net working systems. 2 System Mo del and Problem F orm ulation Consider a quan tum switc h represen ted b y a graph G = ( V , E ) , where V is the set of vertices (quantum memories) and E represents edges (user pairs). The system op erates in discrete time slots t = 0 , 1 , . . . . En tangled qubits (which w e will call entanglemen ts) arrive and are queued at the v ertices and service requests arrive and are queued at the edges of the graph. In each time slot, a request at an edge can b e serv ed only if an en tanglemen t is a v ailable at eac h of the tw o v ertices of the edge. Eac h no de v ∈ V main tains a lo cal buffer of size B v to store entanglemen ts. W e denote the num b er of en tanglements stored at vertex v at time t b y L v ( t ) ∈ { 0 , . . . , B v } . In each time slot, new entanglemen ts arriv e at no de v according to a Bernoulli pro cess with parameter λ v . Stored en tanglements can b e lost due to decoherence. W e mo del decoherence loss as follows: eac h stored en tanglement at no de v decays indep enden tly with probability µ v p er time slot. 2 F or eac h edge e ∈ E , user requests arrive according to an i.i.d. stochastic pro cess A e ( t ) with mean rate E [ A e ( t )] = ν e and v ariance V ar ( A e ( t )) = σ 2 e . These requests are stored in an infinite-buffer queue denoted b y R e ( t ) . A sc heduling algorithm determines whic h requests to serv e in eac h time slot. A schedule is represented b y a matching M ( t ) ⊂ E , i.e., a collection of edges such that no tw o edges in M ( t ) share a common no de, resp ecting the ph ysical constraint that a node can serv e at most one request p er slot. W e assume the following order of ev en ts within a time slot: 1. Sche dule: A sc heduler chooses a sc hedule for the time slot. 2. Servic e: If an edge is included in the sc hedule, then if a request is waiting in the request queue at the edge, it is served if the en tanglement queues at its vertices are non-empty . If a service o ccurs at an edge, then a request is remo ved from its request queue and an entanglemen t is remo v ed from eac h of the corresp onding v ertex entanglemen t queues. 3. De c oher enc e: Next, the remaining entanglemen ts at each v ertex u indep endently decohere with prob- abilit y µ u . 4. A rrivals: At eac h v ertex u, a new entanglemen t arriv es with probability λ u . And arriv als occur at eac h of the edges according to their respective arriv al processes. The queue ev olution equations are giv en b y: R e ( t + 1) = R e ( t ) − S e ( t ) + A e ( t ) , (1) L v ( t + 1) = min   B v , L v ( t ) − X e ∈ δ ( v ) S e ( t ) − D v ( t ) + Y v ( t )   , (2) where: • S e ( t ) ∈ { 0 , 1 } indicates if edge e w as served. Service occurs only if e ∈ M ( t ) , R e ( t ) > 0 and the required en tanglements are a v ailable at the b eginning of the slot ( L u ( t ) > 0 , L v ( t ) > 0 ). Due to the fact that eac h sc hedule is a matc hing in the graph and S e ( t ) is the actual service, P e ∈ δ ( v ) S e ( t ) ≤ L v ( t ) ∀ v . • D v ( t ) represents the num b er of en tanglements lost to decoherence from L v ( t ) − P e ∈ δ ( v ) S e ( t ) , i.e., D v ( t ) ∼ Binom ( L v ( t ) − P e ∈ δ ( v ) S e ( t ) , µ v ) • Y v ( t ) represen t the arriv als of entanglemen ts, which occur after service and decoherence. Newly arriving en tanglements are sub ject to decoherence only in subsequen t slots. • A e ( t ) represent the arriv als of requests, whic h occur after service. 2.1 Mark o vian Sc heduling and Throughput Optimalit y A Markovian sche dule is a randomized algorithm that selects a matching M ( t ) based on the current state of the system, sp ecifically the vector of queue lengths Q ( t ) = ( L ( t ) , R ( t )) . As shown in [3], a throughput- optimal sc heduling policy for suc h a switch can b e obtained by solving a Mark ov Decision Process (MDP). Roughly sp eaking, the MDP is obtained b y assuming a time-scale decomp osition of the system, where the request queue states are fixed and one con trols the entanglemen t queues by choosing schedules to maximize a long-term exp ected rew ard. F ormally , this corresp onds to the fluid-limit used to characterize throughput- optimal p olicies, where request queue lengths enter the MDP only as fixed weigh ts [3]. The instantaneous rew ard is the sum of the request queue lengths of all the edges incl uded i n the schedule that are successfully serv ed. More precisely , the MDP is defined b y the tuple ( S , A , P , R ) : • State Sp ac e ( S ): The set of all p ossible entanglemen t queue configurations: S = Q v ∈ V { 0 , . . . , B v } . W e will denote a specific state in the MDP b y s . 3 • A ction Sp ac e ( A ): In eac h slot, the controller selects a matching M ∈ M , where M is the set of all v alid matc hings in the graph. • R ewar d F unction ( R ): The immediate reward for taking action M in state s is defined as the sum of the request queue lengths of the edges included in the matching: r ( s , M ) = X e ∈ M w e · I ( Resources a v ailable for e ) , (3) where w e = R e ( t ) is fixed for the purp oses of defining the MDP under a time-scale separation as in [3]. Note that resources a v ailable for e means ( L u ( t ) > 0 , L v ( t ) > 0 , R e ( t ) > 0) . • T r ansition Dynamics ( P ): The system evolv es according to the entanglemen t queue evolution equations defined previously , gov erned by sto chastic arriv als and decoherence up dates. The goal of the MDP is to find a p olicy π that maximizes the long-term exp ected a verage reward. The optimal v alue function W ∗ satisfies the Bellman Optimalit y Equation: W ∗ ( s ) + r ∗ M D P = max M ∈M ( r ( s , M ) + X s ′ ∈S P ( s ′ | s , M ) W ∗ ( s ′ ) ) (4) where r ∗ M D P is the optimal a verage rew ard. Solving the MDP is computationally in tractable b ecause state space size is exp onential in the num b er of entanglemen t queues. F or a netw ork with | V | no des, |S | is O ( B | V | ) . Our ob jective is to design a low- complexit y sc heduling algorithm with a prov able performance guarantee. Sp ecifically , we seek an algorithm that: 1. Polynomial Complexity: The computation time per slot should b e polynomial in | V | and | E | . 2. Appr oximation Guar ante e: W e aim to prov e that our algorithm is α -throughput optimal. Definition ( α -Throughput Optimality): An algorithm is α -throughput optimal (for 0 < α ≤ 1 ) if it stabilizes the system for any arriv al rate vector ν such that ν ∈ α Λ , where Λ is the capacit y region achiev able b y the MDP solution. In other words, the algorithm guaran tees stable op eration for at least an α -fraction of the theoretical maxim um throughput. 3 Prop osed Algorithm and Its A c hiev able Throughput As discussed in the previous section, a throughput–optimal scheduling rule can in principle be obtained b y solving the Marko v decision pro cess (MDP) defined by the en tanglement-buffer state s ∈ S and the set of feasible matchings M . One w a y to compute the optimal a verage rew ard of a finite-state MDP is through a linear programming (LP) form ulation in terms of stationary state–action frequencies. Let x ( s, M ) denote the steady-state fraction of time that the system is in state s ∈ S and action M ∈ M is chosen. The av erage-rew ard optimal control problem can then b e written as the follo wing L P: maximize X s ∈S X M ∈M r ( s, M ) x ( s, M ) (5) sub ject to X M ∈M x ( s, M ) = X s ′ ∈S X M ∈M P ( s | s ′ , M ) x ( s ′ , M ) , ∀ s ∈ S (6) X s ∈S X M ∈M x ( s, M ) = 1 (7) x ( s, M ) ≥ 0 , ∀ s, M . (8) The v ariables x ( s, M ) represen t the stationary occupancy measure of the MDP . The first set of constrain ts enforces flow conserv ation of the stationary distribution, while the normalization constrain t ensures that the total probability mass is one. The optimal v alue of this LP is equal to the optimal av erage rew ard r ∗ MDP . 4 Although this formulation is conceptually useful, it is computationally in tractable for quan tum switches of practical size. The state space of the MDP is |S | = Y v ∈ V ( B v + 1) , whic h grows exp onentially with the num b er of entanglemen t buffers. Consequen tly , the ab ov e LP con tains exp onen tially man y v ariables and constrain ts. T o obtain a tractable algorithm, we therefore in tro duce a relaxation of this LP . Instead of tracking the full state of the entanglemen t queues, we consider the long-term av erage rate at whic h each ed ge is activ ated. Let x e denote the long-run service attempt rate assigned to edge e ∈ E . These rates m ust satisfy t wo fundamen tal constrain ts. First, b ecause a no de can participate in at most one entanglemen t swap in a given slot, the total service rate of edges incident on no de v cannot exceed the long-term rate at whic h entanglemen ts are generated at that node. Since en tanglements arrive at no de v with probability λ v p er slot, this implies the no de-capacity constrain t X e ∈ δ ( v ) x e ≤ λ v , ∀ v ∈ V . (9) Second, since only matchings can be sc heduled in any given slot, the v ector x = ( x e ) e ∈ E m ust lie in the con vex hull of the set of matchings of the graph. This leads to the following linear programming relaxation of the MDP con trol problem: maximize X e ∈ E w e ( t ) x e (10) sub ject to X e ∈ δ ( v ) x e ≤ λ v , ∀ v ∈ V (11) X e ∈ E ( S ) x e ≤ | S | − 1 2 , ∀ S ⊆ V with | S | odd , | S | ≥ 3 (12) x e ≥ 0 , ∀ e ∈ E , (13) where E ( S ) denotes the set of all edges in S. W e note that (12) are blossom inequalities used to characterize the matching p olytop e [8]. How ever, due to (11), our constrain t set is not a matching p olytop e, but a p olytop e which is a subset of the matching p olytop e since λ v ≤ 1 ∀ v . The optimal solution x ∗ to the LP defined by (10)–(13) represen ts an upp er b ound on the maxim um v alue achiev able under the assumption that the switc h can support fractional edge activ ations, i.e., the optimal solution x ∗ ma y b e suc h that, for eac h edge e, x ∗ e ma y not lie in { 0 , 1 } but will more generally lie in [0 , 1] . Since the vector x ∗ is guaran teed to lie within the matching p olytop e of the graph G , defined as the con vex hull of all v alid integral matchings, b y Carathéo dory’s theorem, x ∗ admits a decomp osition into a con vex combination of at most | E | + 1 in tegral matc hings. This decomposition yields a set of matc hings M = { M 1 , M 2 , . . . , M k } and associated probabilities { p 1 , p 2 , . . . , p k } such that: k X j =1 p j 1 M j = x ∗ and k X j =1 p j = 1 , p j ≥ 0 ∀ j. (14) Prop osed Sc heduling Algorithm Our scheduling algorithm is as follows: at time slots t = 0 , T , 2 T , ... observ e the request queue length v ector R ( t ) . Obtain probabilities { p M } as ab o v e and select matchings according to these probabilities o v er time slots t, t + 1 , t + T − 1 . In other w ords, a probabilistic schedule is c hosen once ev ery T slots using the request queue lengths at the b eginning of the T slots. W e will sho w that the parameter T can b e c hosen appropriately dep ending on ho w far the arriv al rates are from the b oundary of the capacit y region. 5 3.1 Solving the LP and Decomp osing the Solution In to Matc hings The LP in (10)–(13) can b e solved in p olynomial time using the ellipsoid method [10] due to the fact that there is a separation oracle for the constraints [19, 13]. Given the solution x , a p olynomial-time algorithm to compute an explicit conv ex decomposition into matchings is pro vided in [5]. How ever, as in m uch of LP theory , algorithms that are not theoretically polynomial-time are often practically more efficient. F or our problem, this means that the LP should b e solv ed using the simplex method and the decomp osition should b e p erformed using column generation [7] whic h w e present in Algorithm 1. Algorithm 1 Decomp osition Using Column Generation Require: V ector x ∈ C o ( M ) Ensure: Set of pairs { ( p k , M k ) } such that P p k 1 M k = x ∗ , P p k = 1 , and p k ≥ 0 1: Initialize: K ← {{ e } | x ∗ e > 0 } ∪ {∅} ▷ Initialize with singleton matchings 2: lo op 3: // Step 1: R estricte d Master Pr oblem 4: Solv e the follo wing LP o ver the current set K : find { p M } s.t. X M ∈K p M 1 M = x ∗ , X M ∈K p M = 1 , p M ≥ 0 5: Let ( y , z ) be the dual v ariables asso ciated with the equality constraints. 6: // Step 2: Pricing Or acle 7: M ∗ ← MaxW eightMatc hing ( G, w eights = y ) 8: W ∗ ← P e ∈ M ∗ y e 9: // Step 3: Che ck T ermination Condition 10: if W ∗ ≤ − z then 11: Break ▷ No violating matc hings exist in M all 12: else 13: K ← K ∪ { M ∗ } 14: end if 15: end lo op 16: return { ( p M , M ) | M ∈ K , p M > 0 } T o understand the algorithm, w e formulate the decomp osition as a Linear Program (LP): • The Primal Pr oblem: W e treat the probabilities p M as decision v ariables. minimize 0 sub ject to X M ∈M p M 1 M = x ∗ X M ∈M p M = 1 p M ≥ 0 ∀ M ∈ M • The Dual Pr oblem: W e introduce dual v ariables y ∈ R | E | for the equality constrain ts and z ∈ R for the normalization constrain t. maximize y · x ∗ + z sub ject to y · 1 M + z ≤ 0 ∀ M ∈ M The primal problem requires the solution of a set of linear equalities and non-negativity constraints. W e p ose it as an LP so that it is easy to see the dual form and use the column generation idea used in linear programming. Since the primal problem has an exp onential num b er of v ariables, w e start the algorithm b y p osing the primal problem with a restricted set of matchings K , whic h is a muc h smaller set than M . In the 6 LP terminology , this problem is called the restricted master problem. W e solv e for the dual v ariables of this problem which gives a candidate y and z . Then we try to find a matching that violates the dual constraint. In LP language, this part of the pro cedure is called a pricing oracle. In our problem, the pricing oracle is equiv alent to finding a maximum weigh t matching in G with weigh ts y and chec king if the weigh t of the maxim um w eigh ted matc hing is greater than − z . If so, w e add the maxim um w eigh ted matc hing (a column in LP terminology) to the set K and rep eat the pro cess. Else, the column generation pro cess terminates. When the process terminates, w e solv e the primal problem with M replaced by K to obtain p M ∀ M . 3.2 A c hiev able Throughput F or the algorithm to b e able to provide service to a request queue, we need to ha ve an en tanglement in eac h of the entanglemen t queues at the vertices of the edge corresp onding to the request queue. T o characterize the probabilit y with which this ev ent happ ens, w e consider a vertext u and define a reference discrete-time Mark ov c hain { ˜ L u ( t ) } t ≥ 0 with state space { 0 , 1 , . . . , B u } and a fixed service attempt probability: p u = λ u . The transition probabilities P ij for the Mark o v c hain are deriv ed by considering its state evolution through the three phases: service, decoherence and then arriv als, in that order. Let L b e the state at the start of the slot. First, the in termediate state after service is K serv = max(0 , L − S ) , where S ∈ { 0 , 1 } is the service indicator. Second, the num b er of entanglemen ts surviving decoherence is K dec ∼ Binom ( K serv , 1 − µ u ) . Finally , the state at the start of the next slot is j = min( B u , K dec + A ) , where A ∈ { 0 , 1 } is the arriv al indicator. The probability of transitioning from state i to j is giv en by: P ij = 1 X S =0 1 X A =0 P ( S ) P ( A ) X K dec  max(0 , i − S ) K dec  (1 − µ u ) K dec µ max(0 ,i − S ) − K dec u · I ( j = min( B u , K dec + A )) (15) where P ( S = 1) = p u and P ( A = 1) = λ u . if π is the vector of stationary probabilities for this Marko v chain, then the stationary probabilit y of a entanglemen t b eing a v ailable is C u = 1 − π 0 . Lemma 1 (A v ailability Lo w er Bound via Coupling) . L et P ( L u > 0 , L v > 0) b e the stationary pr ob ability that the buffers at b oth endp oints of e dge e = ( u, v ) ar e non-empty under the r andomize d de c omp osition p olicy with fixe d weights { w e } . L et C u and C v b e the ste ady-state availabilities of the r efer enc e chains ˜ L u and ˜ L v as define d ab ove. Then: P ( L u > 0 , L v > 0) ≥ ( C u + C v − 1) + . (16) Pr o of. Fix a node u . Under our randomized decomposition p olicy , the probabilit y that u is sc heduled in a giv en slot is a u := X e ∈ δ ( u ) x ∗ e ≤ λ u =: p u , where the inequalit y follows from the LP constrain ts. F or eac h no de u and slot t , let U u ( t ) ∼ Unif [0 , 1] b e i.i.d. ov er u, t , independent of arriv als and decoher- ence. Define Bernoulli indicators S u ( t ) := I { U u ( t ) ≤ a u } , ˜ S u ( t ) := I { U u ( t ) ≤ p u } . Then S u ( t ) ≤ ˜ S u ( t ) almost surely for all t . Construct the reference c hain ˜ L u ( t ) to ev olve with the same arriv al and decoherence even ts as L u ( t ) at no de u , but with remov als driven by ˜ S u ( t ) (i.e., whenev er ˜ S u ( t ) = 1 and ˜ L u ( t ) > 0 , one entanglemen t is remov ed b efore decoherence). Since the actual pro cess can remov e at most one entanglemen t from u per slot, and S u ( t ) ≤ ˜ S u ( t ) path wise, this coupling implies the sample-path dominance L u ( t ) ≥ ˜ L u ( t ) ∀ t. Th us, P ( L u > 0) ≥ C u . The result follo ws from the fact 1 ≥ P ( L u > 0 or L v > 0) = P ( L u > 0) + P ( L v > 0) − P ( L u > 0 , L v > 0) . 7 Next, as in [3], w e define a request-agnostic p olicy π as one that selects matching M with probability π ( M | L ) when the entanglemen t queue length vector is in state L . Let d π b e the stationary distribution of the entanglemen t queues under such a policy . Recall that ν is the v ector of request arriv al rates. F ollowing [3], the capacit y region of the switch is defined as follo ws: C = { ν : ∃ π suc h that ν e < X L : l u e > 0 ,l v e > 0 X M ∋ e π ( M | L ) d π ( L ) ∀ e ∈ E } , where we ha ve used the notation u e , v e to denote the v ertices asso ciated with edge e. Define Γ coh := min e =( u,v ) ∈ E ( C u + C v − 1) + . Next, w e show that our proposed algorithm achiev es at least Γ coh fraction of the capacity region. Note that Γ coh can p otentially be zero for certain v alues of the problem parameters. Later, w e will demonstrate using a combination of theory and sim ulations that for switch parameters of in terest, Γ coh is a non-trivial fraction. Theorem 1. Supp ose that (1+ ϵ ) Γ c oh ν ∈ C and c onsider the sche duling algorithm pr op ose d at the end of Se ction 3. Define the Lyapunov function V ( R ) = 1 2 X e R 2 e . Then, ∃ T ϵ , δ > 0 and a finite set B such that ∀ k ∈ { 0 , 1 , 2 , . . . } E ( V ( R (( k + 1) T ϵ )) − V ( R ( k T ϵ )) | L ( k T ϵ ) , R ( k T ϵ )) ≤ − δ whenever R ( k T ϵ ) ∈ B c . Pr o of. W e will present the pro of in sev eral steps. Step 1: Uniform Mixing of Entanglement Queues. W e use the uniform mixing property of the en tangle- men t queues pro ved in [3]. F or any fixed matc hing p olicy π , the entanglemen t pro cess { L ( t ) } is a Marko v c hain on the finite state space S = Q v { 0 , . . . , B v } . Let P π b e its transition matrix. Since µ v , λ v ∈ (0 , 1) , the state 0 (all buffers empt y) is reac hable from an y state L ∈ S in one time slot via the the decoherence of all en tanglements after service and the absence of arriv als. Specifically , for all π ∈ Π and L ∈ S : P π ( L , 0 ) ≥ Y v ∈ V (1 − λ v ) µ B v v ≜ η > 0 , (17) indep enden t of π . Th us, by the standard Doeblin condition, the c hain is uniformly ergo dic with a con traction co efficien t ρ := 1 − η < 1 . Consequently , ∥ P t π ( L , · ) − d π ( · ) ∥ T V ≤ ρ t ∀ L , t ≥ 0 , π. Step 2: T -Step Drift De c omp osition. Since departures o ccur b efore arriv als and S e ( t ) denotes actual service (departures), w e hav e S e ( t ) ≤ R e ( t ) a.s., and hence the request queue evolv es as R e ( t + 1) = R e ( t ) − S e ( t ) + A e ( t ) , without the need for a pro jection operator. Fix a frame index k and a frame size T . Iterating ov er the frame yields R e (( k + 1) T ) = R e ( k T ) + ( k +1) T − 1 X τ = k T  A e ( τ ) − S e ( τ )  . Squaring b oth sides and subtracting R 2 e ( k T ) giv es R 2 e (( k + 1) T ) − R 2 e ( k T ) = 2 R e ( k T ) ( k +1) T − 1 X τ = k T  A e ( τ ) − S e ( τ )  +  ( k +1) T − 1 X τ = k T  A e ( τ ) − S e ( τ )   2 . Summing o ver all edges and taking conditional expectations given ( L ( k T ) , R ( kT )) , the T -step Lyapuno v drift ∆ T := E [ V ( R (( k + 1) T )) − V ( R ( k T )) | L ( k T ) , R ( k T )] 8 satisfies ∆ T ≤ X e R e ( k T ) E   ( k +1) T − 1 X τ = k T  A e ( τ ) − S e ( τ )    + 1 2 X e E    ( k +1) T − 1 X τ = k T  A e ( τ ) − S e ( τ )   2   . (18) By Cauch y–Sch warz,  ( k +1) T − 1 X τ = k T ( A e ( τ ) − S e ( τ ))  2 ≤ T ( k +1) T − 1 X τ = k T ( A e ( τ ) − S e ( τ )) 2 . Since S e ( τ ) ∈ { 0 , 1 } and E [ A 2 e ] = σ 2 e + ν 2 e , we ha ve E [( A e ( τ ) − S e ( τ )) 2 ] ≤ 2 E [ A e ( τ ) 2 ] + 2 E [ S e ( τ ) 2 ] ≤ 2( σ 2 e + ν 2 e ) + 2 . Therefore, 1 2 X e E    ( k +1) T − 1 X τ = k T ( A e ( τ ) − S e ( τ ))  2   ≤ K T , K T := T 2 2 X e  2( σ 2 e + ν 2 e ) + 2  = O ( T 2 ) . Define the p otential service indicator ˆ S e ( t ) := 1 { e ∈ M ( t ) , L u e ( t ) > 0 , L v e ( t ) > 0 } , whic h ignores request-queue emptiness. Then S e ( t ) = ˆ S e ( t ) 1 { R e ( t ) > 0 } ≤ ˆ S e ( t ) . Moreov er, since S e ( t ) ≤ 1 and arriv als are nonnegativ e, we hav e the deterministic b ound R e ( k T + τ ) ≥ R e ( k T ) − τ , τ = 0 , 1 , . . . , T − 1 . Hence, if R e ( k T ) ≥ T , then R e ( k T + τ ) > 0 for all τ ∈ { 0 , . . . , T − 1 } and therefore S e ( k T + τ ) = ˆ S e ( k T + τ ) throughout the frame. It follows that for every edge e , T − 1 X τ =0 S e ( k T + τ ) ≥ T − 1 X τ =0 ˆ S e ( k T + τ ) − T 1 { R e ( k T ) < T } . (19) Multiplying (19) b y R e ( k T ) and using R e ( k T ) 1 { R e ( k T ) < T } ≤ T gives R e ( k T ) T − 1 X τ =0 S e ( k T + τ ) ≥ R e ( k T ) T − 1 X τ =0 ˆ S e ( k T + τ ) − T 2 . (20) Summing (20) ov er e yields an additive p enalt y of at most | E | T 2 , which can b e absorb ed into the O ( T 2 ) term K T . Conditional on R ( kT ) , the distribution { p M } is fixed o ver the frame. Th us, for τ ′ = 0 , 1 , . . . , T − 1 , P ( ˆ S e ( k T + τ ′ ) = 1 | L ( kT ) , R ( kT )) = X M ∋ e p M P ( L u e ( k T + τ ′ ) > 0 , L v e ( k T + τ ′ ) > 0 | L ( kT )) . Let µ e ( R ( k T )) := X M ∋ e p M d π ( L u e > 0 , L v e > 0) denote the corresp onding stationary p otential service probability under the entanglemen t-chain stationary distribution d π induced b y these fixed probabilities { p M } . By the uniform mixing b ound from Step 1, there exists a constan t C ′ > 0 such that, uniformly ov er L ( kT ) , T − 1 X τ ′ =0 E [ ˆ S e ( k T + τ ′ ) | L ( k T ) , R ( k T )] ≥ T µ e ( R ( k T )) − C ′ . 9 Com bining this with (20) and summing ov er e yields X e R e ( k T ) E   ( k +1) T − 1 X τ = k T S e ( τ )   ≥ T X e R e ( k T ) µ e ( R ( k T )) − C ′ X e R e ( k T ) − | E | T 2 . Substituting the ab ov e b ound (and E [ P τ A e ( τ )] = T ν e ) into (18), and, abusing notation and absorbing the additional | E | T 2 term into K T , we obtain the drift b ound ∆ T ≤ T X e R e ( k T )  ν e − µ e ( R ( k T ))  + C ′ X e R e ( k T ) + K T , where K T = O ( T 2 ) is independent of L ( k T ) and R ( k T ) . Step 3: L ower b ounding the p olicy servic e r ate. Fix a frame k and consider the matc hing distribution { p M } computed at time k T . Let µ e ( R ( k T )) := X M ∋ e p M d π ( L u e > 0 , L v e > 0) denote the stationary p otential service probabilit y for edge e = ( u e , v e ) under the entanglemen t-chain sta- tionary distribution d π induced by { p M } . By construction of the decomp osition in Section 3, X M ∋ e p M = x ∗ e , where x ∗ is the optimal solution of the LP (10)–(13) with w eights R ( k T ) . By Lemma 1, d π ( L u e > 0 , L v e > 0) ≥ C u e C v e ≥ Γ coh , and therefore µ e ( R ( k T )) ≥ Γ coh x ∗ e . (21) Multiplying (21) b y R e ( k T ) and summing ov er e yields X e R e ( k T ) µ e ( R ( k T )) ≥ Γ coh X e R e ( k T ) x ∗ e . (22) Step 4: R elating the r elaxe d LP value to the arrival r ates. By the assumption of the theorem, Γ coh (1 + ϵ ) ν ∈ C . Hence there exists a request-agnostic stationary p olicy π † whose steady-state suc c essful service rates s † = { s † e } e ∈ E satisfy s † e ≥ (1 + ϵ ) Γ coh ν e , ∀ e ∈ E . The v ector s † represen ts the steady-state successful service rates under p olicy π † . In eac h slot, the set of successfully serv ed edges forms a matc hing; therefore the long-run a verage service rate v ector lies in the con vex hull of matchings, i.e., s † ∈ C o ( M ) . Moreo ver, since at most one entanglemen t can b e consumed p er no de p er slot, w e hav e P e ∈ δ ( v ) s † e ≤ λ v for all v . Hence s † is feasible for the LP (10)–(13). By optimalit y of x ∗ for weigh ts R ( k T ) , X e R e ( k T ) x ∗ e ≥ X e R e ( k T ) s † e ≥ (1 + ϵ ) Γ coh X e R e ( k T ) ν e . (23) Step 5: Completing the drift ar gument. Combining (22) and (23) giv es X e R e ( k T ) µ e ( R ( k T )) ≥ (1 + ϵ ) X e R e ( k T ) ν e . 10 Substituting this bound in to the T -step drift inequality from Step 2, ∆ T ≤ T X e R e ( k T )  ν e − µ e ( R ( k T ))  + C ′ X e R e ( k T ) + K T , yields ∆ T ≤ − ϵT X e R e ( k T ) ν e + C ′ X e R e ( k T ) + K T . Let ν min := min { ν e : ν e > 0 } . Then P e R e ν e ≥ ν min P e : ν e > 0 R e . Cho osing T ϵ suc h that ϵT ϵ ν min > C ′ , there exist γ , δ > 0 and a finite set B for whic h ∆ T ϵ ≤ − δ whenev er R ( kT ϵ ) / ∈ B . The ab ov e theorem can then b e used to show positive recurrence of the Mark ov c hain or other notions of stabilit y used in the literature using the F oster-Lyapuno v theorem and related techniques [22]. F urther, the Lyapuno v drift can also b e useful to obtain b ounds on queue lengths as in [9, 15]. W e note that, instead of using a T ϵ that dep ends on ϵ, one can also c ho ose an adaptive frame size based on the queue length as prop osed in [4]. The same throughput guaran tees would contin ue to hold follo wing argumen ts similar to [4]. In the next subsection, we will explore ho w Γ coh v aries as a function of the switc h parameters: the arriv al rates of en tanglemen ts, the decoherence probabilities and buffer sizes. 3.3 A Low er Complexit y Algorithm F or concreteness, w e will call the algorithm presen ted earlier Algorithm I. No w, we preen t a lo wer complex- it y algorithm, whic h w e will call Algorithm II. T o describ e Algorithm I I, w e onsider the degree-only LP relaxation, without the blossom inequalities: maximize X e ∈ E w e ( t ) x e (24) sub ject to X e ∈ δ ( v ) x e ≤ λ v , ∀ v ∈ V (25) x e ≥ 0 , ∀ e ∈ E . (26) Let x frac denote an optimal solution. In general, x frac need not lie in the matching p olytop e C o ( M ) because the blossom inequalities ma y b e violated. Nevertheless, a standard argument in p olyhedral combinatorics implies that the scaled vector x app := 2 3 x frac is feasible for the matc hing p olytop e, i.e., x app ∈ C o ( M ) . While this appears to b e a well kno wn result, a pro of do es not app ear in commonly used texts in the field; hence, w e sk etch it here for completeness: • The extreme p oints of the degree-only fractional matc hing p olytop e are half-in tegral, and the edges with v alue 1 / 2 form no de-disjoint odd cycles [2, 1]. • On any odd cycle of length 2 k + 1 , the unscaled solution has total weigh t (2 k + 1) / 2 on the cycle edges, whic h violates the o dd-set (blossom) bound k ; scaling b y 2 / 3 reduces this to (2 k + 1) / 3 ≤ k , hence all blossom inequalities X e ∈ E ( S ) x e ≤ | S | − 1 2 , ∀ S ⊆ V odd , | S | ≥ 3 are satisfied, and therefore x app ∈ C o ( M ) [8]. Since x app ∈ C o ( M ) , it admits a conv ex decomp osition in to integral matc hings, and we may implement the same randomized matching schedule as in Section 3, with x ∗ replaced by x app . The resulting no de activ ation probabilities satisfy a v := X e ∈ δ ( v ) x app e ≤ 2 3 λ v , 11 so the coupling argumen t of Lemma 1 applies with reference service-attempt probabilit y p v = 2 3 λ v , yielding a coherence factor Γ ′ coh defined analogously . The main adv antage of the simpler algorithm is computational: (24)–(26) contains only the | V | degree constrain ts, and can therefore b e solv ed efficiently in practice using standard LP solvers. Whether this leads to a significant reduction in throughput compared to Algorithm I is unclear. Specifically , let Γ ′ coh denote the coherence factor computed using the reference en tanglement chain with service-attempt probabilit y p v = 2 3 λ v at each no de v . Then, b y the same Lyapuno v-drift argumen t as in Theorem 1, the simpler algorithm stabilizes all arriv al-rate vectors ν satisfying (1 + ϵ ) 2 3 Γ ′ coh ν ∈ C . In other words, this algorithm guarantees stability for a constant 2 3 Γ ′ coh of the capacity region C defined in Section 3.2. It should b e noted that if Γ coh → 1 , then Algorithm I can p otentially achiev e the full capacit y region. On the other hand, Algorithm I I is limited to achieving 2 / 3 of the capacit y region. How ever, it is not obvious whether 2 3 Γ ′ coh is alwa ys less than Γ coh b ecause the service attempt probability of the reference en tanglement chain under Algorithm II is 2 3 λ v is low er than the corresp onding λ v for Algorithm II. Numerical results presented later indicate that 2 3 Γ ′ coh is typically less than Γ coh . 4 Throughput P erformance as a function of Switc h P arameters As technology impro ves, it is exp ected that entanglemen t arriv al rates will increase and decoherence rates will decrease. F urther, buffer sizes are also exp ected to increase o v er time. In this subsection, w e will consider a single node and study its a v ailability C u as a function of the switch parameters λ u , µ u , B u . Since Γ coh is going to b e determined by the vertices with smallest C u in the switch, we will consider one vertex and drop the subscript u in this section. A dditionally , we will add a sup erscript C B to indicate the low er b ound on a v ailability at a vertex with buffer size B . Motiv ated by results for traditional switches in [27, 6], w e will c haracterize the set of problem parameters for whic h Γ coh ≥ 0 . 75 so that (2 C − 1) + ≥ 0 . 5 . Our study of Γ coh as a function of the problem parameters will b e carried out in three steps. • First, we will show that C B is an increasing function of B for a fixed λ, µ. • Next, w e will show that C B approac hes C B exp onen tially fast in B , This implies that one can achiev e C ∞ with a small buffer size. • Finally , b y numerically calculating the stationary distribution of the reference chain presen ted in Sec- tion 3.2, we will c haracterize Γ coh as a function of λ, µ, B . The reasons will show that Γ coh ≥ 0 . 5 for practical v alues of the system parameters. The n umerical results will also show that C ∞ can b e ac hieved with relatively small buffer sizes as the theory suggests. W e start b y establishing the monotonicity of C B , which follo ws from a standard coupleing argumen t. Prop osition 1. F or B 1 ≤ B 2 , we have C B 1 ≤ C B 2 . Mor e over, C B ↑ C ∞ as B → ∞ . Pr o of. Fix i.i.d. randomness { S ( t ) , A ( t ) } and the binomial thinning coins across time slots. Run ˜ L B 1 and ˜ L B 2 with the same randomness and the same initial state. The one-step up date is monotone in the state and the truncation lev el, hence ˜ L B 1 ( t ) ≤ ˜ L B 2 ( t ) for all t almost surely . Therefore, 1 { ˜ L B 1 ( t ) = 0 } ≥ 1 { ˜ L B 2 ( t ) = 0 } for all t , and taking stationary exp ectations yields π B 1 0 ≥ π B 2 0 , i.e. C B 1 ≤ C B 2 . The limit C B ↑ C ∞ follo ws b y coupling ˜ L B ( t ) = min { B , ˜ L ∞ ( t ) } and applying monotone conv ergence to stationary exp ectations. The next theorem sho w that C B approac hes C ∞ exp onen tially fast. Theorem 2 (Exp onential rate in B ) . Fix λ ∈ (0 , 1) and µ ∈ (0 , 1) . L et { L ( t ) } b e the infinite-buffer r efer enc e chain, and let { L B ( t ) } b e its trunc ation at level B . L et π and π B denote their stationary distributions, and set C ∞ := 1 − π 0 , C B := 1 − π B 0 . 12 Then ther e exist c onstants θ > 0 and M < ∞ (dep ending on ( λ, µ ) but not on B ) such that 0 ≤ C ∞ − C B ≤ M e − θB , ∀ B ≥ 1 . (27) In p articular, C B ↑ C ∞ and the c onver genc e is exp onential ly fast in B . Pr o of. The pro of is based on regeneration at state 0 . Step 1: W e exploit the relationship b etw een return times and stationary probabilities. Let τ := inf { t ≥ 1 : L ( t ) = 0 } , τ B := inf { t ≥ 1 : L B ( t ) = 0 } , with b oth chains started from L (0) = L B (0) = 0 and coupled using the same arriv als, service-attempt coins, and decoherence-thinning randomness. F or a p ositiv e recurrent irreducible Marko v c hain, the stationary probabilit y of state 0 is the recipro cal of the mean return time to 0 . Hence π 0 = 1 E 0 [ τ ] , π B 0 = 1 E 0 [ τ B ] . Therefore π B 0 − π 0 = E 0 [ τ ] − E 0 [ τ B ] E 0 [ τ ] E 0 [ τ B ] . Since b oth return times are at least 1 , the denominator is at least 1 , so 0 ≤ π B 0 − π 0 ≤ E 0 [ τ − τ B ] . Th us, our goal is to b ound E 0 [ τ − τ B ] . Step 2: No w, w e relate the return times to zero of the finite and infinite-buffer reference chains. Due to the coupling mentioned in Step 1 and monotonicity of the queue update rule, we ha ve L B ( t ) ≤ L ( t ) for all t ≥ 0 a.s. Hence τ B ≤ τ a.s. No w define the ov erflow ev ent E B :=  max 0 ≤ t<τ L ( t ) ≥ B + 1  . If E B do es not o ccur, then the infinite-buffer chain never exceeds lev el B b efore returning to 0 , so the truncation is nev er activ ated and the t wo chains ev olve iden tically throughout the cycle. Therefore τ B = τ on E c B . Th us 0 ≤ τ − τ B ≤ τ 1 E B . (28) Step 3: Next, we argue that the ov erflow cycles in the infinite-buffer system are exponentially rare. F or the infinite-buffer reference c hain, a standard F oster–Lyapuno v drift condition holds outside a finite set. By applying the exp onential b ound for hitting times in [11], this implies that the return time to a finite set has an exponential momen t. In particular, the stationary distribution of the infinite-buffer c hain has an exp onen tial tail: there exist constan ts ϑ > 0 and M 0 < ∞ suc h that the stationary distribution π of the infinite-buffer chain satisfies π { L ≥ m } ≤ M 0 e − ϑm , ∀ m ≥ 0 . (29) No w use the standard regenerativ e o ccupation iden tity for the chain started from 0 : for any set A , E 0 " τ − 1 X t =0 1 { L ( t ) ∈ A } # = π ( A ) π 0 . 13 Applying this with A = { B + 1 , B + 2 , . . . } gives E 0 " τ − 1 X t =0 1 { L ( t ) ≥ B + 1 } # = π { L ≥ B + 1 } π 0 . Th us, P 0 ( E B ) ≤ π { L ≥ B + 1 } π 0 ≤ M 0 π 0 e − ϑ ( B +1) . (30) Th us the probabilit y of an ov erflow cycle is exp onentially small in B . Step 4 : W e now complete the proof. Using (28) and Cauch y–Sc hw arz, E 0 [ τ − τ B ] ≤ E 0 [ τ 1 E B ] ≤  E 0 [ τ 2 ]  1 / 2 P 0 ( E B ) 1 / 2 . The same Lyapuno v argument men tioned earlier will also show that E 0 ( τ 2 ) < ∞ . Hence, from (30), there exist constants θ 1 > 0 and M 1 < ∞ such that 0 ≤ E 0 [ τ − τ B ] ≤ M 1 e − θ 1 B . (31) F rom Step (1), using (31), 0 ≤ π B 0 − π 0 ≤ M 1 e − θ 1 B . Finally , since C ∞ − C B = (1 − π 0 ) − (1 − π B 0 ) = π B 0 − π 0 , w e obtain (27). 4.1 Numerical results W e n umerically compute the s tationary distribution π ( B ) of the reference Marko v chain ˜ L u ( t ) ∈ { 0 , 1 , . . . , B } with service attempt probabilit y p u = λ u . The steady-state av ailability is C B u = 1 − π ( B ) 0 , and a low er b ound on the fraction of the capacity region achiev able by our algorithm is min e =( u,v ) ∈ E ( C B u + C B v − 1) + . Considering a no de u with the smallest v alue of C B u , and dropping the subscript, Γ coh is further low er b ounded by (2 C B − 1) + . W e provide n umerical results for b oth Algorithms I and I I, where we recall that Algorithm I I has lo wer complexity . Our numerical exp eriments are presented for these set of parameter v alues: λ ∈ [0 . 1 , 0 . 5] , µ ∈ { 0 . 05 λ, 0 . 1 λ } , and buffer sizes B ∈ [5 , 25] . Since viable quantum switc hes are still under developmen t, it is difficult to know what suitable switch parameters are consisten t with quantum netw orking hardware in the next five years or so. Ho wev er, based on the a v ailable literature, these ranges of parameters seem feasible for the follo wing reasons: • Entanglement gener ation r ates ( λ ). State-of-the-art quan tum netw ork testb eds achiev e entanglemen t generation rates ranging from a few Hz to tens of Hz o ver metrop olitan distances, with pro jections to ward O (10 2 ) Hz in the near term [20, 26]. When normalized to a time slot corresponding to a swap or scheduling ep o ch, this corresp onds to per-slot arriv al probabilities in the range λ ∈ [0 . 1 , 0 . 5] . • De c oher enc e r ates ( µ ). Quantum memories in current systems exhibit coherence times ranging from milliseconds to seconds depending on the platform [14, 12]. Giv en en tanglement generation times on the order of 10 – 100 ms, this yields a p er-slot decoherence probabilit y that is ty pically an order of magnitude smaller than the arriv al probability , i.e., µ ≈ (0 . 05 – 0 . 1) λ , consistent with the regimes considered here. • Buffer sizes ( B ). Near-term quantum rep eaters are exp ected to supp ort a small num b er of simulta- neously stored entangled pairs per link due to hardw are constraints such as limited memory qubits and control ov erhead [26, 24]. Exp erimen tal platforms curren tly demonstrate storage of a handful of qubits, with pro jections to ward tens of qubits p er no de in early deploymen ts. Thus, buffer sizes in the range B ∈ [5 , 25] appear to be feasible for near-term systems. 14 Figure 1: Results for Algorithm I. 15 Figure 2: Results for Algorithm II. 16 Figure 1 shows the numerical results for Algorithm I and Figure 2 shows the corresp onding results for Algorithm I I. The follo wing observ ations can be made from the figures. • F or any fixed ( λ, µ ) , Γ coh increases rapidly with B and saturates after a mo dest buffer size. The same holds for Γ ′ coh as well. This is consisten t with the theory in Section 3.2. • Γ coh seems to impro v e more than Γ ′ coh for larger v alues of λ. Ho wev er, for mo derate v alues of λ exp ected in the very near term, Algorithm I I may achiev e nearly the same throughput as Algorithm I, but with m uch low er complexit y . 5 Conclusion In this pap er, w e considered the problem of sc heduling in a quan tum switch with sto chastic en tanglement generation, finite memory , and decoherence. Our ob jective was to design a sc heduling algorithm with p oly- nomial computational complexit y that stabilizes a nontrivial fraction of the capacit y region. Our results pro vide a tractable alternativ e to throughput-optimal dynamic-programming-based scheduling for quantum switc hes and help quan tify the impact of en tanglemen t generation rates, decoherence, and memory size on achiev able throughput. F uture w ork includes extending the framew ork to richer entanglemen t-request mo dels, and dev eloping delay and queue-length performance guaran tees. A ckno wledgments: • Researc h supported b y NSF Gran t CCF 22-07547. • The author gratefully ackno wledges the use of ChatGPT and Gemini for brainstorming, literature searc h, editorial assistance, and obtaining n umerical results. References [1] E. Balas. Integer and fractional matchings. In North-Hol land Mathematics Studies , volume 59, pages 1–13. Elsevier, 1981. [2] M. L. Balinski. 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